Mathematics and Physics

VOLUME 10 NUMBER 1 2019 ISSN 2218-7987 eISSN 2409-5508

International Journal of Mathematics and Physics

Al-Farabi Kazakh National University The International Journal of Mathematics and Physics is a peer-reviewed international journal covering all branches of mathematics and physics. The Journal welcomes papers on modern problems of mathematics and physics, in particular, Applied Mathematics, Algebra, Mathematical Analysis, Differential Equations, Mechanics, Informatics, Mathematical Modeling, Applied Physics, Radio Physics, Thermophysics, Nuclear Physics, Nanotechnology, and etc.

International Journal of Mathematics and Physics is published twice a year by al-Farabi Kazakh National University, 71 al-Farabi ave., 050040, Almaty, Kazakhstan website: http://ijmph.kaznu.kz/

Any inquiry for subscriptions should be send to: Prof. Tlekkabul Ramazanov, al-Farabi Kazakh National University 71 al-Farabi ave., 050040, Almaty, Kazakhstan e-mail: [email protected]

The most significant scientific achievements are attained through joint efforts of different sciences, mathematics and physics are among them. Therefore publication of the Journal, which shows results of current investigations in the field of mathematics and physics, will allow wider exhibition of scientific problems, tasks and discoveries. One of the basic goals of the Journal is to promote extensive exchange of information between scientists from all over the world. We propose publishing service for original papers and materials of Mathematical and Physical Conferences (by selection) held in different countries and in the Republic of Kazakhstan. Creation of the special International Journal of Mathematics and Physics is of great importance because a vast amount of scientists are willing to publish their articles and it will help to widen the geography of future dissemination. We will also be glad to publish papers of scientists from all the continents. The Journal will publish experimental and theoretical investigations on Mathematics, Physical Technology and Physics. Among the subject emphasized are modern problems of Applied Mathematics, Algebra, Mathematical Analysis, Differential Equations, Mechanics, Informatics, Mathematical Modeling, Astronomy, Space Research, Theoretical Physics, Plasma Physics, Chemical Physics, Radio Physics, Thermophysics, Nuclear Physics, Nanotechnology, and etc. The Journal is issued on the base of al-Farabi Kazakh National University. Leading scientists from different countries of the world agreed to join the Editorial Board of the Journal. The Journal is published twice a year by al-Farabi Kazakh National University. We hope to receive papers from many laboratories which are interested in applications of the scientific principles of mathematics and physics and are carrying out researches on such subjects as production of new materials or technological problems. This issue of the journal is dedicated to the memory of the outstanding scientist, teacher, organizer of science and education of the Republic of Kazakhstan, doctor of physical and mathematical sciences, professor, academician of the Engineering Academy of the Republic of Kazakhstan Shaltai Smagulov. The issue of the journal contains selected scientific reports of the expanded city scientific seminar dedicated to the 70-th anniversary of Shaltai Smagulov. International Journal of Mathematics and Physics 10, №1, 4 (2019)

IRSTI 27.41.41 The report is dedicated to the memory of my close friend and classmate D. SC.MD, Professor Smagulov S.S.

Abylkairov U.U.

al-Farabi Kazakh National University, Almaty, Kazakhstan e-mail: [email protected]

The Problem of Single – Determinability Equations of Navier-Stokes

Abstract. The Navier-Stokes equations describing a viscous incompressible fluid have for many decades attracted the attention of scientists working on the problem of solvability of partial differential equations and specialists in the field of numerical analysis due to numerous applications. Despite such interest, the question of the existence and uniqueness of the “on the whole” solution of the non-stationary Navier-Stokes equations in the case of three spatial variables still remains open. S. Smagulov made a great contribution to the development of the theory and numerical methods for solving initial-boundary value problems for the Navier- Stokes equations. The situation with the numerical solution of these equations is more complex. The fact is that numerical methods that have proven themselves in solving one class of problems are ineffective in solving another class. From the point of view of justification of numerical methods, there is no possibility of using a number of results. The theory of the equation of mathematical physics, since, as mentioned above, they are open to the Navier-Stokes system. Therefore, the young scientist S. Smagulov of the 1970s of the last century, in order to work successfully in this field of mathematics, combined in himself a specialist in differential equations and also in computational mathematics. Key words: Navier-Stokes equations, regularization, E-approximation, initial-boundary problem.

1.1. Problem statement. The systems of area Ω with Lipshitz border Ω three-dimensional Navier-Stokes, which describes motion of viscous space incompressible fluid, have the following form S. Smagulov,� already studying� at the NSU at the 4th year� . under the guidance of Ph.D., associate + professor B.G. Kuznetsova, began to study methods for the numerical analysis of problems in

�� =�� hydrodynamics. Academician N.N. Yanenko closely Ω followed his scientific growth, posed fundamental

� � problems for solving boundary value problems for Here�� = 0� ��(�� 0)-velocity=�� (�) vector,��| =0 p-function the Navier – Stokes system of equations, namely, of hydrostatic �pressure,� � and = -vector- questions of the approximation of the Navier-Stokes fucntion �� of= (�sources,�� ) 0, - viscosity equations by evolutionary type equations and the coefficient (value inversely proportional�� (��) to the substantiation of a number of difference methods for Reynolds number), solution�� is� sought= � in a limited solving stationary and nonstationary hydrodynamic

equations. According to the proposed schemes, This restrictive condition has been removed by numerical calculations were carried out, the results О.А. Ladyzhenskii [6]. of which were published and transferred to practical We also note other regularizations of the Navier- application. One of the directions of the study of Stokes equations, for example, Е.G.Dyakonova, approximate solutions of the Navier-Stokes D.P.Kaushilayte, V.Ya.Rivkind, А.P.Oskolkov and equations is based on the use of difference schemes etc. for which the difference energy a priori estimates Since 1975, a series of papers began to appear, are valid. In this case, estimates are used for the then still a young graduate student THE VC SOAN equations themselves, as well as for some of their ε- SSSR S.Smagulov [7],[8],[9]. Two explicit finite- approximations. difference schemes are proposed that approximate a 1.2. The currently known approximations of the quasilinear parabolic system with a small parameter Navier-Stokes equations can be mainly divided into ε, which when ε = 0 degenerates into a non- two types: stationary non-linear Navier-Stokes system. The  approximations of the continuity equation conditions under which the solutions of these finite- (moreover, for the correctness of the approximating difference problems converge to the exact (fairly problem, it is necessary to make adjustments to the smooth) solution of the Navier-Stokes equations are equations of motion) clarified. The conditions obtained are less restrictive  approximation (regularization) of only one than the similar conditions given in the papers equation of motion [6],[10]. The convergence of difference schemes is ε-approximations of the Navier-Stokes equations investigated by the method of energy estimates. for a viscous incompressible fluid, which were S.Smagulov in 1975 proposes an approximation of derived from physical considerations, were the Navier-Stokes equations, which is obtained by proposed in the papers of academician of N.N. replacing the continuity equation with another Yanenko (by the way the author of the method of equation. The convergence of the approximate fractional steps), B.G. Kuznecov, N.N. Vladimirov solution that is obtained by solving the replacing [1],[2]. This idea was quickly picked up by system of equations, to solving the Navier-Stokes prominent French mathematicians Jacques-Louis equations with the speed Lyons, Roger Themes and others. R. Temam [3],[4] proposed a different method of ε-approximation of ∞ Ω � � the Navier-Stokes equations. For these equations, he � � � � studied the behavior of the solution at , a �� −�� (��)=� � −�� ( )�+ difference scheme was constructed, for which it was 0 (4) shown that, under certain conditions on ��0� the � � �W 1  + �� − �� 2 �� solution of the difference problem converges to the solution of the Navier-Stokes equations. ��An attempt�� If the solution of the Navier-Stokes equation has was made in [5] to substantiate difference schemes the following property like fractional steps. We also note other regularizations of the Navier-Stokes equations. Ω , (5) We present an ε-approximation of the Navier- Stokes equations. where m- the exact�� lower−�>�>0 bound of the eigenvalues of the matrix of the strain rate tensor matrix, then

� � , (1) the estimate (5) is uniform. �� Apparently, for the first time the behavior of a �� − �� +�� + � � �� = � − ∇� strong solution of problem (1), (2) with + =0. (2) � was considered in the works of S.Smagulov [11], �� [12] and at the same time in work P.Е. �Sobolevskii = 0� �� > �� At the� convergence+� � �� of difference schemes and0 V.V. Vasilev [13] for the two-dimensional case is proved under the Sh. Smagulov introduced the following system: � assumption,� =0 that � and ��

�� →0 . (3) � � � � � � � - 1 , (6) �� + (� ·∇)� + � �� = � � � 2 � √� →0 = �� ∇� − ∇� Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 4 (2019) 6 The Problem of Single – Determinability Equations of Navier-Stokes

, (7) impenetrable and there is no mass transfer between � � the solution and the external medium: = ��,� + �� =0 , =0 (8) � � � �� Υ= 0; =0; , (11) � | � (�) � | =� (�) � | �� Υ Where a smooth � �| �� [0, �] � � ��(�) where - unit vector of external normal characteristic � function�� on (0, -∞), � = ∑ �� , �(�) − Υ Along with the task of mass flow through the is found from the Navier-Stokes equation � � � � �( ) boundary�� =� Υ,�, ,�another physically reasonable � � � � = (unperturbed).� condition. can also be considered, when the density �� − �� Using the obtained a priori estimates for the values are known on Υ: higher derivatives and for other structural elements of the problem, we proved the following. Υ= Υ ; γ Theorem. Let ∞ Ω Ω , . In addition�| to� the(�, �)boundary�� ; ��[0, conditions, �] we supplement the problem with the given by Cauchy: Then there is �a strong solution� to problem�� (6) – �(8) � �� and� �� for�0, this �; � solution( )�, � the�� following�0, �; � estimate( )� �� takes ; ; x (12) place:

�� Regarding�| =� (�) (also �| Υ=�х (�) we will� � assume Ω + ∞ that this is a positive bounded function: � � � � � � � � � � � (�) � ( , �)) � � (�,�; �� ( )) � � (� ) ∥�By∥ introducing an auxiliary∥ �� function∥ ≤�< which provides the conditions for matching at the ∞ (13) initial time the solution of the Navier-Stokes��(�) 0 < �� (�) = equations and the solution of a parabolic system =�≤��(�) ≤ � = �� (�) < degenerate at ε = 0. That is, ensures equality: We formulate the main� � results for this problem obtained by Sh. Smagulov Theorem 1. Let Н х Ω � � = � , � � = � � � � � � (� ��) � � х � � � � � � � � �� (�)� ; � ( )�� ( ),� ≤ �� Next, we study the internal smoothness of spatial � ( ) ≤ �; Ω где variables. � � Sh. Smagulov in 1976-1977, together with �(�) =�(�, �) �� 0, �; � ( )�, Professor Kasimov A.V. the strongest result was q � � � � obtained on the unambiguous generalized and strong ��[1,2]; � �� ,2� ; � + �� ≤ � solvability of the diffusion model of an Then there is at least one weak solution to inhomogeneous fluid, which was later called the problem (9) – (13) if the constants M, m, Kazhikhov-Smagulov equation. satisfy We formulate the initial-boundary problem for � �� this system: (14) �� ≤ 2�(� − �) Theorem 2. If Ω and �� + (��)� − flow is plane-parallel, i.е.� � � � � �� , (9) then, �if (condition�)��; � (14)(�)�� is �fulfilled( ),�( �on)�� any(� finite) time interval (0, T), there is a �=unique(� �strong,��),� solution =(��,� �to) −�[(�� )�+(��)��] = �� − �� + �� (10) problem (9) – (13). �� And this task is brought by Shaltay Smagulovich Let the�� mixture �� = 0; move �� + in(�� a limited)� = ��area to practical results, i.е an approximation of the Ω with a fairly smooth border Υ (for problem (9) – (13) is proposed, the theorems of example, �twice continuously differentiable). For convergence, the stability of difference schemes are simplicity,�� let us assume that the boundary Υ is proved.

International Journal of Mathematics and Physics 10, №1, 4 (2019) Int. j. math. phys. (Online) U.U. Abylkairov 7

1.3. The important step in the scientific career of fundamentally new approach to solving the problem Sh.Smagulov is an adoptation of ideas of fictitious of setting boundary conditions for the current area method for the first time for nonlinear function. For the first time carried out the equations of mathematical physics, namely for the substantiation mf. for problems of flow of a viscous Navier-Stokes system of equations during 1978- incompressible fluid with inhomogeneous boundary 1979 in papers [14]-[15]. Fictitious areas method conditions in regions with curvilinear boundaries (briefly f.a.m.) are pioneers for simple models and complex geometry. (Saulev1963 г; lebedev 1964 y, and etc.) reduces the Numerical algorithms based on the method of solution of the boundary in regions of complex fictitious areas were developed, which allow shape to the solution of boundary-value problems in studying the flow characteristics of a viscous regions of simple shape (for example, a rectangle or incompressible fluid around an obstacle in a channel parallelepiped). The reduction is due to the fact that with curvilinear boundaries, characteristics of a in the domains of simple form, boundary value convective motion of a viscous incompressible fluid problems and / or their grid analogues can be solved in areas of complex geometry [16-18]. by efficient, cost-effective methods. There are 1.4 The next stage of scientific research of several fundamentally different approaches for such Smagulov Shaltay is the problem of the existence of information. The method of fictitious areas for the global solutions of the Navier-Stokes equation of nonlinear Navier-Stokes system. Consider a compressible continuous media. boundary value problem for a nonlinear The first theorem on the existence of a solution equation. в Ω for the Navier-Stokes equations of compressible � � � � viscous fluid was obtained by John Nash [19] in (� ·∇)� = �� − ∇� +� � (15) 1962, the future winner of the Nobel Prize in � economics. He proved the existence theorem for the �� =0 в Ω classical solution of the Cauchy problem “in the small” in time. Somewhat different methods his � � � � � � � � � result was repeated and summarized in the works (� ·∇)� = �� − � �, �� − � � N. Itay [20], А.I. Wolpert, S.I. Hudyaev [21] � � � � � Significant development of non-local theory � |� =� |� � |� = 0, received in the works А.V. Kozhihov [22-23] and � = � . �� his students V.V. Sheluhin [24], V.А. Vaigant [25] � � � � �� | � �� и др. The behavior of the solution of problem (15) is The first theorems of solvability “on the whole” investigated as ε → 0. An a priori estimate of the with respect to time for the model of magnetic gas solution to the problem (15) was obtained : dynamics were obtained in the work of Sh. Smagulov [26] in the case of a barotropic motion of Ω Ω Ω (16) a viscous gas. In the work of his student � � � � � �� А.А.Durmagambetov [27], [28], [29], essential � � �� ( ) Further,�‖� the‖ existence+ � ||� ‖ of at� least ��‖ one generalized results on the solvability of boundary value solution of problem (15) and its weak convergence problems and the Cauchy problem for degenerate in Ω) to a generalized solution of the stationary equations of a viscous heat-conducting gas have Navier� – Stokes equation is proved. The rationale been obtained. Also, another talented student of for� the� ( method of fictitious domains for the Navier- Smagulova, Sh. Z.E. Konysbaev, proved the Stokes equations and equations for the current correctness of the initial-boundary value problem function and velocity vortex with inhomogeneous for a system of equations of a viscous barotropic gas boundary conditions is carried out for the first time. with an initial density that has degeneracy and takes The obtained estimates of the convergence of the the magnetic field into account [30–32]. solution of the auxiliary problem to the solution of 1.4.1 Formulation of the problem. The equations the original problem. of magnetic gas dynamics in Lagrangian coordinates Professor Sh. Smagulov and his students for are: the first time used the method of fictitious areas to describe incompressible fluid flows in multiply

connected areas based on the equations for the � current function and velocity vortex, which is a �� − = 0, � = , �� Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 4 (2019) 8 The Problem of Single – Determinability Equations of Navier-Stokes

Then there is a unique generalized solution to the problem (17)-(19), where strictly positive, � �� 1 �� bounded function, -limited function. � =� � � − −�� , �� The proof of the� theorem�( �,is �)carried out by the method of regularizations.� (�) � (An�, �interesting) issue is the � � study of the correctness of difference schemes of the � = �� , � model of magnetic gas dynamics in Euler variables. In the one-dimensional case, the correctness of difference schemes in Lagrange variables is well � �� 1 �� studied in the work of Sh. Smagulov and B. � =� � � − �� + � ��( (17) Rysbayev [33]. � � �� −� �� + � (��) � ��) Conclusion (vH)= ( ). � � � �� Scientific works of Sh.S.Smagulova are devoted Initially, Cauchy�� data is� known:�� to the creation and study of efficient algorithms of computational mathematics, the development of ; ; numerical methods for the Navier-Stokes equations � � of great practical and theoretical value, a rigorous �|�� =� (;�) �|�� =�; x (�) � | �� = (18) mathematical analysis of the solvability of initial- � boundary value problems for nonlinear equations of where =� (�) �|��) =1– continuous,� [0,1], - composite, non-classical, degenerate types limited, non-negative� � � and� has degeneration �when mathematical physics. Sh.S.Smagulov paid much х=1, i.е. (� ,� ,� ,� � (� , �) attention to the problems of mathematical modeling � � and computer technologies in oil production, �х(1) =0∞ (x) ∞ information technologies for solving practical � � problems of science and technology. 0��The required( ) ��< functions,0<� satisfy� �� the boundary�� < In the field of computational mathematics, conditions: Sh.S.Smagulov made a significant contribution to the development of difference schemes. He constructed and investigated difference schemes for = = classes of equations:Для стационарных и �� нестационарных уравнений навье-Стокса �|�� =�|�� = �� = 0 �� (19)  For heat convection system  For equations of gas dynamics and magnetic �� and the conditions=� of| approval=�| are met. gas dynamics. Theorem. Let the initial data (18) has the Sh. Smagulov is the author of the method of following properties of smootheness: fictitious domains for the non-linear Navier-Stokes equation. For the first time, the method of fictitious Ω regions was applied to describe incompressible fluid � � � � flows in one or multiply connected domains, and the (� (�), � (�), � (�))�� ( ), method of fictitious domains with non-uniform � Ω , ��(�) boundary in regions with curvilinear boundaries was � � � � � � ( ) substantiated. (� ) (�) Developed numerical algorithms based on the � method of fictitious areas. � (�)  Sh. Smagulov made a great contribution to � < �� (�) the development of the theory of solvability of the Ω Ω non-linear Navier-Stokes equations, the equations of � � � � � thermal convection, magnetic gas dynamicsю �, �� , �� ( ),� (�)�� ( ). �� 0��  The correctness of the problems of degenerate equations of magnetic gas dynamics ��

International Journal of Mathematics and Physics 10, №1, 4 (2019) Int. j. math. phys. (Online) U.U. Abylkairov 9

 The correctness of the problems of degenerate of fluid flow. ITAM SB AS USSR Novosibirsk equations of a reacting mixture of gases (1978): 158-175. Solvability and convergence of the ε- 12. Sh. Smagulov. “On parabolic approximation approximation of the Navier-Stokes equations, the of the Navier-Stokes equations.” Numerical temperature model of homogeneous and Methods of Continuum Mechanics, VTS and ITPM inhomogeneous liquids; and taking into account the of Soan USSR vol. 10, no. 1979: 137-149. dissipation of energy. 13. P.Е. Sobolevskii, V.V. Vasilev. On one E- approximation of the Navier-Stokes equations. References “Numerical Methods of Continuum Mechanics” vol. 9, no. 5 (1978): 115-139. 1. N.N. Yanenko. The method of fractional 14. А.N. Bugrov, Sh. Smagulov. “The method steps for solving multidimensional problems of of fictitious domains in boundary value problems mathematical physics, Novosibirsk, Science (1967): for the Navier-Stokes equations.” Mathematical 225. models of fluid flow (1978): 79-90. 2. N.N. Vladimirova, B.G. Kuznecov, N.N. 15. Sh. Smagulov. “The method of fictitious Yanenko. Numerical calculation of symmetric flow domains for boundary value problems of the Navier- of a plate around a plane flow of a viscous Stokes equations.” VTS SO AN USSR. Preprint. 68: incompressible fluid. Some questions of 22. computational and applied mathematics. 16. А.V. Kazhihov, Sh. Smagulov. “On the Novosibirsk (1966): 29-35. correctness of boundary problems of a diffusion 3. Temam R. “Une method d’approximation model of an inhomogeneous fluid.” DAN SSSR vol. dela solution des-equations de Navier-Stokes.” 234, no. 2 (1377): 330-332. Bull.Soc.Math.France 96 (1968): 115-152. 17. Sh. Smagulov. “Approximation of the 4-5. Temam R. Sur l’approximation de la equations of one model of an inhomogeneous fluid.” solution des equations de Navier-Stokes parla Numerical Methods of Continuum Mechanics vol. method de pas fractionnaries (I)(II),Arzhive 8, no. 2 (1977): 112-124. Rat.mech.Anal.,32. 18. Sh. Smagulov. “On a variant of 6. O.A. Ladyzhenskaya. Mathematical approximation of the Navier-Stokes equations.” problems of the dynamics of a viscous Partial differential equations, Novosibirsk (1980): incompressible fluid. Science (1970): 288. 57-62. 7. А. Smagulova. “On a non-linear equation 19. J. Nesh. “The problem de Cauchy pour les approximating the Navier-Stokes equation.” equation diffeentielles d'un fluide general.” Proceedings of the Vth All-Union Seminar on Bull.Soc.Math.France vol. 90, no.4 (1962): 487-497. Numerical Methods of Viscous Fluid Mechanics 20. Itaya N. “The existence and uniqueness of Novosibirsk (1975): 123-134. the solution of the equations describing 8. B.G. Kuznecov, Sh. Smagulov. “On compressible viscous fluid flow.” proc. Jap. Acad. difference schemes with a small parameter vol. 46, no. 4 (1970): 379-382. approximating the Navier-Stokes equations.” 21. Wolpert А.I., Hudlev S.I. “On the Cauchy Proceedings of the Vth All-Union Seminar on problem for composite systems of nonlinear Numerical Methods of Viscous Fluid Mechanics, differential equations.” Mathematical collection Novosibirsk (1975): 109-122. vol.87, no.4 (1972): 504-528. 9. B.G. Kuznecov, Sh. Smagulov. 22. Каzhihov А.V. “The correctness “on the Approximation of the hydrodynamic equation. “ whole” of mixed boundary value problems for a Numerical Methods of Continuum Mechanics” no. model system of equations of a viscous gas.” 2, (1975): 158-175. Dynamics of continuous media no. 21(1975): 18-47. 10. А.P. Оskolkov. “On some convergent 23. Kajihov А.V. On the Cauchy problem for difference schemes for the Navier-Stokes viscous gas equations.” Siberian mathematical equations.” Proceedings of the Steklov journal vol. 23, no.1 (1982): 60-64. mathematical Institute 125 (1973): 164-172. 24. Kajihov А.V., Sheluhin V.V. 11. B.G. Kuznecov, Sh. Smagulov. “On the rate “Unambiguous solvability of the initial-boundary of convergence of solutions of a single system of value problems for the one-dimensional viscous gas equations with a small parameter to the solution of equations.” Applied mathematics and mechanics the Navier-Stokes equations.” Mathematical models vol. 41, no. 2 (1977): 282-291.

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25. Vaigant V.А., Каzhihov А.V. “On the of Computational Mathematics and Mathematical existence of global solutions of the two-dimensional Physics. – 1993. – T. 33, no. 8. – C. 1251- Navier-Stokes equations compressible viscous 1259. fluid” Siberian mathematical journal vol. 36, no.6 30. Smagulov Sh., Коnysbaev Zh.Е. “On the (1995): 1283-1316. correctness of the initial-boundary value problem 26. Smagulov Sh. “On the correctness of some for the equations of a viscous heat-conducting gas problems for the equations of magnetic with a degenerate density.” Continuum dynamics hydrodynamics.” Mate.models of fluid flows. vol. 93 (1989):119-130. Novosibirsk : In-t Theor. And PM .mechanics of 31. Smagulov Sh. Iskenderova J.A. “The USSR Academy of Sciences (1978): 257-266. Caushy problem for the equations of a viscous heat 27. Durmaganbetov А.А. Solvability of some – conducting gas with degenerate density.” nonlinear problems of gas dynamics: Thesis of Comput.math.and math.phys vol 29, no.8 (1993): candidate of Phys.-math. sciences': 01.01.02. 1109-1117. Аlmaty (1987): 75. 32. Smagulov Sh., Ourmaganbetov A.A., 28. Smagulov Sh., Durmaganbetov А.А., Iskenderova J.A. “The Caushy problem for the Iskenderova D.А. “The Cauchy problem for the equations of magnetic-gas dynamics.” Different. equations of magnetic gas dynamics.” Differential Equations vol. 29, no. 2 (1993): 278-288. equations vol. 29, no.2 (1993): 337-348. 33. Smagulov Sh., Rysbaev B.R. “On 29. Smagulov Sh., Iskenderova D.А. The convergent difference schemes for viscous gas Cauchy problem for the equations of a viscous heat- equations.” Doc.USSR ACADEMY OF SCIENCES conducting gas with a degenerate density // Journal vol. 287, no. 3 (1988): 558-559.

International Journal of Mathematics and Physics 10, №1, 4 (2019) Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 11 (2019)

IRSTI 27.31.15

1,*S. Aisagaliev, 1Zh. Zhunussova, 2H. Akca

1Al-Farabi Kazakh National University, Almaty, Kazakhstan *e-mail: [email protected], 2Professor of Applied Mathematics, Abu-Dhabi University, Abu-Dhabi, UAE e-mail: [email protected]

Construction of a solution for optimal control problem with phase and integral constraints

Abstract. A method for solving the Lagrange problem with phase restrictions for processes described by ordinary differential equations without involvement of the Lagrange principle is supposed. Necessary and sufficient conditions for existence of a solution of the variation problem are obtained, feasible control is found and optimal solution is constructed by narrowing the field of feasible controls. The basis of the proposed method for solving the variation problem is an immersion principle. The essence of the immersion principle is that the original variation problem with the boundary conditions with phase and integral constraints is replaced by equivalent optimal control problem with a free right end of the trajectory. This approach is made possible by finding the general solution of a class of Fredholm integral equations of the first order. The scientific novelty of the results is that: there is no need to introduce additional variables in the form of Lagrange multipliers; proof of the existence of a saddle point of the Lagrange functional; the existence and construction of a solution to the Lagrange problem are solved together. Key words: immersion principle, feasible control, integral equations, optimal control, optimal solution, minimizing sequence.

Introduction problems based on the Lagrange principle is described in [4]. One of the methods for solving the variational In the classical variational calculus, it is assumed calculus problem is the Lagrange principle. The that the solution of the differential equation belongs Lagrange principle makes it possible to reduce the to the space С1(I,Rn), and the control u(t), t  I is solution of the original problem to the search for the from the space С1(I,Rm), in optimal control problems extremum of the Lagrange functional obtained by 1 n [5] the solution  and the control introducing auxiliary variables (Lagrange x(t) KC (I, R ), multipliers). u(t) KC(I, Rm ). In this work, the control u(t), The Lagrange principle is the statement about the m t  I is chosen from L2(I,R ) and the solution x(t), existence of Lagrange multipliers, satisfying a set of t  I is an absolutely continuous function on the conditions when the original problem has a weak local minimum. The Lagrange principle gives the interval I = [t0,t1]. For this case solvability and necessary condition for a weak local minimum and uniqueness of the initial problem for differential it does not exclude the existence of other methods equation are given in [4, 6-8]. for solving variational calculus problems unrelated The purpose of this work is to create a method to the Lagrange functional. for solving the variational calculus problem for the The Lagrange principle is devoted to the works processes described by ordinary differential [1-3]. A unified approach to different extremum equations with phase and integral constraints that

© 2019 al-Farabi Kazakh National University Int. j. math. phys. (Online) 12 Construction of a solution for optimal control problem with phase and integral constraints differ from the known methods based on the continuous with respect to the variables Lagrange principle. It is a continuation of the (x,u,t)  Rn  Rm  I, satisfies the Lipschitz research outlined [9-10]. condition by x, i.e.

Problem statement | f (x,u,t)  f ( y,u,t) | l(t) | x  y |, (1.7) We consider the following problem: minimize (x,u,t),( y,u,t) Rn  Rm  I the functional and the condition J (u ( ), xx01 , ) = t1 2 (1.1) |f (xut , , ) | c01 (| x | | u | ) c ( t ),   (1.8) F0(x (),(), t u t x01 , x ,) t dt inf (xut , , ), t 0

  1 at conditions where l(t) 0, l(t) L1(I, R ), c0 = const > 0, 1 c1(t)  0, c1(t)  L1(I, R ). x = A(t)x  B(t) f (x,u,t), t  I = [t0 ,t1] (1.2) The vector function

F(x,t) = (F1(x,t),, Fs (x,t)) is continuous with with boundary conditions  n  respect to the variables (x,t) R I. Function 2n f (x,u, x , x ,t) = ( f (x,u, x , x ,t), (x(t0 )) = x0 , x(t1 ) = x1 )  S0  S1 = S  R (1.3) 0 0 1 01 0 1 , f (x,u, x , x ,t)) satisfies the condition 0m2 0 1 in the presence of phase constraints |f ( xux , , , x , t ) | c (| x | | u |2  | x | | x |)  x()t Gt (): Gt ()= 0 01 2 0 1 ct3 ( ), n {x R / () t Qxt ( ,) (), t t I }, (,,x ux , x ,),(,, t yux , x ,) t Rn  01 01 and integral constraints mnn R RRI  ,

  guj ( ( ), xx01 , ) 0, 1 c2 = const  0, c3 (t)  0, c3 (t)  L1(I , R ). (1.4) j= 1, m1 ; guj ( ( ), xx01 , ) = 0, Scalar function F0 (x,u, x0 , x1,t) is defined and continuous with respect to the variables together jm= 1, m , 12 with partial derivatives by variables (x,u, x , x ), 0 1 (t), (t), t  I – are given s– dimensional guj ( ( ), xx01 , ) =

t1 functions. S is given bounded convex closed set of  2n  f0j (x ( t ), u ( t ), x01 , x , t ) dt , (1.5) R , the time moments t0,t1 are fixed. t0 In particular, the set jm= 1, . 2n 2 S = {(x0 , x1)  R /H j (x0 , x1)  0, j = 1, p1; where the control  < a j ,(x0 , x1 ) >= 0, j = p1 1, p2}, where m u()  L2 (I, R ). (1.6) H j (x0 , x1 ), j = 1, p1 are convex functions, a  R2n , j = p  1, p are given vectors. Here A(t), B(t) are matrices with piecewise- j 1 2 Note, that if the conditions (1.7), (1.8) are continuous elements of orders n  n, n  r, satisfied for any control   m and the respectively, a vector function u( ) L2 (I, R ) initial condition x(t0 ) = x0 of the differential f (x,u,t) = ( f1(x,u,t),, fr (x,u,t)) is

International Journal of Mathematics and Physics 10, №1, 11 (2019) Int. j. math. phys. (Online) S. Aisagaliev et al. 13 equation (1.2) has a unique solution. x(t), t  I. Note, that the optimal control problem (1.1) – (1.6) has a solution if and only if the boundary value This solution has derivative   n and x L2 (I, R ) problem (1.2) – (1.6) has a solution. satisfies equation (1.2) for almost all t  I. Problem 1.3. Find an admissible control It should be noted that integral constraints **     (utxx* ( ), 01 , ) U S0 S 1. t1 If problem 1.2. has a solution, then there exists gu( ( ), xx , ) = f ( xtutxxtdt ( ), ( ), , , ) 0, an admissible control. j 01 0j 01 (1.9) t0 Problem 1.4. Find the optimal control * * jm=1,1 , u** () t Ut (), the point (x0 , x1 )  S0  S1 = S * and the optimal trajectory x (tt ; , x0 ), t I, by introducing additional variables d j  0, *0 * j = 1, m , can be written in the form 1 where x* ()t Gt (), t I, x*1(t )= xS1  1 , ** ** gu( ( ), xx01 , ) 0, j = 1, m , gu( ( ), xx01 , ) = 0, g j (u(), x0 , x1 ) = d j , j = 1, m1. j * 1 j * **   01  0 1 Let the vector be j = m1 1,m2, J(u* ( ), xx , ) = inf J (u( ), x , x ), m m2 (u(), x0 , x1)  L2 (I, R )  S0  S1. c = (d1,,dm ,0,0,,0) R , where d j  0, 1 One of the methods for solving the problem of j = 1, m1. Let a set be variation calculus is the Lagrange principle. The m Q = {c  R 2 /d  0, j = 1, m }, where d  0, Lagrange principle allows to reduce the solution of j 1 j the original problem to the search for an extremum j = 1, m1 are unknown numbers. of the Lagrange functional obtained by introducing Definition 1.1. The triple auxiliary variables (Lagrange multipliers). * * In the classical variation calculus, it is assumed (u* (t), x0 , x1 ) U  S0  S1 is called by admissible that the solution of the differential equation (1.2) control for the problem (1.1) – (1.6), if the boundary belongs to the space and the control u(t), t problem (1.2) – (1.6) has a solution. A set of all I of the space � �in the optimal control admissible controls is denote by , 1 n problems [5], the solution� �� x � ) KC (I, R ) and control 1 m   U  S0  S1. u(t)∈ KC (I, R ). In�� this � paper,) the control u(t), t I From this definition it follows that for each is chosen from L2(I, ), and∈ the solution x(t), t I element of the set Σ the following properties are is an∈ absolutely continuous� function on the interval∈ I = [t0, t1]. For this case,� the existence and uniqueness∈ satisfied: 1) the solutions x*(t), t  I of the differential equation (1.2), issuing from the point of the solutions of the initial problem for equation (1.2) are presented in the references [4, 6, 7, 8]. *  *  x0 S0, satisfy the condition x*(t1) = x1 S1, and The purpose of this paper is to create a method * * also (x0 , x1 )  S0  S1 = S; 2) the inclusion for solving the problem of the variation calculus for processes described by ordinary differential   x*(t) G(t), t I holds; 3) for each element of equations with phase and integral constraints that the set Σ we have the equality g(u(), x0 , x1 ) = c, differ from the known methods based on the where Lagrange principle. It is a continuation of the scientific research presented in [9-16]. gu( ( ), x** , x ) = ( g ( u ( ), x** , x ), , **01 1 01 g( u ( ), xx** , )). m2 * 01 Existence of a solution We consider the following optimal control The following problems are set: problem: minimize the functional Problem 1.2. Find the necessary and sufficient conditions for the existence of a solution of the Iu1( ( ), p ( ), v1 ( ), v 2 ( ), x 01 , xd , ) = t boundary value problem (1.2) – (1.6). 1 (2.1)   F1( qt ( ), t ) inf t0

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 11 (2019) 14 Construction of a solution for optimal control problem with phase and integral constraints

nm nm z = Atz1 () Btvt 1 () 1 () Bvt 22 (),( zt 0 )=0, q=(,(),,,, zzt u pv v , x , x ,) d R22 R  (2.2) 1 12 01 , tI Rmsr RRR  mm21 RRR nn 

r m2 v1()  L2 (I, R ), v2 ()  L2 (I, R ), (2.3) q = (z,z(t1), u,p,v1,v2 ,x0 ,x1,d ) , l = const> 0 . pt( ) V ( t ), u ( ) L ( I , Rm ), 2 (2.4) (,)x01 x S 0 S 1 =, Sd  . Then the functional (2.1) under the conditions (2.2) – (2.4) is continuously Frechet differentiable,           We introduce the following notations: I1()=( III 11u (), pv (), 1 (), I 1 v (), the gradient 12at H= L (, IRmsr ) L (, IR ) L (, IR ) III (),  (),())    H 2 22, 11xx011d mm21nn LIRRRR2 (, )   any point   X is calculated by the formula mr X= LIR22 (, ) V LIR (, ) m , vector function 2 I11uu()= F ((),), qt t I 1 p ()= F 1 p ((),), qt t I 1 v ()= L2(,IR ) S01 S   H 1

(t) = (u(t), p(t),v1(t),v2 (t), x0, x1,d) X  H , *  F11v ( qt ( ), t ) B ( t ) ( t ), q(t) = (z(t),z(t1),(t)) . 1 The optimization problem (2.3) – (2.6) can be t1 I ()= F ((),) qt t B* (), t I ()= F ((),), qt tdt represented in the form: 1vv22 1 21x0 1 x 0 t t1 0        tt11 I1( ( )) = F1(q(t),t) inf , ( ) X H. I11xx()= F ((),), qt tdtI11dd ()= F ((),), qt tdt t0 11  tt Let the set be 00 (2.6)        X* = { *( ) X | I1( *( )) = inf I1( ( ))}. X Lemma 2.1. Let the matrix be positive definite where z(t) , t  I is the solution of the differential   T(t0,t1) > 0 . In order to the boundary value equation (2.2), and the function (t) , t I is the problem (1.2) – (1.6) have a solution, it is necessary solution of the conjugate system and sufficient that lim I1(n ) = I1* = inf I1( ) = 0 , n X *  =F1z (qt (),) t A 11 () t , ( t )= where    is a minimizing sequence in the t { n ( )} X 1 (2.7)   F( q ( t ), t ) dt . problem (2.1) – (2.4).  1(zt1 ) Proof of the lemma follows from Theorem 2.3. t0 and Lemmas 2.4. and 2.5. [9].

Theorem 2.2. Let the matrix be T(t0,t1) > 0 , the In addition, the gradient I1() ,   X satisfies function F (q,t) be defined and continuous in the the Lipschitz condition 1 set of variables (q,t) together with the partial ||II11 ( ) 12 ( )|| K | | 1 2 || , 1 , derivatives with respect to q and satisfies the (2.8)   X, Lipschitz conditions 2

where K = const > 0 . | F1q (q  q,t)  F1q (q,t) | l | q |, t  I , (2.5)      Proof. Let (t), (t) (t) X , z(t,v1,v2 ) , where z(t,v1  v1,v2  v2 ) , t  I be a solution of the system (2.2), (2.3). Let F1q(,)=(qt F 1 z (,), qt F1(zt ) (,), qt F11up (,), qt F (,), qt 1 z(t,v1  v1,v2  v2 ) = z(t,v1,v2 )  z(t) , t  I . F (qt , ), F ( qt , ), F ( qt , ), F ( qt , ), F ( qt , )), 11vv12 1 x 01 1 x1d Then

International Journal of Mathematics and Physics 10, №1, 11 (2019) Int. j. math. phys. (Online) S. Aisagaliev et al. 15

t1 | z(t) | C1 || v1 || C2 || v2 ||. (2.9)    | R9 | l9  | z(t1) || q(t) | dt by the Lipschitz The increment of the functional (see (2.5)) t0 condition (2.5). We note that (see (2.7), (2.9))

II11= (   ) I1 ( )= t t1 1 z* ( t ) F ( q ( t ), t ) dt =  [F11 ( qt ( )  qt ( ), t ) F ( qt ( ), t )] dt =  1 1(zt1 )  t t0 0 t t1 1 **   [v** () t B () t  v ** () t B ] () t dt  (2.11) = [u () tF11up ( qt (),) t  p () tF ( qt (),) t   11 22  t t0 0 t * 1 v( tF ) ( qt ( ), t )  * 11v1    z( t ) F1z ( q ( t ), t ) dt . t0 v**() tF ( qt (),) t  xF ( qt (),) t  2 1vx2001 From (2.10) and (2.11) we get

** t1 xF11x ( qt ( ), t )  dF ( qt ( ), t )  1 1d   **   I11= { u () tFup ( qt (),) t p () tF 1 ( qt (),) t t0 * * ** z() tF11z (qt ( ), t )  z ( t ) F1(zt ) ( qt ( ), t )] dt vtF()[ ( qtt (),) Bt () ()] t 1 11v1 1

9 v**( t )[ F ( qt ( ), t ) B ( t )] 21v2 2 Ri , i=1 (2.10) t ** 1 x01Fxx( qt ( ), t )  xF11 ( qt ( ), t )     01 where | R1 | l1  | u(t) || q(t) | dt, 9 t *  0  d F1d (q(t),t)}dt  Ri =< I1( ), >H R, t1 i=1

| R2 | l2 | p(t) || q(t) | dt,  9 t0 2 | R | where R = Ri , | R | C ||  || ,  0, t  3 1 i=1 ||  ||    | R3 | l3  | v1(t) || q(t) | dt, at ||  || 0 . t0 This implies the relation (2.6). Let t1            | R4 | l4  | v2 (t) || q(t) | dt, 1=(u u , p pv ,1 v 12 , v t0 , t1 v20, x  x 01 , x  xd 1 ,)  d    | R5 | l5  | x0 || q(t) | dt, t0 2 = (u, p,v1,v2 , x0, x1,d)  X . Since t 1 | R6 | l6 | x1 || q(t) | dt, 2 2 2 2  | I1(1 )  I1(2 ) |  l10 | q(t) | l11 |  (t) | l12 |  | , t 0 t 1 | q(t) | l ||  ||,|  (t) | l ||  ||,    13 14 | R7 | l7  | d || q(t) | dt, t that 0 2 ||II11 ( ) 12 ( ) || = t1 t | R | l | z(t) || q(t) | dt, 1 8 8        22   |()I1 1 I 1 ()| 2 dt l15 || ||, t0 t0 Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 11 (2019) 16 Construction of a solution for optimal control problem with phase and integral constraints

u= Pu [  I ( )], p = Pp [  I ( )], where li = const > 0 , i = 10,15 . This implies the n1 Unnunn 11 Vnnpn 1 estimation (2.8), where K = l15 . The theorem is vPvInn11= [   ( )], vPvI n = [ n   ( )], proved. 1V11 11 nv n 2V 2 21 nv 2 n

Lemma 2.3. Let the matrix be T(t0,t1) > 0 , the n11 n nn  xPxI0=S [ 01  nx ( n )], xPxI 1 =S [ 11  nx ( n )], function F1(q,t) be convex, with respect to the 0 0 11 N variable q  R , N = 4n  m  s  r  m1 , i.e. d = Pd [  I ( )], n = 0,1,2, , n111 n nd n

F1(q 1 (1 ) q2 )  Fqt 11 ( , ) (1 ) Fq12 ( , t ), 2 0< 0  n  , > 0, qq, RN , ,  [0,1]. K  2 12 (2.13) (2.12)

Then the functional (2.1) under the conditions where P[] is the projection of the point on the set (2.2) – (2.4) is convex.  , K = const > 0 from (2.8). Theorem 2.4. Let the conditions of Theorem 2.2. Proof. Let 1,2  X ,  [0,1]. It can be shown, that be satisfied, in addition, the function F1(q,t) be convex with respect to the variable q  R N and the zt(,(1), v12   v12 v (1))=  v sequence{n}  X1 be determined by formula   ztv(, , v ) (1  ) ztv (,12 , v ), 12 (2.13). Then: 1) the lower bound of the functional (2.1) is  r m2 (v1,v2 ), (v1,v2 )  L2 (I, R ). reached under the conditions (2.2) – (2.4) Then t 1 inf I1( ) = I1(*) = min I1( ), *  X1;   I(  (1 ) ) =F ( qt ( )  X1 X1 11 2 11 t0   (1 )q2 ( t )) dt  I11 ( ) (1 )I12 ( ), 2) the sequence { n} X1 is minimizing

lim I1(n ) = I1* = inf I1() ;    ,  X , = ( u , pvv , , , x , xd , ), n X1 1 2 1 1 112 0 1   2 = (u1 , pvv1 , 12 , , x 01 , xd , ). 3) the sequence { n} X1 weakly converges to

the point *  X1 , uu , pp , The lemma is proved. n * n * n * n * n * n * The initial optimal control problem (2.1) – (2.4) vv11 , vv22 , x0  x0 , x1  x1 , can be solved by numerical methods for solving d  d at n   , where = (u , p ,v*,v*, extremal problems [9,10]. We introduce the n * * * * 1 2 * * m x0 , x1 ,d* )  X 1 ; following sets U = {u()  L2 (I, R )/ || u || }, 4) in order to the problem (1.2) – (1.6) have a r r solution, it is necessary and sufficient that V1 (I, R ) = {v1 ()  L2 (I, R ) / || v1 || }, lim I1(n ) = I1* = 0 ; n m2 m2 5) the following estimation of the rate of V2 (I, R ) = {v2 ()  L2 (I, R ) / || v2 || }), convergence holds

m1 1 = {d  R /d  0, | d | }, C 0 II ( )  0 , n =1,2, , 1 n 1* n (2.14)  > 0 is a sufficiently large number. We C0 = const > 0. construct sequences { } = {u , p ,v n , v n , x n , x n ,d }  X , n n n 1 2 0 1 n 1 Proof. Since the function F1(q,t) , t  I is n = 0,1,2, by the algorithm convex, it follows from Lemma 3.3. that the

International Journal of Mathematics and Physics 10, №1, 11 (2019) Int. j. math. phys. (Online) S. Aisagaliev et al. 17

Proof of the theorem follows from Theorem 2.4. functional I1() ,   X1 is convex on a weekly 1 From the results it follows that 1) if bicompact set X . Consequently, I ( )C (X ) * * * * 1 1 1  = (u , p ,v ,v , x , x ,d )  X is the solution of is weakly lower semicontinuous on a weakly * * * 1 2 0 1 * 1 optimal control problem (2.1) – (2.4), bicompact set and reaches the lower bound on X1 . for which I1(*) = 0 , then This implies the first statement of the theorem. * * Using the properties of the projection of a point (u* = u* (t), x0 , x1 )    U  S0  S1 is admissible * on a convex closed set X1 and taking into account control; 2) the function x* (t;t0 , x0 ) , t  I is the 1,1 solution of differential equation (1.2), satisfies the that I1( ) C (X 1 ) it can be shown that * * * 2 conditions: x(t1;t0 , x0 ) = x1 , x* (t;t0 , x0 )  G(t) , I1 ( n )  I1 ( n1 )   || n   n1 || , n = 0,1,2,, t  I , the functionals  * *  ,  > 0. It follows that: 1) the numerical sequence g j (u* ( ), x0 , x1 ) 0 * * {I1(n )} strictly decreases; 2) ||n  n1 || 0 at j = 1, m1 , g j (u* (), x0 , x1 ) = 0 , j = m1  1, m2 ; 3) n   . the necessary and sufficient condition for the existence of a solution of the boundary value Since the functional is convex and the set X 1 is bounded, the inequality holds problem (1.2) – (1.6) is I1(*) = 0 where *  X1 is the solution of problem (2.1) – (2.4); 4) for the

0 IIC11 (nn ) (* ) 1 || n 1 ||, admissible control, the value of the functional (1.1) (2.15) equals to C1 = const > 0, n = 0,1,2, . ** Ju(* ( ), x 01 , x ) = Hence, taking into account that t 1 ||   || 0 at n   ,, we have: the ** (2.16) n n1  F( x ( t ), u ( t ), x , x , t ) dt = ,   0* * 01 * sequence { n} is minimizing. t 0 * lim I1(n ) = I1(*) = inf I1( ) .   where x* (t) = x* (t;t0 , x0 ) , t  I . In the general n X1 case, the value Since{n}  X1 , X1 is weakly bicompact, that, ** ** weakly Ju((),,)* x 01 x Ju (,,)=*01 x x  n   * at n   . As it follows from Lemma 3.1., if the value infJ (u ( ), xx01 , ), m I1(*) = 0 , then the problem of optimal control (u(), x0 , x1 )  L2 (I , R )  S0  S1. (1.1) – (1.6) has a solution. The estimation (2.14) follows directly from the Construction of an optimal solution inequalities (2.15), 2 I1 ( n )  I1 ( n1 )   || n   n1 || . We consider the optimal control problem (1.1) – We briefly outlined above, the main steps in (1.6). We define a scalar function  (t) , t  I as: proof of the theorem. Detailed proof of an t analogous theorem is given in [16]. The theorem is       (t) = F0(x( ),u( ), x0, x1, )d , t I. proved. t0 For the case when the function F1(q,t) is not convex with respect to the variable q, the following Then (t) = F0 (x(t),u(t), x0 , x1,t) ,  (t0 ) = 0 , theorem is true.  (t ) =  = I(u(), x , x ),   {  R1 / Theorem 2.5. It is supposed, that the conditions 1 0 1 of Theorem 2.2. are satisfied, the sequence 0,}, 0  whereinf I ( u (  ), x01 , x )  0 , {n}  X1 is determined by formula (2.13). Then: the value  0 is bounded from below, in particular 1) the value of the functional I1(n ) strictly  0 = 0, if F0  0. decreases for n = 0,1,2,; 2) ||n  n1 || 0 at Now the problem of optimal control (1.1) – (1.6) n   . can be written in the form (see (2.1))

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 11 (2019) 18 Construction of a solution for optimal control problem with phase and integral constraints

 (t1) =  = I(u(), x0, x1)  inf (3.1)      =A2 () t BF 00 ( P 1 , ux , 0 , x 1 ,) t at conditions (3.7)

C0() t f ( P 1 , ut ,) D00 f ( P 1 , ux , 0 , x 1 ,), t  (t ) = F ( xt ( ), ut ( ), x , x , t ), 0 01 (3.2)  (tt )=0, ( )= , 01  ()t0    ()=t00 = xt ()= 0 x =At () x Bt () f ( xut , ,),( xt (00 )= x ,   (3.3) ()t0  xt( )= x ) S S , 1 1 01   (3.8)  O1,1   =f (xt ( ), ut ( ), x , x , t ),   x0 O 1,1 SO 0 m ,1=, T 0 0 01 (3.4)   2 O  (t01 )=0, ( t )= cQ ,  m2,1 

x(t)G(t), u() L (I, Rm ), t  I. (3.5)  ()t  2  1  ()=t11 = xt () 1 = We introduce the notations   ()t1  (3.9)     ()t            x1 SQT 11=, ()=t xt () ,     c ()t   

m P1(t) G(t),u()  L2 (I, R ),d  , (3.10)  OOO  1,1 1,nm 1, 2  A()= t O At () O  ,      2  n,1 nm, 2  where x(t) = P1 (t), (t) = P0 (t), t I, is OOO determined by formula (3.6).  m2,1 mn2,, mm 22 The immersion principle. We consider the boundary value problem (3.7) – (3.10). The  1  corresponding linear controlled system has the form     BO0=,n ,1     =A () t  Bw12 () t C () tw () t   O  20 0 (3.11)  m2,1  Dw3 ( t ), t I , 0

   O  O1,m 1 r  1,r   2  w12()  L22 ( IR , ), w ()  L ( IR , ),   (3.12) C0()= t Bt () , D 0, ()= t Onm , m2 2       w3 ( ) L2 ( IR , ), O mr2 ,  I     m2   (t0 ) = 0 T0 ,  (t1) = 1 T1. (3.13)

P= 1, O , O , PO = , IO , , 0  1,n 1, m22 1 n ,1 n nm,  We introduce the following notations: where P (t ) =  (t ) , P = x . B0 ()=( t B , C (), t D ), wt ()= 0 1 1 1 00 0 1 Then the optimal control problem (3.1) – (3.5)  (wtwtwt123 ( ), ( ), ( )), ( t , ) = KtK ( ) ( ), has the form: a= (,) tt   ,(,)= Rtt 01 1 0 01 t1     * P0 (t1) = = I(u( ), x0, x1) inf, (3.6)   *  (t00 ,) tB00 () tB () t ( t ,) tdt , t0 at conditions

International Journal of Mathematics and Physics 10, №1, 11 (2019) Int. j. math. phys. (Online) S. Aisagaliev et al. 19

t where * * Rt(00 ,)= t ( t ,) B00 () B ()  ( t0 ,) d ,  v(t) = (v1 (t), v 2 (t), v 3 (t)), z(t) = z(t,v), t  I is t 0 the solution of the differential equation  Rt(,)=(,)01 t Rt 0 t Rtt (,), 1  z= A () t z Bv12 () t C () tv () t * * 1 20 0 1(t, 0 , 1 ) = B0 (t) (t0 ,t)R (t0 ,t1 )a = (3.17)

Dv003 ( t ), zt ( ) = 0, * * 1  B  (t ,t)R (t ,t )a   11(t, , )  0 0 0 1   0 1  1 r * * 1 v12( ) L22 ( IR , ), v ( ) L ( IR , ), = C0 (t0,t)R (t0,t1)a  = 12 (t,0,1), (3.18) m2  * * 1    v3 ( ) L2 ( IR , ). D  (t ,t)R (t ,t )a 13(t,  ,  )  0 0 0 1   0 1  Solution of the system (3.11) – (3.13) has the * * 1 form K (t) = B0 (t)(t ,t)R (t ,t )(t ,t ) = 1 0 0 1 0 1

 ()=t ztv (, )  2 (, t01 , )   B**(t ,t)R1(t ,t )(t ,t )   K (t) (3.19)  0 0 0 1 0 1  11 K( tzt ) ( , v ), t I . * * 1   21 =   C0  (t0 ,t)R (t0 ,t1)(t0 ,t1)  =  K12 (t),   ** 1     The proof of the analogous theorem is presented  D0 (t0 ,t)R (t0 ,t1) (t0 ,t1)  K13 (t) in the work [10].

1 Lemma 3.2. Let the matrix be R(t0,t1) > 0 . Then  2 (,t0 , 1 )= (, tt 0 ) Rtt (, 1 ) R ( t 01 , t ) 0 the boundary value problem (3.7) – (3.10) is  1   (,)(,)ttRttRtt0 0 (,) 01 (,) tt 01 1 , equivalent to the following problem

1 K 2 (t) = (t,t0 )R(t0 ,t)R (t0 ,t1 )(t0 ,t1 ), t  I. wt1() W 11 , wt ()= FP ( , uxxt , , ,), 01 01 (3.20) tI ,

Theorem 3.1. Let the matrix be R(t0 ,t1 ) > 0.

Then the control w2 (t) W 2 , w2 (t) = f (P1 ,u,t), t  I , (3.21)   1 r m2 w(t) = (w1(t), w2 (t), w3 (t))  L2 (I, R ) w3() t W 33 ,()=(,,,,), w t f P ux x t transforms the trajectory of the system (3.11) – 01 01 (3.22)   1 n m2 tI , (3.13) from any initial point 0  R to any   1 n m2 given finite state 1  R if and only if s p()t Vt ()={ p ()  L2 ( I , R )/ pt ()= 1 (3.23) wt1() W 11 ={ w ()  LIR2 ( , )/ wt1 ()=        F(P1 ,), t () t pt () (), t t I },  v1() t 11 (, t0 , 1 ) K 11 () tzt ( 1 , v ),  z= A () t z Bv12 () t C () tv () t 1 20 0 v1()  L2 (I , R ), t  I}, (3.14) (3.24)  Dv003 ( t ), zt ( ) = 0, t I , r wt2( ) W 2 ={ w 2( )  LIR2 ( , )/ wt2 ( )= 1 r       v12( ) LIRv22 ( , ), ( ) LIR ( , ), v2 () t12 (, t0 , 1 ) K 12 () tzt ( 1 , v ), (3.25) m2 v3 ( ) L2 ( IR , ), r v 2 ()  L2 (I , R ), t  I}, (3.15) m m2 (x01 ,x ) S 0 S 1 , u ( ) L 2 ( IR , ), w3( t ) W 33 ={ w ()  L2 ( IR , )/ w3 ( t )= (3.26)   ,,d    v3 () t 13 (, t , ) K () tzt ( , v ), 0 1 13 1

where  (t), t  I is determined by formula (3.19), m2 v3()  L (I, R ),t  I}, (3.16) 2 z(t,v) is the solution of system (3.17), (3.18).

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 11 (2019) 20 Construction of a solution for optimal control problem with phase and integral constraints

We consider the following optimal control for any point   X is calculated by the formulas problem: minimize the functional

F( qt ( ), t ) t1 J ( ) = 2  Bt* ( ),  2v1  0 J2(vupxxd , , , 01 , , , ) = Fqttdt2 ( ( ), ) = v1 t0 t1 2 F2 ( qt ( ), t ) *  [|w1 ( t ) F ( P ( t ), ut ( ), x , x , t ) |  J ( ) =  Ct ( ),  01 01 2v2 0 v2 t0 2  |w2 ( t ) f ( P ( t ), ut ( ), t ) |  1 F( qt ( ), t ) J ( ) = 2  Dt* ( ), 2v3 0  3 2 v  |wt3 () f ( P (),(), tutxxt , ,)|  01 01

F (qt ( ), t ) F( qt ( ), t )  | p(t)  F(P (t),t) |2 ]dt  inf (3.27) JJ ( )=22 , ( )= 1 22upup under the conditions (3.24) – (3.26), where t 1 F( qt ( ), t ) w1(t) W 1, w2 (t) W 2 , w3 (t) W 3, J  ( )= 2 dt, 2x0  t x0 v = (v1, v 2 ,v 3 ), 0 qt()=( v123 , v , v , u , px , , x , d , , zt (),( zt )). t 01 1 1 F( qt ( ), t ) J  ( )= 2 dt, Note, that the optimization problem (3.27), 2x1  x1 (3.24) – (3.26) is obtained on the basis of relations t0 (3.20) – (3.23). t Theorem 3.3. Let the matrix be, the derivative 1 F( qt ( ), t )   2 J 2d ( )= dt, F2 (q,t) d satisfies the Lipschitz condition. t0 q Then: t 1 F( qt ( ), t ) 1. The functional (3.27) under conditions J  ( )= 2 dt, 2   (3.24) – (3.26) is continuously differentiable by t0 Frechet, gradient of the functional where  (t), t  I is the solution of the adjoint J()=( J (), J (), J (), JJ (),  (), 2 22vv123 2 v22up system

J ( ), J ( ), JJ ( ), ( )), F( qt ( ), t ) * 2x01 2 xd 22  2  =  At2 () , z

 = (v1,v2 ,v3,u, p, x0 , x1,d, )  X ,

t 1 X= L (, IR1 ) L (, IRr ) L (, IRm2 ) F( qt ( ), t ) 22 2  (t )= 2 dt; 1  t zt()1 L(, IRm ) V S S    0 2 01

2. gradient J  ( ),  X satisfies the H= L (, IR1 ) L (, IRr ) L (, IRm2 ) 2 12 2 2 Lipchitz condition

 m s nn  m1  1 L22(, IR ) L (, IR ) R R RR, ||JJ22 ( 1 )  ( 2 ) || l || 12 ||, (3.28)  '   XH12, J () H 1  12,.  X

International Journal of Mathematics and Physics 10, №1, 11 (2019) Int. j. math. phys. (Online) S. Aisagaliev et al. 21

The proof of the analogous theorem can be found 1. the numerical sequence {J 2 ( n )} is strictly in the work [16]. We construct the following n n n decreasing || n   n1 || 0 , at n   . sequences { n } = {v1 , v 2 , v 3 , u , p , n n If, in addition,, F (q,t) is a convex function with n n 2 x , x , d , }  X 2 by the algorithm 0 1 n n respect to a variable q , then:

nn1 2. the lower bound of the functional (3.27) is v11= Pv [ J ( n )], Vv11n 2 obtained under the conditions (3.24) – (3.26)

nn1 J (* )=inf J (  )= min J ( )= J ; 22   n 2 2 2 2* v= PvVv [n J2 ( )], X 22 X 2 2

nn1 v33= Pv [ J ( n )],  n  2 Vv33n 2 3. the sequence { } X is minimizing

lim J 2 ( n ) = J 2* = inf J 2 ( ); n  X 2 un12= Pu Un [ nu J ( n )], 4. the sequence { { n }  X 1 weakly converges  pn12= Pp V [ n np J ( n )], to the point  *  X 1*,

X 2* = { */J 2 ( *) = J 2* = inf J 2 () = min J 2 ()} X1 nn1 X1 x= Px [ J ( n )], 0S00 02 nx n * n * , where vv11 , vv22 , nn1 x= Px [ J ( n )], n * 1S11 12 nx vv33 , uun  *, pn  p*, * * n  n       x0 x0 , x1 x1 , dn d *, n * at dn12= Pd [ n nd J ( n )], * * * * * n   ,  * = (v1 ,v 2 ,v3,u*, p*, x0 , x1 ,d *, * );  n12=PJ [  nn  ( n )], n = 0,1, 2,..., 5. if J 2 ( * ) = 0 , then the optimal control for * * problem (1.1) – (1.6) is u* U, x0  S0, x1  S1, 2 0   n  , > 0,l = const > 0, (3.29) and the optimal trajectory l  2

1 xtPtPztv**()=11 * ()= [ (, ) where V 1 = {v1 ()  L (I , R )/ || v1 || }, 2 r V 2 = {v 2 ()  L2 (I , R )/ || v 2 || }, **  2 (t ,01 , ) K 2 ( tzt ) ( 1 , v* )], t I , m2 V 3 = {v 3 ()  L2 (I, R )/ || v 3 || }, where m *** * * U = {u()  L (I , R )/ || u || }, v*= ( vvv 123 , , ), = ( O , xO0 , ), 2 0 1,1 m2 ,1 m1 *  = {d  R /d  0, | d | }, * m2 1 = ( * ,xc1* , ), c * Q = { c * R / cj* = 1 * * = { Ra / * }, = c j  d j ,d j  0, j = 1,m1;c j* = c j , j = m1 1,m2},            X 2 = V 1 V 2 V 3 U V S0 S1 H1, m U = {u()  L2 (I, R ) / || u || },  > 0 is a the inclusion x* (t)  G(t) and limitations (1.4) – sufficiently large number. * * (1.6) J (u* , x 0 , x1 ) =  ; hold. Theorem 3.4. Let the conditions of Theorem 3.3. * 6. The following estimation of the rate of be satisfied X 1 is a bounded convex closed set, the convergence holds sequence { n }  X 2 is determined by the formula c0 (3.29). Then: 0  J ( n )  J  ,n =1,2,...,c0 = const > 0. 2 2* n

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 11 (2019) 22 Construction of a solution for optimal control problem with phase and integral constraints

Proof of the analogous theorem is given above. constraints for processes described by ordinary A more obvious method for solving problem differential equations. The particular cases of which (1.1) – (1.6) is the method of narrowing the domain are the simplest problem, the Bolz problem, the of admissible controls. isoperimetric problem, the conditional extremum Theorem 3.5. Let the conditions of Theorem 3.3. problem. be satisfied, In contrast to the well-known method for solving the problem of the variation calculus on the basis of 3        X= V123 V V UVS01 S be a the Lagrange principle, an entirely new approach an bounded convex closed set, the sequence "immersion principle" is proposed. The immersion { n }  X 2 be defined by (3.28) with the exception principle is based on the investigation of the Fredholm integral equation of the first kind. For the of the sequence { }  . Then: n Fredholm integral equation of the first kind, the

1. the numerical sequence {J 2 ( n )}, existence theorem for the solution as well as the theorem on its general solution are proved.  n  is strictly decreasing; { } X 3 The main scientific results are: 2. || n   n1 || 0 at n  , { n }  X 3; - reduction of the boundary value problem connected to the conditions in the Lagrange problem If, in addition, the function F (q,t) is convex 2 to the initial optimal control problem with a specific with respect to a variable q for fixed  , then: functional; - necessary and sufficient conditions for the 3. the sequence { n }  X 3, for a fixed existence of the admissible control;  =  is minimizing; - method of constructing an admissible control on the limit point of the minimizing sequence; 4.  n  * 3 at n  ,  =; X - necessary and sufficient conditions for the

5. J 2 ( *) = inf J 2 ( n ) = min J 2 ( n ); existence of a solution of the Lagrange problem;  X  nX 3 n 3 - method for constructing the solution of the 6. the following estimation holds Lagrange problem.

c1 0 JJ () n  () *  , 22n References

c= const > 0, n = 1,2,...,{ n } X 3 . 1 1. Bliss Dzh. Lectures on variation calculus, – The proof of the analogous theorem is presented M.: IL (1950):348. in the work [10] for a fixed  ,  =. 2. Lavrentev M.A., Lyusternik L.A. “Basics of variation calculus.” ONTI 1, Ch.1, 2 (1935): 148, Let the solution of the problem be  *  X 2 400. (3.27), (3.24) – (3.26) with  . There are = * 3. Yang L. “Lectures on variation calculus and the possible cases: optimal control theory.” Mir (1974): 488.

1. the value J 2 (* ) > 0; 4. Alekseev V.M., Tihomirov V.M., Fomin S.V. “Optimal control.” Nauka (1979): 430. 2. the value  . J 2 ( * ) = 0 5. Pontryagin L.S., Boltyanskiy V.G., Gamkrelidze R.V., Mischenko E.F. “Mathematical Note, that J 2 ( )  0,  X 3. theory of optimal processes.” Nauka (1969): 384.   If J 2 ( * ) > 0 , then a new value of is 6. Varga Dzh. “Optimal control by differential and functional equations.” Nauka (1977): 624. selected as =2* , and if J 2 ( * ) = 0 , then a new  7. Gamkrelidze R.V. “Basics of optimal control.” value  = * . According to this scheme, by – Tbilisi: Izd-vo Tbilisskogo n-ta, (1977): 253. 2 8. Li Ya.B., Markus L. “Basics of the optimal dividing the uncertainty segment in half, the control theor.” Nauka (1972): 574. smallest value of the functional (1.1), under the 9. Aysagaliev S.A., Sevryugin I.V. conditions (1.2) – (1.6) can be found. “Controlabillity and speed of the process described by linear system of ordinary differential equations.” Conclusion Mathematical journal 13, no. 2(4) (2013): 5–30. 10. Aysagaliev S.A., Kabidoldanova A.A. The Lagrange problem of the variation calculus “Optimal control by dynamic systems, Palmarium is investigated in the presence of phase and integral Academic Publishing.” Verlag (2012): 288.

International Journal of Mathematics and Physics 10, №1, 11 (2019) Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 23 (2019)

IRSTI 27.31.15

3M. Ruzhansky, 1,2D. Serikbaev, 1,2N. Tokmagambetov

1 Al-Farabi Kazakh National University, Almaty, Kazakhstan 2Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan 3Ghent University, Ghent, Belgium e-mail: [email protected] An inverse problem for the pseudo-parabolic equation for Laplace operator

Abstract. A class of inverse problems for restoring the right-hand side of the pseudo-parabolic equation for 1D Laplace operator is considered. The inverse problem is to be well-posed in the sense of Hadamard whenever an overdetermination condition of the nal temperature is given. Mathematical statements involve inverse problems for the pseudo-parabolic equation in which, solving the equation, we have to find the unknown right-hand side depending only on the space variable. We prove the existence and uniqueness of the classical solutions. The proof of the existence and uniqueness results of the solutions is carried out by using L-Fourier analysis. The mentioned results are presented as well as for the fractional time pseudo–parabolic equation. Inverse problems of identifying the coefficients of right hand side of the pseudo-parabolic equation from the local overdetermination condition have important applications in various areas of applied science and engineering, also such problems can be modeled using common homogeneous left-invariant hypoelliptic operators on common graded Lie groups. Key words: Pseudo-parabolic equation, 1D Laplace operator, fractional Caputo derivative, inverse problem, well-posedness.

Introduction and boundary conditions

In this paper we study inverse problem for the (3) time-fractional pseudo-parabolic equation for one dimensional Laplace operator. We consider is self-adjoint in �(�) =. �,�(�) The problem = �, (2)-(3) has following equation the following eigenvalues� � (�, �) (1) � � �� = � � , � � �, for� [�(�, �) −�Ω (�, �)] −� (�, �) =�∞(�), and the corresponding system� of eigenfunctions , where is the Caputo derivative which is (�, �) �� = �(�, �)�� ,�� defined in the� next section. The operator � �� which is participating in the equation(1) is the well� known 1D Laplace operator and we will denote− �� it 2 �� It is known��(�) that = �the sinself-adjoint(�), � problem � �� has real further by . We know the second order differential � � operator in generated by the differential eigenvalues and their eigenfunctions form a complete orthonormal basis in . expression ℒ � � (�, �) The study of inverse problems� for pseudo � (�, �) (2) parabolic equations began in the 1980s. The first result obtained by Rundell [2] refers to the inverse ℒ�(�) = −��(�), � � (�, �)

identification problems for an unknown sourse derivative of order ( function in a following equation ) is defined� as �� ∈� ��1<�<�,�∈ � ℕ � (4) �� (�) = � �� [�(�, �) � ℒ�(�, �)] � ℒ�(�, �) = �. Where is even order linear differential =�� (�) ��(�) ′ operator. Rundell proved global existence and �� uniqueness theoremsℒ for cases when depends only (� � �) (��) (� � �) �� (�) �...�� (�) �. or only . In a series of articles [6], [7], [8], [9], If 1! then, the Caputo(� � 1)!fractional [10], [12], [13], [14], [15],[16],[17]� some recent derivative � of order ( �work has been� done on inverse problems and spectral �∈�) is �defined[�, �] as problems for the di usion and anomalous di usion �� ∈� ��1<�< equations. �, � ∈ ℕ ﬀ ﬀ � �� (�) Definitions of fractional operators �� [�](�) =� � (�) = Γ �� (�) We begin this paper with a brief introduction of 1 = � (��) � (�)��. several concepts that are important for the further (��) � Formulation of the problem studies. Definition 1. [5] The Riemann-Liouville Problem 1.We aim to find a couple of functions fractional integral of order for an satisfying the equation(1), under the integrable function is defined� by conditions I α>0 (u(t, x), f(x)) (5)

Γ � �(0, �) = �(�), � ∈ [0, �] (6) � 1 ��

� [�](�) = �� (� � �) �(�)��, � ∈ [�, �], where Γ denotes(�) the Euler gamma function. and the homogeneous�(�, �) = Dirich �(�),let � ∈boundary [0, �]. conditions

Definition 2. [5] The Riemann-Liouville (7) fractional derivative of order of a continuous function is defined� by By using�(�,0) –Fourier = �(�,�) analysis = 0,� ∈we [0,�]. obtain existence D α ∈ (0,1) and uniqueness results for this problem. We say aℒ solution of Problem 1 is a pair of

functions such that they satisfy � � � � [�](�) = � [�](�), � ∈ [�, �]. equation(1) and conditions(5)-(7) where Definition 3. ��[5] The Caputo fractional (�(�, �), �(�)) and . derivative of order of a differentiable � � �(�, �) ∈ function is defined by � ([0,Main �]� results � ([0, �])) �(�) ∈ �([0, �]) 0<α<1 ′ For Problem 3.1, the following theorem holds. � � Theorem 1.Assume that . �∗ [�](�) = � [� (�)], � ∈ [�, �]. Definition 4.[5] (Caputo derivative). Let Then the solution � � ∞ ∞and of the Problem�(�),� 3.1 �(�) exists, ∈ �is � unique,[0, �] � The Caputo fractional�∈ �(�, �) ∈� �[0, �],�([0, �])�, �� and can be written in the form � [�,�,� �], � ��<�<�� ∗� (�) ∈ �(�) ∈ �([0, �]) � [�, �], � = [�], � > 0.

International Journal of Mathematics and Physics 10, №1, 23 (2019) Int. j. math. phys. (Online) M. Ruzhansky et al. 25

�� ∞ (�) (�) � � � � � � � �,� �� −� ��1−� �− �� (�, �) = ��(�) +� �� sin (�), �� ,� �� 1−� �− �� (8)

∞ ′′ (�) (�) �� (9) �� (�) = −� (�) + ∑ � � sin (�), �,� ��

where ′′ ′′ where the constants and are unknown. To find these constants, we use conditions(5), (6). Let and (�) is the Mittag-Leffler� (�) type function� (see � � � (�,�) � � � (�,�) and be the coefficients�� of the expansions of [4]): � = (� ,� ) ,� = (� ,� ) �,� and : � (��) � � ∞ � � �(�) �(�)

Γ � � � 2 �� ,�(�) = � . �� = � � �(�)sin (�)��, � � �, � � � First of all, we start�� by(�� proving + �) an existence result. � 2 �� Proof. Let us seek functions and � � � = �� (�)sin (�)��, � � �. in the forms: We first find � : � �(�, �) �(�) � ∞ � �� (10) �� and �(�, �) = ∑ � (�)sin � (�), � � �, � � � � � (0) = �� +� =� , � � � ∞ (11) �� ,� � �� (�) = �� +� � �− �� =� . where �(�) =and∑ � aresin unknown.(�), � � �,Substituting Equations (10) and (11) into Equation (1), we obtain Then � � � 1+� � � the following��(�) equation�� for the functions and the constants : ��(�) � �� −�� = � � � . � � �� ,� �� 1−� �− �� The constant is represented as � � (�) + �� (�) = �� .

Solving these1+ equation,we� � � obtain1+ � � � �� =�� −�� . �� Substituting �and into �expansion (10), �� we find � � � �,� � � � � (�) = �� +� � �− �� , � (�) � � � � 1+� � �

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 23 (2019) 26 An inverse problem for the pseudo-parabolic equation for Laplace operator

∞ �� (12) � � � �� ,� �� (�, �) = �(�) + ∑ � �� − �� −��sin (�). By the supposition of the theorem we know

(�) (�) � (�) = �, � (�) = �, � = �,�,�, then using they we have (�) (�) � (�) = �, � (�) = �, � = �,�,�.

(�) (�) � � � � � −�� −� � �� = � � � =− . � �� ,� �,� �� −� �− �� −� �− �� Putting this into equations (10) and (11) we obtain

�� (�) (�) � � ∞ �,� (13) �� (�, �) = ��(�) + ∑ � � � sin � (�). �� ,� � � � �� Similarly, �

∞ ′′ (�) (�) �� (14) �� (�) = −� (�) + ∑ � � sin (�). �,� ��

The following Mittag-Leffler function’s estimate is known by [11]: � � �� (��,�� (��,��))=��,�� max � ∞ ∞ � �(�,�) �� (��,��)+��,�� max � � � (15) � � (��,��) ��,�(�)� � �� , ��(�) = �, �� . �� (�,�) � < , and Now, we show that ∞ , , that is ��(��,��)< . � � �(�, �) � By using (15), we get following estimates � (��, �� (��, ��)) �(�) � �(��, ��)

∞ (�) (�) �� (�, �)� � ��(�)� + ∑ � � � �� (16) �,� � � � �� ∞ (�) (�) �� (�)� + ∑ � � � ,

International Journal of Mathematics and Physics 10, №1, 23 (2019) Int. j. math. phys. (Online) M. Ruzhansky et al. 27

∞ ′′ �� (17) �� + ∑ � � �,� �� ′′ ∞ � �� + ∑ �� + �� . Where, $L denotes for some trigonometric series (see [1]) and by the Weierstrass positive constant independent of and . M-test (see [3]), series (16) and (17) converge By supposition�� of the theorem� � we C� know absolutely and uniformly in the region Ω. Now we and are continuousC on . � � �� show. �� Then� by the Bessel��, inequality �� for the

∞

′′ �� + �� , �� + � � � � �� ∞ �,� �� 1�� ′′ ∞ �� + � �� + �� < , �� ∞

�� + �� , �� ,� �� ∞ �1+� � ��1��

�� ∞ �� + �� < , �� 1+� � �

∞

�� + �� , �� ,� �� ∞ �1+� � ��1��∞ ��

�� ∞ �� + �� + �� +� �� < . �� 1+� � � Finally, we obtain Now, we start proving uniqness of the solution.Let us suppose that and ∞ are solution of the Problem 1. Then � � and � � and�� ,��,�� ,��,� ��,�� <∞ , � � ��, �� , ��,are �solution�� of following problem: � � � ��, �� , �� , �� ,�� Existence of the�� solution is< proved.. � �� (18) � � �� Int. j. math. phys. (Online) International� ��,Journal �� of � M �athematics��, �� and � Physics��, �� 10, � №1, ��, 23 (2019) 28 An inverse problem for the pseudo-parabolic equation for Laplace operator

(19) 8. I. Orazov, M.A. Sadybekov. “One nonlocal problem of determination of the temperature and (ݑሺͲǡ ݔሻ ൌ Ͳǡ (20) density of heat sources.” Russian Mathematics 56(2 (2012): 60–64 and Computation. 218(1):163–170, .By using (13) andݑሺܶǡ (14) Ͳሻ for ൌ (18)-(20) ͲǤ we easily see 2011 . Uniqness of the solution of 9. Orazov I., Sadybekov M.A. “On a class of the Problem 1. problems of determining the temperature and density ”.ݑሺݔǡ ݐሻ ؠ Ͳǡ ݂ሺݔሻ ؠ Ͳ of heat sources given initial and final temperature Acknowledgement Sib Math J. 53(1) (2012): 146-151. 10. K.M. Furati, O.S. Iyiola, M. Kirane. “An Theauthors were supported by the Ministry of inverse problem for a generalized fractional Education and Science of the Republic of di�usion.” Applied Mathematics and Computation Kazakhstan Grant AP05130994. 249 (2014): 24–31. 11. Z. Li, Y. Liu, M. Yamamoto. “Initial – boun- References dary value problems for multi-term time-fractional diffusion equations with positive constant coefficients.” 1. Zygmund, A., “Trigonometrical Series.” Appl. Math. Comput. 257 (2015): 381–397,. Cambridge (1959). 12. H.T. Nguyen, D.L. Le, V.T. Nguyen. 2. Rundell W. “Determination of an unknown “Regularized solution of an inverse source problem nonhomogeneous term in a linear partial differential for a time fractional di�usion equation.” Applied equation from overspecified boundary data.” Appl. Mathematical Modelling 40(19) (2016): 8244–8264. Anal. 10 (1980): 231-242. 13. M.I. Ismailov, M. Cicek. “Inverse source 3. Knopp K. “Theory of Functions Parts I and II, problem for a time-fractional di�usion equation with Two Volumes Bound as One, Part I.”New York: nonlocal boundary conditions.” Applied Dover; (1996): 73. Mathematical Modelling 40(7) (2016): 4891–4899. 4. Y. Luchko, R. Gorenflo. “An operational 14. M, Al-Salti N. “Inverse problems for a non- method for solving fractional differential equations local wave equation with an involution perturbation.” with the Caputo derivatives.” Acta Math. Vietnam. J Nonlinear Sci Appl. 9(3) (2016): 1243-1251. 24 (1999): 207–233. 15. N. Al-Salti, M. Kirane, B. T. Torebek. “On a 5. Kilbas A.A, Srivastava H.M, Trujillo J.J. class of inverse problems for a heat equation with “Theory and Applications of Fractional Differential involution perturbation.” Hacettepe Journal of Equations.” Mathematics studies 204. Mathematics and Statistics. (2017) Doi: North-Holland: Elsevier; (2006): vii–x. 10.15672/HJMS.2017.538 6. Kaliev I.A, Sabitova M.M. “Problems of 16. M. Kirane, B. Samet, B. T. Torebek. determining the temperature and density of heat “Determination of an unknown source term sources from the initial and final temperatures.” J temperature distribution for the sub-diffusion Appl Ind Math. 4(3) (2010): 332-339. equation at the initial and final data.” Electronic 7. M. Kirane, A.S. Malik. “Determination of an Journal of Differential Equations (2017): 1–13. unknown source term and the temperature 17. B.T. Torebek, R. Tapdigoglu. “Some inverse distribution for the linear heat equation involving problems for the nonlocal heat equation with Caputo fractional derivative in time.” Applied Mathematics fractional derivative.” Mathematical Methods in the and Computation 218(1) (2011):163–170. Applied Sciences 40(18) (2017): 6468–6479.

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IRSTI 27.31.15

1*S.Ya. Serovajsky, 2A.A. Azimov, 1M.O. Kenzhebayeva, 2D.B. Nurseitov, 1A.T. Nurseitova, 1M.A. Sigalovskiy

1al-Farabi Kazakh National University, Almaty, Kazakhstan 2Satbayev University, Almaty, Kazakhstan *e-mail: [email protected]

Mathematical problems of gravimetry and its applications

Abstract. Gravimetry is associated with analysis of the gravitational field. The gravitational field is characterized by its potential. This is described by the Poisson equation, the right side of which includes the density of the environment. There exists direct and inverse problems of gravimetry. Direct gravimetry problems involve the determination of the potential of the gravitational field in a given region. The inverse problems of gravimetry imply the restoration of the structure of a given area from the results of measuring the characteristics of the gravitational field. Such studies are needed to assess on the basis of gravimetric geodynamic events occurring in oil and gas fields. The relevance of such research is necessary, because with prolonged development of the oil and gas fields, negative consequences may occur. This paper discusses some of the features of direct and inverse gravimetry problems. A description of the mathematical model of the processes under consideration is given. Different direct and inverse gravimetry problems are posed. Describes the methods of its solving. Based on the analysis of the results of a computer experiment, appropriate conclusions are made. Key words: gravimetry, inverse problems, mathematical model.

Introduction The relevance of the inverse problems of gravimetry is due to the fact that in the process of Gravimetry is a science related to the study of long-term operation of deposits of different minerals gravitational fields. The gravitational field is (in particular, oil and gas), significant changes occur potential, i.e. the work expended on movement in that have undesirable consequences [3–5]. In this this field along a closed curve is zero. The main regard, monitoring of existing fields is regularly function characterizing a potential field is the conducted. In this case, we are interested in potential. The potential of the gravitational field is gravimetric monitoring. With the help of described by the Poisson equation, the right-hand gravimeters, measurement of the acceleration of side of which includes the density distribution in a gravity, corresponding to the gradient of the given region [1, 2]. potential of the gravitational field, is carried out [6, Mathematical problems of gravimetry are 7]. This experimental information can be used as a divided into direct and inverse. Direct gravimetry basis for the formulation of inverse problems of problems involve finding the distribution of the gravimetry. potential of a gravitational field over a known It is known that the inverse problems are ill- density distribution in a given region. This is posed, in principle [8]. However, the inverse achieved by solving the corresponding boundary problems of gravity are essentially ill-posed. In value problem for the Poisson equation. In the particular, the values of the acceleration of gravity inverse problems of gravimetry, on the contrary, it determined during the measurement process may be is necessary to reconstruct the structure of the due to various gravity anomalies. Thus, the solution considered set by measuring the gravitational of the inverse problem of gravimetry in the full field. formulation is not the only solution. Naturally, in

© 2019 al-Farabi Kazakh National University Int. j. math. phys. (Online) 30 Mathematical problems of gravimetry and its applications the numerical solution of such a problem, the natural to choose it as a reference, i.e. the zero value algorithm outputs to one of these solutions. of the vertical coordinate z. Then we can assume However, this result may not correspond to reality. that the given function is h = h(x, y), which Mathematically correct, it will have no practical characterizes the height of the terrain at the point x, meaning. In this connection, in practice, only partial y of the surface S with respect to the chosen system. inverse gravimetry problems are actually solved Thus, the system is considered in three-dimensional (see, for example, [9–18]). These works differ, volume firstly, in the volume of measured information, secondly, in the amount of identifiable information V = {(x,y,z) | 0 < z < h(x,y), (x,y)S}. and, thirdly, in research methods. In this paper, we discuss some peculiar In practice, using gravimeters, the gravitational properties of the formulation of direct and inverse acceleration is measured, which, up to a sign, gravimetry problems based on the available coincides with the vertical derivative of the potential experimental information, as well as methods for of the gravitational field. Thus, the following solving these problems. We characterize some of the condition holds difficulties encountered in solving direct and inverse gravimetry problems and discuss ways to overcome.   xyhxy, ,(, )  gxy(, ), (,) xy S , (2) Statement of the problem z

At first, give the general direct gravimetry where g is the experimentally measured value of the problem (see, for example, [1,2]). The gravitational gravitational acceleration. This information, which field in the given volume is described by the is the result of gravimetric monitoring, can be used Poisson equation as the basis for the formulation of inverse gravimetry problems. (x,y,z) = –4 G(x,y,z), (1) The purpose of the gravimetric monitoring of the existing field is largely to clarify the geological and where  is the gravitational potential,  is the tectonic structure and geological field information density, G is the gravitational constant. of the study area in order to highlight the risk of It is necessary to add the boundary conditions. In geodynamic processes. Such information can be principle, we can have some results of measuring on obtained by knowing the density distribution in a the ground surface. Unfortunately, we do not, as a given region. Thus, the object of the search in the rule, any information about the gravitational field process of solving the inverse problem of underground. However, it is known that the gravimetry is the density function, which is in the influence of the object to the gravitational field right-hand side of the considered equation (1). decreases with distance from the object and tends to Therefore, the general gravimetry problem is finding zero with unlimited distance from it. Then we can the density distribution  = (x,y,z) in the volume V extend the given set such that the gravitational such that the solution of the homogeneous Dirichlet potential on the boundary of the extended set will be problem for the Poisson equation (1) in the extended zero. Thus, the general direct gravimetry problem is set satisfies the additional condition (2). finding the gravitational potential  = (x,y,z) from As is known, the most natural way to solve the homogeneous Dirichlet problem for the Poisson inverse is to reduce them to optimization problems. In particular, the stated inverse problem can be equation (1), using known density distribution  = reduced to the problem of minimizing the functional (x,y,z).

For formulating inverse problems of gravimetry, 2  xyhxy, ,(, ) it is necessary to determine what specific    information we can directly get into the process of I I()  g(, x y ) dS ,(3) S z gravimetric monitoring and what exactly we would like to find on the basis of this information. Note that when analyzing deposits, we have some where  is a solution of the considered direct territory S in the x, y plane. The terrain in this area is problem corresponding to the given function . known. In addition, the maximum depth that is of Naturally, the solution of the inverse problem also interest to the research is usually specified. It is turns out to be the solution of the optimization

International Journal of Mathematics and Physics 10, №1, 29 (2019) Int. j. math. phys. (Online) S.Ya. Serovajsky et al. 31

2 the numerical solution of such a problem, the natural to choose it as a reference, i.e. the zero value problem, and the solution of the optimization M  xyhxy, ,(, )  algorithm outputs to one of these solutions. of the vertical coordinate z. Then we can assume problem under the condition of its existence will be ii ii II()     gi  . (5) However, this result may not correspond to reality. that the given function is h = h(x, y), which the solution of this inverse problem. The practical i1  z  Mathematically correct, it will have no practical characterizes the height of the terrain at the point x, solution of the obtained optimization problem is meaning. In this connection, in practice, only partial y of the surface S with respect to the chosen system. carried out using numerical optimization methods, Results and discussion inverse gravimetry problems are actually solved Thus, the system is considered in three-dimensional for example, the gradient method [19–21]. (see, for example, [9–18]). These works differ, volume Note that in reality the measurement of the For simplicity, we perform the analysis for the firstly, in the volume of measured information, gravitational acceleration is carried out not two-dimensional case, considering the horizontal secondly, in the amount of identifiable information V = {(x,y,z) | 0 < z < h(x,y), (x,y)S}. everywhere in a given area S, but only at certain coordinate x and the vertical coordinate z. For the and, thirdly, in research methods. points (xi,yi), i = 1,2,…,M, where gravimeters are direct problem in the simplest case, we consider a In this paper, we discuss some peculiar In practice, using gravimeters, the gravitational located. Thus, in fact, instead of (2), we have the rectangular area in which the gravitational anomaly properties of the formulation of direct and inverse acceleration is measured, which, up to a sign, condition is located, i.e. an object significantly different in gravimetry problems based on the available coincides with the vertical derivative of the potential  xyhxyii, ,(, ii ) density from the environment. Figure 1 shows the experimental information, as well as methods for of the gravitational field. Thus, the following gi, 1,..., M (4) z i density distribution over a given area, as well as the solving these problems. We characterize some of the condition holds calculated distribution of the gravitational potential with known values gi. Thus, in practice, either by difficulties encountered in solving direct and inverse and its derivative in the upper part of the region interpolation, the transition to condition (2) is gravimetry problems and discuss ways to overcome.  corresponding to the earth's surface if the anomaly is  xyhxy, ,(, ) performed, followed by minimization of the  gxy(, ), (,) xy S , (2) located in the center of the considered set (Figure z functional (3), or we solve the minimization Statement of the problem 1a) and near its boundary (Figure 1b). problem for the functional. where g is the experimentally measured value of the At first, give the general direct gravimetry problem (see, for example, [1,2]). The gravitational gravitational acceleration. This information, which field in the given volume is described by the is the result of gravimetric monitoring, can be used Poisson equation as the basis for the formulation of inverse gravimetry problems. (x,y,z) = –4 G(x,y,z), (1) The purpose of the gravimetric monitoring of the existing field is largely to clarify the geological and where  is the gravitational potential,  is the tectonic structure and geological field information density, G is the gravitational constant. of the study area in order to highlight the risk of It is necessary to add the boundary conditions. In geodynamic processes. Such information can be principle, we can have some results of measuring on obtained by knowing the density distribution in a the ground surface. Unfortunately, we do not, as a given region. Thus, the object of the search in the rule, any information about the gravitational field process of solving the inverse problem of underground. However, it is known that the gravimetry is the density function, which is in the influence of the object to the gravitational field right-hand side of the considered equation (1). decreases with distance from the object and tends to Therefore, the general gravimetry problem is finding zero with unlimited distance from it. Then we can the density distribution  = (x,y,z) in the volume V extend the given set such that the gravitational such that the solution of the homogeneous Dirichlet potential on the boundary of the extended set will be problem for the Poisson equation (1) in the extended set satisfies the additional condition (2). zero. Thus, the general direct gravimetry problem is finding the gravitational potential  = (x,y,z) from As is known, the most natural way to solve the homogeneous Dirichlet problem for the Poisson inverse is to reduce them to optimization problems. In particular, the stated inverse problem can be equation (1), using known density distribution  = reduced to the problem of minimizing the functional (x,y,z).

For formulating inverse problems of gravimetry, 2  xyhxy, ,(, ) it is necessary to determine what specific    information we can directly get into the process of I I()  g(, x y ) dS ,(3) S z gravimetric monitoring and what exactly we would like to find on the basis of this information. Note where  is a solution of the considered direct that when analyzing deposits, we have some a) anomaly is in the center b) anomaly is near the boundary territory S in the x, y plane. The terrain in this area is problem corresponding to the given function . known. In addition, the maximum depth that is of Naturally, the solution of the inverse problem also Figure 1 – Density distribution, potential, and its vertical derivative interest to the research is usually specified. It is turns out to be the solution of the optimization for the case of rectangular anomaly

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As can be seen from Figure 1a, at the location of At the next stage of the study, we are already the anomaly (point x), the potential and its vertical repelling ourselves from the geological and derivative have their maximum value. As one moves lithographic section of the real field. Figure 2 away from the anomaly, these values decreases to depicts the density distribution in the considered zero equally in both directions, which is the area, with the yellow color indicating the area filled corollary to zero boundary conditions. However, with clay – the predominant environmental material when the anomaly is located near the boundary of and relatively high density, blue – the air that inside the region under consideration (Figure 1b), the the field corresponds to the existing voids with potential distribution and its derivative are no longer significantly lower density, and green – oil, more symmetrical, which is not satisfactory. The results lighter than clay, but heavier than air. When suggest that, in order to eliminate the influence of extending a given area, it is assumed that air is the boundaries, the area under consideration should located above the surface of the earth, and clay is be significantly extended. located outside the initial area.

Figure 2 – Density distribution, potential, and its vertical derivative for the extended set with real parameters and usial density outside the initial set

As can be seen from the results obtained, the the boundaries. This is explained by the fact that potential on the surface of the earth over the zone of there is heavy clay outside the initial region, and predominance of oil and voids is lower compared to zero potential values are rigidly set at the boundary. the neighboring zones where clay is predominant. Besides, the potential derivative greatly increases in This is due to the fact that the clay has a greater the neighborhood of the boundaries. Such a result density. In this case, the vertical derivative of the cannot be considered satisfactory, and suggests that potential is negative, since as the distance from the the density in the extended part of the region should object increases, the potential value decreases, and be continued to zero. Indeed, we consider the the larger, the larger the potential value itself. We gravitational field created by objects located in a draw attention to the fact that outside the initial given area. region, the potential value turns out to be rather At the next stage of the analysis, we carried large and decreases sharply to zero in the vicinity of out calculations with the extension of the set so

International Journal of Mathematics and Physics 10, №1, 29 (2019) Int. j. math. phys. (Online) S.Ya. Serovajsky et al. 33 that the density outside the initial region is boundaries approach. The value of the vertical assumed to be zero. In this case, the distribution derivative potential also tends to zero, see of the potential gradually decreases to zero as the Figure 3.

Figure 3 – Density distribution and vertical derivative of the potential for the extended set with real parameters and usial density outside the initial set

The question arises, how we can determine the density of the environment (see Figure 4a). Then we size of the extended set. First, some extension is solve the Poisson equation with given boundary selected, and the value of the derivative potential on condition and calculate the vertical derivative of the the earth's surface in the given region is calculated. potential at the ground surface. Now we put the Then the area extends again and calculations are result to the minimizing functional and solve the carried out. If the newly found value of the minimization problems by means of the gradient derivative potential practically does not differ from method. The iterative method converged, and the that found earlier, then the calculations are sequence of functional tends to zero. Thus, we terminated. Otherwise, a new extension is carried found the solution of the optimization problem that out. is the solution of inverse problem too. The obtained Now consider the inverse problem. At first, we result is shown in the Figure 4b). Unfortunately, this try to solve the general inverse problem for the two- result is significantly different from the real. This is dimensional case. We determine the gravitational the corollary of the non-uniqueness of the anomaly as the square with a higher density than the considered inverse problem.

a) real position b) obtained position

Figure 4 – Position of the gravitational anomaly of the inverse problem

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 29 (2019) 34 Mathematical problems of gravimetry and its applications

Its is clear that the general inverse problem of 7. The general inverse problem of gravimetry has gravimetry is not very interesting because of its essentially not the only solution, as a result of which significantly non-uniqueness. We had a few data the value of the density distribution found using (boundary value of potential derivative) for finding standard optimization methods may differ from its the many information (density as a function of real value. spatial variables). Then we consider two partial 8. Some particular inverse problems of cases. For the first case, we suppose that we know gravimetry have been solved, in particular, the the position of the homogeneous anomaly, its form restoration of constant density and coordinates of and size, but its density is unknown. For the second the location of the gravitational anomaly. case, we consider inverse situation. The density of 9. Optimization problems corresponding to the anomaly with given form and size is known, and inverse gravimetry can be characterized by a non- its position is unknown. The first problem was be differentiable functional. In this case, non-smooth solved by gradient methods [19–21] with good optimization methods can be used, in particular, the enough exactness. The second (geometric) inverse subgradient method, the Nelder – Mead method, and problem has the peculiarity. The minimizing genetic algorithms. functional is not Gateaux differentiable. It is 10. The obtained results can be used in subdifferentiable only. Then we use the methods of monitoring oil and gas fields. non-smooth optimization, particularly, the subgradient method [22], the Nelder – Mead method Acknowledgment [23], and genetic algorithms [24]. The exactness of the results was be good enough too. This is clear, The research was supported by the project because for both partial inverse problems, we AP05135158 “Development of geographic determine one (constant density) or two (coordinates information system for solving the problem of of the anomaly) parameters, using the knowledge of gravimetric monitoring of the state of the subsoil of the function (vertical derivative of the potential at oil and gas regions of Kazakhstan based on high- the ground surface). performance computing in conditions of limited experimental data” Conclusion References Based on the obtained results, the following conclusions can be drawn: 1. C. Misner, K. Thorne. Gravitation. – W.H. Free- 1. The direct problem of gravimetry is based on man & Company, Princeton University Press, 1973. the Poisson equation with respect to the potential of 2. D.D. Ivanenko, G.A. Sardanashvili the gravitational field with a density included in the Gravitation. – M .: URSS, 2008. right-hand side of the equation. 3. Yu.M. Nesterenko, M.Yu. Nesterenko, V.I. 2. To find the potential distribution in a given Dnistryanski, A.V. Glyantsev. “Influence of the region, the given region should be extended by development of hydrocarbon deposits on setting uniform Dirichlet boundary conditions on the geodynamics and water systems of the Southern extended set. Urals region.” Bulletin of the Orenburg Scientific 3. The density value outside the source region is Center, Ural Branch of the Russian Academy of assumed to be zero. Sciences no. 4 (2010): 28–41. 4. Experimentally measured the acceleration of 4. Yu.O. Kuz’min “Modern geodynamics and gravity, which corresponds to the vertical derivative geodynamic risk assessment for subsoil use.” AEN of the gravitational potential. (1999). 5. The method of choosing the size of the 5. V.A. Chernyh. “Hydro-mechanics of oil and extended set is proposed. gas production.” VNIIGAZ (2001). 6. The general inverse problem of gravimetry is 6. H. Virtanen. “Studies of earth dynamics with to find the density distribution in a given area, using superconducting gravimeter.” Academic the measure of the potential derivative on the outer Dissertation at the University of Helsinki, surface. Geodetiska Institutet (2006): 45 p.

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7. V.B. Vinogradov, L.A. Bolotnova, 16. V.V. Vasin, E.N. Akimova, A.F. Gravimeters. UGGU publ. (2010). Miniakhmetova. “Iterative Newton Type 8. S. Kabanikhin. “Inverse and Ill-posed Algorithms and Its Applications to Inverse Problems: Theory and Applications.” – Boston, Gravimetry Problem.” Vestnik of the South Ural Berlin: Walter de Gruyter (2012). State University. Series: Mathematical modelling 9. V.N. Strakhov, M.I. Lapina. “The assembly and programming 6, no. 3 (2013): 26–33. method for solving the inverse problem of 17. E.N. Akimova, V.V. Vasin, V.E. Misilov. gravimetry.” RAS 227, no. 2 (1976): 344–347. “Algorithms for solving inverse gravimetry 10. L. Misici, F. Zirilli. “The inverse gravimetry problems of finding the interface between media on problem: An application to the northern San multiprocessing computer systems.” Vestnik of Francisco craton granite.” Journal of Optimization Ufim State Aviation Technical University 18, no. 2 Theory and Applications 63, no. 1 (1989): 39–49. (2014): 208–217. 11. R. Barzaghi, F. “Sansó The integrated 18. P. Balk, A. Dolgal, T. Balk, L. Khristenko. inverse gravimetric-tomographic problem: a “Joint use of inverse gravity problem methods to continuous approach.” Inverse Problems 14, no. 3 increase interpretation informativity.” (1998): 499–520. Geofizicheskiy Zhurnal 37, no. 4 (2017): 75–92. 12. V. Michel. “Regularized wavelet-based 19. E. Polak. Computational methods in multiresolution recovery of the harmonic mass optimization: A unified approach. – New York, density distribution from data of the earth's Academic Press (1971). gravitational field at satellite height.” Inverse 20. R.P. Fedorenko. “Approximated solving of Problems 21 (2005): 997–1025. optimization control, problem.” Nauka (1978). 13. W. Jacoby, P. Smilde. “Gravity 21. A. Parkinson, R. Balling, J. Hedengren. Interpretation: Fundamentals and Application of “Optimization methods for engineering design.” Gravity Inversion and Geological Interpretation.” Provo, UT: Brigham Young University (2013). Springer Science & Business Media (2009). 22. N. Shor “Minimization Methods for non- 14. I. Prutkin, A. Saleh. “Gravity and magnetic differentiable functions.” –Berlin, etc. Springer- data inversion for 3D topography of the Mono Verlag (1985). discontinuity in the northern Red Sea area.” Journal 23. M. Avriel. Nonlinear Programming: Analysis of Geodynamics 47, no. 5 (2009): 237–245. and Methods. – Dover Publishing (2003). 15. P.I. Balk, A. Yeske “Fitting approach of V.N. 24. L.A. Gladkov, V.V. Kureichik, V.M. Strakhov to solving inverse problems of gravity Kureichik. “Genetic Algorithms: Study Guide.” exploration: up-to-date status and real abilities.” Fizmatlit (2006). Geophysical journal 35, no. 1 (2013): 12–26.

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 29 (2019) International Journal of Mathematics and Physics 10, №1, 36 (2019)

IRSTI 27.35.43

1S.I. Kabanikhin, 2M.A. Bektemessov, 3O.I. Krivorotko, 4Zh.M. Bektemessov

1Institute of Computational Mathematics and Mathematical Geophysics of SB RAS, Novosibirsk, Russian, e-mail: [email protected] 2Abai Kazakh National Pedagogical University, Almaty, Kazakhstan, e-mail: [email protected] 3Institute of Computational Mathematics and Mathematical Geophysics of SB RAS, Novosibirsk, Russian, e-mail: [email protected] 4al-Farabi Kazakh National University, Almaty, Kazakhstan, e-mail: [email protected]

Determination of the coefficients of nonlinear ordinary differential equations systems using additional statistical information

Abstract. The study of the clinical and epidemiological features of tuberculosis combined with HIV infection (TB + HIV) is one of the priorities in the prevention of infectious diseases and is necessary to improve the quality of medical care for patients. So this article is devoted to the mathematical model of epidemiology, its’ investigation and analysis. The previous works showed what identifiability analysis is and considered methods of performing them, such as orthogonal method, eigenvalue method and etc., for more precise clarification of model parameters. However, the choice of solving the inverse problem to restore unknown parameters is playing a huge role. So here was showed the combination of two numerical algorithms, as stochastic method of simulating annealing to determine the region of the global minimum and gradient method to determine the inverse problem in a region, of solving the inverse problem that will help to create effective treatment plan for the elimination and treatment of the disease. Key words: epidemiology, inverse problems, ODE, optimization.

Introduction response, as well as determining the optimal treatment, is mathematical modeling. Systems of nonlinear ordinary differential According to the characteristics of the immune equations (ODE) describe processes in biology and response, it is already possible to numerically medicine, namely, immunology, epidemiology, analyze the optimal control programs for treating a pharmacokinetics, sociology, economics and etc. disease. Example of mathematical model of The equations are built on the basis of the law of epidemiology (co-infection of HIV and mass balance and operate in a closed system. The tuberculosis) shows studies on the identifiability of coefficients of ODE systems characterize important mathematical models for ODE systems, stability of parameters of the immune response, the spread rate inverse problems and methods for their numerical of the disease in the region, the absorption rate of solution and computational optimization, which are drugs, etc., which cannot be determined from necessary to develop an algorithm for regularizing statistical data and need to be clarified. Specified the solution of inverse problems. individual parameters will allow you to create the In [1] the deterministic model of TB dynamics most effective treatment plan and action plan for the was observed, they also conducted identifiability elimination and treatment of the disease. One way to analysis by constructing sensitivity matrix to restore identify the extent of damage to the immune system, the identifiable parameters. The identification of namely the parameters of the disease, the immune parameters was conducted by solving the linear least

squares problem using the QR factorization with leads to tuberculosis. This happens if an infected column pivoting. person does not have sufficiently strong immunity to protect against bacteria. A person with active Model tuberculosis feels sick; he may have the following symptoms: cough for several weeks, chest pain, The cause of tuberculosis (TB) is a bacterium blood or sputum when coughing, weakness, fatigue, called Mycobacterium tuberculosis, which usually weight loss, lack of appetite, chills, fever and night affects the lungs, and TB is spread by airborne sweats. droplets (coughing, sneezing, etc.). People living Co-infection of TB and HIV is a situation where with HIV (PLHIV) are at much greater risk of a person lives with HIV and latent or active TB at contracting TB than HIV-negative people. If TB is the same time. Worldwide, TB is the leading cause not treated properly, death is possible. Tuberculosis of death among PLHIV, as it accounts for 25% of is one of the leading causes of death among PLHIV all deaths among PLHIV. Given the detrimental in the world. It can manifest itself in two ways: effects of HIV infection on the immune system, latently infected with TB and active way of TB. PLHIV with TB co-infection are 20 times more Latently infected with TB means that not all likely to develop active TB [2]. In addition, it has people infected with the bacteria become ill with been proven that tuberculosis increases viral TB. When a person is infected with TB, but has no replication in PLHIV and accelerates the symptoms and does not feel sick, it is considered progression of HIV, being unhealed. that he has a “latent infection of TB”. Such a person A mathematical model of the epidemiology of is not infectious and is not able to infect other co-infection with tuberculosis (TB) and HIV is people. In about 5–10% of cases, a latent infection considered [3]:

∗ (� � ��) (� ) �� − �� − �� − ��, � � ∗ � � � � (�� )� (� )� � � � � ��(��) � − �� − (��)�, � �� − (��)�, � ∗ (��) (� ) � �� − �� − �� − ��, (1) � � ∗ � (��) (� ) ∗ �� −�� (��) � − (�� )�� , � � ∗ � (��) (� ) ∗ ∗ � �� − (�� )��, � ∗ � ∗ � � � � � � � � �� − (� �� )� , � �� − (� � �)�, � � � � � � �(0) �� ,�(0) �� ,�(0) ��,�(0) ��, � ��(0) ��,��(0) ��,��(0) ��,�(0) ��. Here S(t) – number of non-infected individuals, number of individuals with AIDS. L(t) – number of individuals latently infected with total population, – TB (without HIV), I(t) – number of individuals with �� - active population, active TB (without HIV), T(t) – number of �� – people infected with HIV [4]. ∗ individuals cured of TB (without HIV), J1(t) – the ��This model� �� contains many parameters,� �� 6�� of� number of individuals infected with HIV (without which are individual in each specific case and need TB), J2(t) – the number of individuals infected with to be clarified . The HIV and latently infected with TB, J3(t) – number of individuals infected with HIV and active TB, A(t) – values of the parameters are∗ given in Table� 1. ��(k, k ,r�,α�,α�,α�)

�

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 36 (2019) 38 Determination of the coefficients of nonlinear ordinary differential equations systems using ...

Table 1 – Definitions of parameters used in the model (1)

Parameter Definition Units Value the rate of development of active TB (without HIV) year 0,05 the rate of development of active TB (with HIV) year 0,25 � ∗ TB treatment rate (without HIV) human/ year 1 � group HIV transition rate J1(t) year 0,1 �� group HIV transition rate J2(t) year 0,2 �� group HIV transition rate J3(t) year 0,5 �� Let additional information about system (1) be eigenvalues [7-8], we will determine only 4 known at time points of only three groups of identifiable parameters from individuals: additional statistical information. � �� In the case of matrices� �of (large�, ��,� dimensions�,��) or non-uniformly filled matrices, the eigenvalue �(��) ��,��(��) ��, method is unstable [9]. In such cases, it is recommended to use the method based on singular The mathematical�(��) �� model�,� ��,�,� of co-infection of numbers [10], since it is known [11] that the tuberculosis and HIV consists of 8 equations, but condition number of the sensitivity matrix measurement data are known only about 3 of them determines the convention (incorrectness) of system once a year during the 5 years. That is, we have M = (2). The greater the condition number, the higher the 3, K = 5. This model contains many parameters, 6 of incorrectness of the inverse problem (for the which are individual in each specific case and need inversion of the sensitivity matrix, the use of to be refined. Based on the analysis of identifiability regularization methods is required). carried out by one of the methods such as the The values of the parameters, depending on the orthogonal method [5], condition numbers, etc. [6], population, and the initial data are known are given more precisely, in this work, by the method of in Table 2.

Table 2– Values of parameters and known initial data used in the model (1)

Parameter Value Unit S(0) 430 in thousands of people (0) 3854.5 in thousands of people (0) 16.875 in thousands of people � (0) 3.412 in thousands of people � 1(0) 3.2757 in thousands of people � 2(0) 27.7 in thousands of people � 3(0) 1.4 in thousands of people � (0) 0.357 in thousands of people � Λ 43 in thousands of people � 4315.76 in thousands of people

0.0143 in thousands of people � * 0.529 Nondimensionalized � * 0.222 Nondimensionalized � � 0.997 Nondimensionalized � * 0.99927 Nondimensionalized � 0.06685 Nondimensionalized � � �

International Journal of Mathematics and Physics 10, №1, 36 (2019) Int. j. math. phys. (Online) S.I. Kabanikhin et al. 39

In vector form, the inverse problem is written as transitions of individual atoms from one cell to follows: another are still permissible. It is assumed that the process proceeds at a gradually decreasing temperature. The transition of an atom from one cell (2) Ф to another occurs with a certain probability, and the �� =�(�(�),�),�(0) =��, � probability decreases with decreasing temperature. � � � � Here is the� (�vector-function) = . – A stable crystal lattice corresponds to the minimum , data – Ф . energy of the atoms, so the atom either enters a state with a lower energy level or remains in place. The inverse problem was reduced� to the problem�= (�,�,�,�,�of minimizing�,�� the,��, objective �) functional� = (�� ,�in�� the,� form:�) By simulating such a process, one finds a point or a set of points at which the minimum of some numerical function is reached, where Ф (3) . The solution is figured out by � � � � �� sequential calculation�(�) of points in space�= ; �(�) = ∑∑��(� ��) − � � � � Due to the fact that the task of minimizing the each(� ,� point,,…,� starting)�� with , “pretends” to better the objective functional is a multiparameter, then the � � solution than the previous ones.� ,� Algorithm, …, takes� determination of the global minimum requires the � point as the raw data. �At each step, the algorithm use of combined numerical methods – method of (which is described below) calculates a new point simulated annealing and gradient method. � and lowers� the value of the “temperature” value

(initially positive). The algorithm stops when it Methods reaches a point that turns out to be at a temperature Simulated annealing – general algorithmic of zero. method for solving the global optimization problem, According to the algorithm the point is obtained on the basis of the current point as especially discrete and combinatorial optimization. �� follows. The operator that randomly modifies� the Is the one examples of Monte Carlo methods. The � algorithm is created by N. Metropolis and it is based point is applied to point , in result we obtain� new point . � on the imitation of the physical process that occurs � during the crystallization of a substance, including The ∗ point becomes�� a point with the during annealing of metals. It is assumed that atoms probability�� ∗ , which is calculated �� are already lined up in the crystal lattice, but according to the�� Gibbs∗ distribution: � �� (� ,� )

∗ �, �(� ) −�(��) <0 ∗ ∗ �(� ��|��) = � �(� ) −�(��) ∗ �� − �,�(� ) −�(��) ≥0 �� Here – elements of arbitrary decreasing, Gradient descent – method for finding a local converging to zero positive sequence, which sets the extremum (minimum or maximum) of a function analogue� of� >0 the falling temperature in the crystal. using motion along a gradient. To minimize the The rate of decrease and the law of decrease can be function in the direction of the gradient, one- set at the request of the creator of the algorithm. dimensional optimization methods are used, for The simulated annealing algorithm is similar to a example, the golden section method. You can also gradient descent, but due to the randomness of the look for not the best point in the direction of the choice of an intermediate point, it should fall into gradient, but some better than the current one. local minima less often than gradient descent. The The easiest to implement of all local simulated annealing algorithm does not guarantee optimization methods. It has rather weak finding the minimum of the function; however, with convergence conditions, but at the same time the the correct policy of generating a random point in rate of convergence is rather small (linear). The the space , usually, the initial approximation gradient method step is often used as part of other improves. optimization methods. �

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 36 (2019) 40 Determination of the coefficients of nonlinear ordinary differential equations systems using ...

The description of the method is as follows: let 1. Set the initial approximation and accuracy of the objective function is . So the the calculation – ; optimization problem is set as: 2. Calculate , where �(�)�� ,� �� ; � � � 3. Then check� the =�stopping−� condition:��(� ) � = �� −��(��)� �  If , or In case when �(�)maximum � �� is needed�(� ) to be find, then �� (choose� one ��of conditions),� then instead of we use . �� and go−� to ��step 2.�� −�(�)� � � �� The basic idea of method is to go in the direction �� Otherwise�� and end. �(�) � �(�) of the fastest descent, and this direction is given by �=��1The stochastic method of simulated annealing �� anti-gradient determines the region�=� of the global minimum, and to determine the inverse problem in this region, the −�� gradient method was used, consisting in the iterative sequence of determining the solution to the inverse �� =�� −��(��) where set the velocity of gradient descent and can problem: be chosen as ′ � (4)  Constant� (in this case method may diverge),  Decreasing in the process of gradient descent, �� =�� −�� (��),�� Here is the descent parameter ′  Guaranteeing the fastest descent: ��(��) characterizes the method of steepest descent, the To find minimum we obtain �� = �� (��)� gradient of the objective functional has the �(�) following form [10]: � �� = �� (� ) 1. To find maximum we obtain = �� −��(��)� ′ (5) �(�) � � � � � � � �� Here the� ( �vector-function) =−� �(� ) � (� is( �the), �)�� solution of �And= ��the algorithm�� of the� gradient method is look = �� −��(��)� the conjugate problem: as follows: �(�)

� � (6) � = −�� (�(� �),�)�(�), � � ⋃��,��, �� = 0, �� = �, Ф � �(�) = 0, � �� = ��(� ��) − �,� =1,�.,�. Jacobi matrices have the following form: Table 3 – Solution of the inverse problem for the mathematical model of co-infection of tuberculosis and HIV (7) �� Parameter Relative decision error q � �� ,��, Results� =� ��,��,�� ,� =� � ��,��

�� The result of solving the inverse problem for a �� 1.0 ∗ 10 �� mathematical model of co-infection with �� 5.7 ∗ 10 tuberculosis and HIV using a combined method of �� 5.3 ∗ 10 simulating annealing with a gradient method is �� presented in Table 3 [13-14]. Note that� for each parameter, the4.1 relative ∗ 10 error is Thus, effective numerical algorithms for solving less than 0.001%, and the parameters are inverse problems for systems of ODE (for problems restored better than the others, as identifiability of epidemiology, pharmacokinetics and immune- analysis showed. The result of solving�� the�� direct� logy), based on a combination of stochastic and problem (1) with the found parameters is shown in gradient methods, have been created. Figure 1a-1c for the three measured functions. Red

International Journal of Mathematics and Physics 10, №1, 36 (2019) Int. j. math. phys. (Online) S.I. Kabanikhin et al. 41 dots describe statistical data that were not involved in inverse problems. It is shown that the obtained solving the inverse problem, but are presented for solution (solid line) is in good agreement with the comparison of the forecast using methods for solving real data and serves as a reliable forecast [15].

Figure 1a. – Numerical solution of the problem of the Figure 1b. – Numerical solution of the problem of the spread of co-infection of tuberculosis and HIV with spread of co-infection of tuberculosis and HIV with specified parameters (solid line). Black dots mean data of specified parameters (solid line). Black dots mean data of the inverse problem, red dots – statistical data taken into the inverse problem, red dots – statistical data taken into account for the prediction. The number of individuals with account for the prediction. The number of individuals active TB (without HIV). All values are in thousand. infected with HIV and the active form of TB. All values are in thousand.

Figure 1c. – Numerical solution of the problem of the spread of co-infection of tuberculosis and HIV with specified parameters (solid line). Black dots mean data of the inverse problem, red dots – statistical data taken into account for the prediction. The number of individuals with AIDS. All values are in thousand.

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 36 (2019) 42 Determination of the coefficients of nonlinear ordinary differential equations systems using ...

Conclusion 5. Vajda S, Rabitz H, Walter E, Lecourtier Y. “Qualitative and quantitative identifiability analysis The research and analysis of the problem arising of nonlinear chemical kinetiv-models.” Chemical in bio-medicine has been carried out, the theoretical Engineering Communications vol. 83 (1989): 191– aspects of this task have been built, including 219. identifiability analysis, which is an important step in 6. Miao H., Xia X., Perelson A.S., Wu H. “On the study of the mathematical model, and is Identifiability of nonlinear ODE models and necessary for the correct solution of the inverse applications in viral dynamics.” SIAM Rev. Soc. problem, since it shows the uniqueness, existence Ind. Appl. Math. vol. 53 (1) (2011): 3–39. and / or stability of the solution . 7. Yao K.Z., Shaw B.M., Kou B., McAuley New combined numerical algorithms for solving K.B., Bacon D.W. “Modeling ethylene/butene the direct and inverse problems of epidemiology copolymerization with multi-site catalysts: have been built. Efficient numerical algorithms for Parameter estimability and experimental design.” solving inverse problems for systems of ODE (for Polymer Reaction Engineering vol. 11 (2003): 563– problems of epidemiology, pharmacokinetics and 588. immunology), based on a combination of statistical 8. Quaiser T., Monnigmann M. “Systematic and gradient methods, have been created. identifiability testing for unambiguous mechanistic Numerical algorithms for solving the problems modeling – application to JAK-STAT, MAPkinase, of determining the coefficients of nonlinear ODE and NF-kB signaling pathway models.” BMC Syst systems using additional statistical information were Biol. no.3 (2009): 50–71. developed and analyzed. 9. Годунов С.К. Лекции по современным Thus, the conducted scientific work opens up аспектам линейной алгебры. – Новосибирск: new directions for the development of research in Научная книга (2002): 216. science and technology, namely, the refinement of 10. Kabanikhin S.I., Krivorotko O.I., Ermolenko mathematical models will improve the prognosis of D.V., Kashtanova V.N., Latyshenko V.A. “Inverse the disease or the development of the epidemic, problems of immunology and epidemiology.” which would entail the need for a plan of measures Eurasian Journal of Mathematical and Computer for treating patients and eliminating the Applications. vol.5 (2017): 14–35. consequences of the disease / epidemic. 11. КабанихинС.И. Обратные и некорректные This work was supported by the grant of the задачи. Новосибирск: Сибирское научное Committee of Science of the Ministry of Education издательство (2009): 460. and Science of the Republic of Kazakhstan 12. Kabanikhin S.I., Krivorotko O.I. (AP05134121 "Numerical methods of identiability “Identification of biological models described by of inverse and ill-posed problems of natural systems of nonlinear differential equations.” Journal science". of Inverse and Ill-Posed Problems vol.23(5) (2015): 519-527. References 13. Kabanikhin S., Krivorotko O., Kashtanova V. “A combined numerical algorithm for 1. Moualeu-Ngangue D.P., Röblitz S., Ehrig R., reconstructing the mathematical model for Deuflhard P. “Parameter identification in a tuberculosis transmission with control programs.” tuberculosis model for Cameroon.” PloS one. 10(4) Journal of Inverse and Ill-Posed Problems vol. 26(1) (2015) e0120607. doi:10.1371/journal.pone.0120607. (2018): 121-131. 2. Pawlowski A., Jansson M., Sköld M., 14. Kabanikhin S.I., Krivorotko O.I., Yermolenko Rottenberg M.E., Källenius G. “Tuberculosis and D.V., Banks H.T. “A numerical algorithm for HIV Co-Infection.” PLoS Pathog. 8(2) (2012) constructing a mathematical model of intracellular e1002464. doi:10.1371/journal.ppat.1002464. dynamics for individual HIV treatment.” Journal of 3. Adams B.M., Banks H.T., Davidian M., Inverse and Ill-Posed Problems vol. 26(6) (2018): Kwon Hee-Dae, Tran H.T., Wynne S.N., Rosenberg 859–873. E.S. “HIV dynamics: Modeling, data analysis, and 15. Miao H, Dykes C, Demeter LM, Perelson AS, optimal treatment protocols.” Journal of Wu H. “Modeling and estimation of kinetic Computational and Applied Mathematics vol. 184 parameters and replication fitness of HIV-1 from (2005): 10–49. Flow-cytometry-based growth competition 4. Perelson A.S., Nelson P.W. “Mathematical experiments.” Bull Math Biol. no.70 (2008): 1749– Analysis of HIV-I: Dynamics in Vivo.” SIAM Rev 1771. vol. 41, no.1 (1999): 3-44.

International Journal of Mathematics and Physics 10, №1, 36 (2019) Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 43 (2019)

IRSTI 27.41.41

1B.T. Zhumagulov, 1,2N.M. Temirbekov, 3Zh.R. Zhaksylykova

1National Engineering Academy, Almaty, Kazakhstan 2Kazakhstan engineering and technological University, Almaty, Kazakhstan email: [email protected] 3Kazakh National Pedagogical University named after Abai, Almaty, Kazakhstan email: [email protected]

Variational method for approximate solution of the Dirichlet problem

Abstract. Several numerical methods can be used to approximate the solution of the problem. In order to determine the most effective of them, it is necessary to carefully study each method. The most efficient approximation method is characterized by properties such as high accuracy of the solution, fewer iterations and parameters in the calculation, calculation speed, etc. In this paper we consider the Dirichlet problem for the Poisson equation described by the initial-boundary value problem for the elliptic type of the second order. As an effective iterative method for its approximate solution, variational methods for constructing difference equations and variational methods for constructing iterative algorithms were used. The article presents the results of calculations developed using the variational method for the selected model problem. Examples of calculations for model problems are given. The results of the computational experiment demonstrate the high efficiency of the proposed iterative method. Key words: Dirichlet problem, difference scheme, Ritz method, conjugate gradient method.

Introduction methods used to solve systems of linear algebraic equations. Finding in an analytical form of the problems In solving the problem, the Ritz method [1-2] solution of mathematical physics is fraught with was used to construct the difference equations, considerable mathematical difficulties. Known iterative algorithms were constructed by the method results apply only to the simplest cases. In other of conjugate gradients [3-4]. Comparing the results cases, are used different numerical methods of the obtained by the variational method with the results approximate solution. obtained in the literature [4-7], it was found that the In this paper, is consider an elliptic differential advantage of the chosen variational method is the equation. At the solution: simplicity and efficiency of memory use. Such 1) To construct difference equations was used advantages of this method will certainly be a variance method, proposed in 1908 by German acceptable when solving large-scale problems. mathematician V. Ritz, which is called Ritz method. The solution found by this method un(x), under Variational methods for constructing certain conditions, tends to exact solution u(x), difference equations. when n   . Questions of convergence of solutions obtained Consideration of the problem in general shape in by the Ritz method are considered in numerous the operator form papers and monographs. 2) To construct iterative algorithms, was used Lu  f , uL (1) the method of conjugate gradients, which stands out for its efficiency among the known iterative

when L – is an operator domain L D x1,x2 :0  x1,x2 1 is a square. This area This task is equivalent to the corresponding is covered with a uniform square grid with a pitch variational problem. 1 h  and then divide each of the squares D N 1 k,l J u  min J v , (2) vL with diagonal. All the internal vertices of the N triangles are numbered throughрk,l k,l1 , the union when Jv  Lv,v  2 f ,v . of all triangles with a point of its vertex рk ,l mark Consideration of the Ritz method use in solving h an elliptic differential equation of the form through Dk,l . h Dk,l can be represented as a union of six 2      u  6  Aij (x)  f x , in D (3) h      triangles Dk ,l,m  , the order of numbering is i, j1 xi  x j  m1 indicated in fig.1 with boundary condition

 u D 0 (4) where D is a bounded domain with a piecewise linear boundary D , Aij x A ji x bounded / functions and for an arbitrary vector   1 , 2 is done inequality

2 2 2 n 2 2 0 i inf Aij x i j  sup Aij x i j 1 i  xD  i1 i,1j1 xD i, j1 i h 6 Figure 1 – Dk ,l,m   triangle designation with positive constants    . m 1 0 1 It can be shown that the operator of problems (3) and (4) is symmetric and positive definite, and the To each internal node (x , y ) , k, l  1, N of problem itself reduces to finding a function that k l  1 the grids is assigned a piecewise linear basic minimizes in space W 2 D a quadratic functional function k,l x, y . We define each of these

functions k,l x, y for the entered grid.  2 u u  J (u)   A (x) dD  2 uf dD (5) To set  x, y analytical, enough for each    ij     k,l D  i, j1 xi x j  D   triangle entering the carrier k,l x, y , to create an To find an approximate solution of problem (5), equation for the plane passing through the unit in is applied the Ritz method with special-type Fh рk,l , and in the other two of its vertices through subspaces that satisfy condition (4). zero. Calculating basic functions k,l x, y in each After, we construct Fh subspaces. To simplify 6 of triangles D h we build a system of basic the presentation, it is illustrated by the example of  k ,l,m m1 piecewise linear approximations, when domain functions

International Journal of Mathematics and Physics 10, №1, 43 (2019) Int. j. math. phys. (Online) B.T. Zhumagulov et al. 45

 1 1 1 x  x  y  y , x, y  Dh ,  h k h l k ,l,1  1 1 x  x , x, y  Dh ,  h k k ,l,2  1 1 y  y , x, y  Dh ,  h l k ,l,3  x, y   k ,l 1 1 (6) 1 x  x  y  y , x, y  Dh ,  h k h l k,l,4  1 1 x  x , x, y  Dh ,  h k k ,l,5  1 1 y  y , x, y  Dh .  h l k,l,6

To build an approximate u h (x) solution of the We consider Dirichlet problem in the area D  problem (1) and (2) we apply Ritz method using with border D with variable coefficients p(x, y) N and q(x, y) basis  x k ,l k ,l1  u  u N  p(x, y)  q(x, y)  f x, y , in D (11) h x x y y u (x)   k,lk,l x (7) k,l1 with boundary condition As a result, comes a system of linear equations u  0 (12) A  g (8) D

The equation (10) is multiplied by the function      T where 1 , 2 ,..., N 2 - vector, made up from u(x, y) and integrated by D in parts, given the N decomposition coefficients     , boundary condition (12) we obtain N k 1 l k ,l k ,l1 T 2 2 g  g , g ,..., g 2 - vector with components 1 2 N  u   u       p(x, y)  dD q(x, y)  dD  f x, y udD DD x   y  D     g N k1 l gk,l  f k,l (x)dD, k,l 1, N (9) Dk ,l According to (5) we construct the functional

2 2 and the elements of matrix A are calculated by the  u   u  formulas Ju  p(x, y)  dD q(x, y)  dD 2 f x, y udD  x   y   DD D ij,   (13) aakl, Nk 1 lNi , ( 1) j

2   (10)  kl, ij,  To apply the Ritz method in (13) the function   Ast () x dD, k , l , i , j 1, N D st,1 xxst u(x, y) replacing by decomposition (7) we get

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 43 (2019) 46 Variational method for approximate solution of the Dirichlet problem

2 2   N    N  J u h   p(x, y)   x  dD  q(x, y)   x  dD  k,l  x  k,l k,l   y  k,l k,l  DD k,l1   k,l1  N     2 f x, y  k,l k,l x dD, k,l 1,..., N D k,l1

Next, the derivatives are found and equating to zero, we get the following equations

h N Ju kl,     xx       kl,,kl   p(, x y )  kl,  dD   xx kl, D  kl,1   N   xx       kl,,kl      q(,) x y   kl,, dD fxy ,  kl xdD 0, kl , 1,..., N . DD kl,1 yy 

Taking  k ,l out from bracket, it is put in the form of

Ju()h   NN  xx        xx        p(,) x y  kl,, kl dD q(,) x y  kl,, kl dD               kl, kl,  DD kl,1 xx  kl,1 yy 

     f x, y kl,  x dD0, k , l 1, N . D

Thus, according to (9) and (10) introducing the notation

 N   x    x ij,   kl, ij,  akl, aN k1 lNi , ( 1) j  p(, x y )   dD D  kl,1 xx  (14) N  kl,  x  ij,  x  q(,) x y    dD, k , l , i , j 1, N , D  kl,1 yy 

k,l k,l1 k1,l k1,l1 ak,l , ak,l , ak,l , ak,l , k , l  1, N . we get a system of linear algebraic equations (8).

Given the symmetry and block tridiagonal of k,l matrix A , it is enough to define I. Accroding (13), where i, j  k,l we find ak,l

   22    kl,   kl,, kl   akl,  p(, x y )  q (, x y )  dD h   xy     Dkl,    22  22  kl,,  kl   kl,,  kl    pxy(,)   qxy (,)   dxdy  pxy(,)   qxy (,)   dxdy   xy     xy  DD1       2       (15)  22  22  kl,   kl,  kl,,  kl    pxy(, )   qxy(, )   dxdy  pxy(, )   qxy (, )   dxdy   x y    xy  DD3       4        22  22  kl,,  kl   kl,,  kl    pxy(,)   qxy (,)   dxdy  pxy(,)   qxy (,)  dxdy.   xy     xy  D5       D6      

International Journal of Mathematics and Physics 10, №1, 43 (2019) Int. j. math. phys. (Online) B.T. Zhumagulov et al. 47

We calculate the integrals in each of the areas h k ,l ~ ~ ~ ~ Dk,l , taking into consideration form of basis ak ,l  P 1  Q 1  P 1  Q 1 . (18) k ,l k ,l k ,l k ,l 2 2 2 2 function  x, y in considered area. k,l  Further, all found six values are substituted in II. To find element ak,l 1 as in the first case we (15). In this case, we combine integrals with the k ,l same values and use the notation use the formula (14), where i, j  k,l 1 and to

~ 1 define the place pk,l1 we down the vertex pk,l P 1  2 px, y dxdy (16) k ,l  h h h lower for one pitch. According to fig.1. it is seen 2 Dk ,l Dk 1,l that the vertexes pk,l and pk,l1 are in triangle D1 ~ 1 k,l1 and D6 . So, it means, ak ,l are defined only in this Q 1  2 qx, y dxdy (17) k,l  h h h 2 Dk ,l Dk ,l1 area.

It comes,

  kl, kl ,1 kl, kl ,1  akl,1  p(, x y )  q(, x y ) dxdy  kl,  xx  yy DD16  

 kl, kl ,1 kl, kl ,1    p(, x y )  q(, x y ) dxdy  (19)  xx  yy D1  

 kl, kl ,1 kl, kl ,1    p(, x y )  q(, x y ) dxdy  xx  yy D6  

k1,l III. To find ak ,l by formula (14), where The found values of the integrals, substituting in (19) and using the notation (17) we get i, j  k  1, l we move vertex pk,l for one pitch to

left and from fig.1. it is seen that p and p are 11 k,l k1,l kl,1     akl, 22 q(, x y ) dxdy q (, x y ) dxdy hh the vertexes of triangles D1 and D2 . It means, DD16 (20) k1,l 1  ak ,l is defined in the triangles of D1 and D2 .   2  q(, x y ) dxdy  Q 1 h  kl,  DD16 2

  klkl,  1, klkl,  1,  akl1,   p(, x y )  q(, x y ) dxdy  kl,  xx  yy DD12  

 kl, k 1, l kl, k 1, l    p(, x y )  q(, x y ) dxdy  (21)  xx  yy D1  

 kl, k 1, l kl, k 1, l    p(, x y )  q(, x y ) dxdy.  xx  yy D2  

k1,l IV. It remains only to identify the element, Found values ak,l , in the triangles D1 and D2 k1,l1 substitute in (21) with (16) and get ak,l . For that the vertex pk,l , needs to be moved firstly for one pitch to left and then for one pitch up. 11 akl1,    pxydxdy(, )   pxydxdy(, )  According to fig.1. it is seen that pk,l and pk1,l1 kl, hh22  DD12 (22) are the vertexes of the triangles D2 and D3 . But 1    2  p(, x y ) dxdy  P 1 h  kl , DD12 2

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 43 (2019) 48 Variational method for approximate solution of the Dirichlet problem

and norm with the selected shift of pk,l the vertex pk1,l1 would not belong to any of six triangles. Therefore u  u,u . a k1,l1  0 (23) k,l   To solve an equation of the form A g we All found values (18), (20), (22), (23) use the conjugate gradient method. The iteration substituting in (8), taking into account the symmetry process is implemented in the following order: of the matrix A we get I. Preparation before the iterative process 0 For given  i, j the residual is calculated        0 0 0  PQ1 1 PQ 1 1kl , Q 1 kl ,1  si, j  ri, j  A i, j  gi, j i  1,..., n1 , j  1,..., n2 ; k ,, l kl k ,, l kl kl,  2 22 2 2 (24)    II. k – integration of the method Q1 kl , 1 P 1 k 1, l  P 1 k 1, l; kl , 1, N k1 k 1 kl, k ,, l k l 22 2 ri, j , ri, j  1) Calculate the parameter:   , k k k 1 Asi, j , si, j Allowing in (24) k k1 k 2) ri, j  ri, j  k Asi, j      3) The following approximation of the solution  A1    P1 k 1, l   PP1  1 kl ,   kl, kl ,kl ,, kl k k 1 k 2 22 (25) is calculated by the formula: i, j  i, j  k si, j  k k  P1 kl 1, ; kl , 1, N kl , ri, j ,ri, j  2   4) Calculate the parameter: k k1 k1 ri, j ,ri, j      5) The auxiliary value is calculated by the  A2    Q1kl ,1   QQ 1 1kl , k1 k k kl, kl, kl ,, kl    2 22 (26) formula: si, j ri, j n si, j  Q1kl ,1 ; kl , 1, N where kl,  2   k 1    As  2  P1 si 1, j s ij ,   P1 sij , s i 1, j  Then, using A  A  А, we get system (8) ij, ij ,,ij 1 2 h1  22

1     Variational methods for constructing  2 Q1 sij ,1  s ij ,  Q 1 ssij , ij ,1 ; i ,, j ij iterative algorithms h2  22 i, j 1, N For numerical solution of system (6), one can apply variational type methods, such as, the method of rapid descent, the method of minimal corrections, \This process continues until the criterion for k 1 k the method of conjugate gradients, etc. stopping the iterations     is satisfied. The conjugate gradient method is most preferable for systems with a self-adjoint positive Calculation examples matrix A  A*  0 . With poor conditioning of the matrix, this method does not always become To illustrate the proposed method, we consider computationally stable. an example of the problem (11) – (12) in a circle. Operator A is self-adjoint and positive definite Let in a rectangular area D is a circle  with 0 operator in the space of grid functions H h . Into R radius and center с1 ,с2 . It is required to find an 0 approximate solution of the Dirichlet problem for H we insert scalar product h the Poisson equation.

N 1  2u  2u u,v  ui, jvi, j h1h2      2 2 x1 x2 i, j1 x1 x2

International Journal of Mathematics and Physics 10, №1, 43 (2019) Int. j. math. phys. (Online) B.T. Zhumagulov et al. 49

In the circle, under the condition problem using an explicit difference scheme with 4  *    10 on a uniform grid with a size of 101  ux1, x2  0 × 101. Therefore, the original system of linear algebraic equations has 101 × 101 unknowns. Then

D  x1 , x 2 , a1  x1  a 2 , b1  x 2  b2 ,

2 2 2   x1, x2 , x1  c1  x2  c2   R ,

2 2 2   x1, x2 , x1  c1  x2  c2   R .

Auxiliary problem of the fictitious domains method

   u    u            px1, x2  qx1,x2  x1 x2 , x1  x1  x2  x2   Figure 2 – Solution of the Dirichlet problem for the  u x ,x  0  1 2 D Poisson equation in a circle

 1, x1, x2 , Conclusion px1, x2  q x1, x2   1/, x1, x2  D / . Today, the task of developing and modifying numerical methods remains relevant. However, the   x1  x2 , x1 , x2   f x , x   development process of computing technology 1 2 shifts the emphasis from the creation of new 0,x1 , x2  D / . numerical methods to the study and classification of

old ones in order to identify the best. Now for Were conducted numerical experiments. Below modern powerful computers, such characteristics as is the table 1 for the number of iterations N and the amount of required memory, and the number of    calculation errors uh u C at n1 n2 101 and arithmetic operations are not necessarily in the various meanings  for Ritz method. foreground. More preferred are those methods that are distinguished by the ease of implementation on a computer, and allow to solve a wider class of Table 1 – Results of numerical experiments problems. The special advantages of this method are its simplicity and low memory costs, which makes it n n 1/  N u  u 1 2 h C effective in solving large-scale problems. 101 101 102 1221 0.003888 The results of the computational experiment 101 101 103 3395 0.003820 confirm the efficiency of the proposed method for 101 101 104 7920 0.003819 solving the Dirichlet problem for the Poisson 101 101 105 12737 0.003818 equation and its rather high efficiency. 101 101 106 20042 0.003817 References 101 101 107 26592 0.003816

1. Marchýk G.I. “Metody vychsltelno matematk.” Naýka (1989): 608. The results show that with an increase in the 2. Mýhambetjanov A.T., Otelbaev M.O., number of grid nodes, the error in the solution Smagýlov Sh.S. “Ob odnom novom prbljennom decreases. In fig. 2 shown the result of solving the

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 43 (2019) 50 Variational method for approximate solution of the Dirichlet problem metode reshene nelnenyh kraevyh zadach.” for Elliptic Equations.” International Journal of Vychsltelnye tehnolog 3, no. 4 (1998). Electronics and Telecommunications 60, no. 3 3. Samarskii A.A. “Teora raznostnyh shem.” (2014): 219-223. Naýka (1983): 616. 8. Temirbekov A.N. “Numerical implement- 4. Bakhvalov N.S, Zhidkov N.P, Kobelkov tation of the method of fictitious domains for elliptic G.M. Сhislennye metody – Izd.: Binom (2008). equations.” 3rd International Conference on Analy- 5. N.M. Temrbekov, J.R. Jaksylykova. “Ob sis and Applied Mathematics, ICAAM 1759 (2016): teratsonnom metode fktvnyh oblaste dla 020053-1–020053-6; doi: 10.1063/1.4959667. reshena ellptcheskogo ýravnena v oblastah so 9. Kireev V.I., Panteleev A.V., Chislennye slojno geometre.” Izvesta MKTÝ m. metody v primerakh i zadachakh. Uchebnoe H.A.Iasav, Sera matematka, fzka, nformatka. posobie. 4-e izd., ispr.-SPb.: Izdatel'stvo «Lan'» 1, no. 1(4) (2018): 128-132. (2015): 488. 6. N.M.Temirbekov, Zh.R.Zhaksylykova. 10. Samarskii A. A., Nikolaev E. S. “Metody “AnIterative Method for Solving Non linear Navier- reshenia setochnyh uravnenii.” Naýka (1978): 592. Stokes Equationsin Complex Domains Taking into 11. Bykov V.M., Nizameev Kh.R. Account Boundary Conditions with Uniform Variatsionnye metody v mekhanike sploshnykh Accuracy.” AIP Conference Proceedings 1997, sred: Metodicheskie ukazaniya po spetskursu. 020036 (2018); doi: 10.1063/1.5049030. Chelyabinsk: Chelyabinskiy gos. un-t (2001): 36. 7. Temirbekov A.N., Wójcik W. “Numerical 12. Berezin I.S., Zhidkov N.P. “Metody Implementation of the Fictitious Domain Method vychisleniy.” GIFML 2 (1959): 620.

International Journal of Mathematics and Physics 10, №1, 43 (2019) Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 51 (2019)

IRSTI 27.31.21

1,2N.M. Koshkarbayev, 1,2,3B.T. Torebek

1Al-Farabi Kazakh National University, Almaty, Kazakhstan 2Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan 3Department of Mathematics, Ghent University002C Ghent, Belgium e-mail: [email protected]

Nonexistence of travelling wave solution of the Korteweg-de Vries Benjamin Bona Mahony equation

Abstract. This paper is devoted to the Korteweg-de Vries Benjamin Bona Mahony equation in an infinite domain. The paper discusses weak solutions of the Korteweg-de Vries Benjamin Bona Mahony equation without any conditions at infinity. This particular problem arises from the phenomenon of long breaking wave with small amplitude in fluid. In fluid dynamics, a breaking wave is a wave whose amplitude reaches a critical level at which some process can suddenly start to occur that causes large amounts of wave energy to be transformed into turbulent kinetic energy. For the Korteweg-de Vries Benjamin Bona Mahony equation, we obtain the conditions of blowing-up of travelling wave solutions in finite time. Moreover, there is an explicit upper bound estimate for the wavelength of the corresponding singular traveling wave, depending on the speed of waves. The proof of the results is based on the nonlinear capacity method. In closing, we provide the numerical examples. Key words: Breaking waves, Korteweg-de Vries-Benjamin-Bona-Mahony equation; blow-up of solution, travelling wave solution.

1. Introduction

1.1. Breaking waves

In fluid dynamics, a breaking wave is a wave whose amplitude reaches a critical level at which some process can suddenly start to occur that causes large amounts of wave energy to be transformed into turbulent kinetic energy. At this point, simple physical models that describe wave dynamics often become invalid, particularly those that assume linear behavior.

Breaking of water surface waves may occur Figure 4 – Types of breaking water waves anywhere that the amplitude is sufficient, including in mid-ocean. However, it is particularly common on beaches because wave heights are amplified in 1.2. Mathematical model the region of shallower water (because the group velocity is lower there). There are three basic types In this section we present the well-known of breaking water waves [1]. They are spilling, mathematical model of the Korteweg de Vries- plunging and surging: Benjamin-Bona-Mahony equation (see. [2]).

One of the well-known non-linear equations that embody both variance and non-linearity and is actively used in applications is the Korteweg-de The last scaled�� + ��KdV� +� equation�� = 0.can be further Vries equation [3] which models the undirectional generalized by using the low-order asymptotic propagation of weakly nonlinear and weakly relations in order to alternate higher order terms as it dispersive waves: was proposed by Bona and Smith [6]. This step is rather standard and we do not provide here the (1) details of the derivation [7]: � �� +��1+ � �� + � �� = 0, (2) where is the vertical excursion of the free surface above the still water level, is the uniform where �� +The �� equation+�� − ��(2)�� is= 0,so-called � undisturbed water depth and is the speed Korteweg-de Vries-Benjamin-Bona-Mahony ℎ of linear gravity waves ( being the gravity equation.� � �. acceleration). �=��ℎ We note that for a particular value of the Stokes- The Benjamin-Bona-Mahony� equation is an Ursell number another simpler scaling is alternative to the Korteweg-de Vries equation [4] possible when all the lengths ( and ) are scaled by which is described as follows: the mean water depth�=1 � � 1.3. Statement of ℎ.the problem

� 3ℎ �ℎ We consider one of the mathematical problem of �� +��1+ �� − �� = 0. We consider the 2following scaled6 dependent and the breaking water waves, the Korteweg-de Vries- independent variables: Benjamin-Bona-Mahony equation:

� � �� + �� +�� −�� −�� = �� , �� , �� , (3) where is the� �characteristic� wave amplitude� and is the characteristic wavelength. In dimensionless The Korteweg-de=0,��0,��. Vries-Benjamin-Bona-Mahony variables�� KdV equation (1) reads: � equation has important application in different physical situations such as waves on shallow water, and processes in semiconductors with differential � conductivity. 3� � In [8], traveling-wave solutions �� + �1+ �� + �� = 0, where parameter 2 measures6 the nonlinearity are sought for the equation (3) which �� describes wave the processes in semiconductors�(�, �)�= and is the dispersion parameter. The relative �= � with��(�� strong ��) spatial dispersion. In [8-12] the authors importance� of these two effects is measured by the obtained sufficient conditions for the finite time so-called�= �Stokes-Ursell number [5]: blow-up of solutions of time and space initial problems for the Korteweg-de Vries and Benjamin– Bona–Mahony type equations. � In this paper, based on the method of nonlinear � �� capacity [13-15], the existence of singular travelling �= � ≡ � . The last equation can� be furtherℎ simplified if we wave solutions of the equation (3) is proved. perform an additional change of variables: 2. Singular travelling wave solutions

We consider the traveling wave type solutions � 3� √6 √6 of the Korteweg-de Vries-Benjamin Bona Mahony �� , � � (�−�), �� , equation (3): �� which yields the following simple equation including explicitly the Stokes-Ursell number : �(�, �) =�(�), International Journal of Mathematics and Physics 10,�� №1, 51 (2019) Int. j. math. phys. (Online) N.M. Koshkarbayev, B.T. Torebek 53

where and is the wave velocity. Then satisfies � � �� = � − �� |� � (�) − � (�)| ( ) �=�� , � � (4) and � (�) �� Equation(1+� (4)) admits� + ��the −(1+�)�following integrals:= 0. �

�=� ��(�)�� . (5) �� Then, if the inequality (7) implies �� where (is1+� an arbitrary)� + � constant.−(1+�)�+�=0, �=�+1

� � 2.1. Nonexistence of travelling wave solution (1+�)(2 + �) � ≥ 2��. From this it directly follows� that if there exist A weak solution of (5) is a function such that the inequality (7) holds, then there is no that satisfies the integral identity such bounded travelling wave solution of equation� � �� (5). � (�),� �� Then the following results are true

� Theorem 1. The equation (4) with support � � (�)�(�)�� = satisfying the inequality � �≥ �� 2, = −2(1+�) �� (�)�� (�) −�(�)�� − (8) (��)(��)� (6) � �� for does not admit a solution. −2� � �(�)�� We multiply� equation� (5) by a nonnegative test Thus, a su cient condition for the existence function�� (�). with compact support. Then of an unbounded traveling wave with a wavelength after integration �we obtain (6). Hence, by the Young is the fulllmentﬃ of the inequality inequality�� with parameter� (�) , we find that �∗

��0 + � 2� � � (�� ) � �� (�)�(�)�� ≤ � �� (�)�(�)�� with . (�+1)(� + 2) �

� �� 2.2.�� Numerical∗ examples �� (�) −�(�)� +�(1+�) � �� − � �(�) (7) In this subsection we consider some numerical examples for equation (5) with different viscosities. We now take −2�the test�� function:�(�)��. We consider some initial data (at ) for a traveling wave. In this case, we note that the nonexistence of a solution to equation� =(5) �� depends on the conditions (8). where is a free parameter �and the function First, consider an example where the wave �(�) =��(�),� = , velocity is small enough. That is, consider a uid such that � with a velocity between zero and one. Then, as seen �≥2� 0≤�� (�) from the Figure 1, the traveling wave breaks relatively quickly. Let us now study a uid with a velocity between � 1��|�| ≤ 1, � (�) =� 50 and 100. In this case, the time of breaking the 0��|�| ≥ 2. Let the function satisfies the following wave slightly increases. It is easy to see from the properties Figure 2. ��

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 51 (2019) 54 Nonexistence of travelling wave solution of the Korteweg-de Vries Benjamin Bona Mahony equation

Figure 2 – Singular travelling waves

Figure 3 – Breaking travelling waves

Now let the uid velocity be large enough. That thousand. In this case, as seen from the Figure 3 the is, consider a uid with a velocity of about one time of breaks of traveling waves will be quite large.

Figure 4 – Breaking travelling waves

International Journal of Mathematics and Physics 10, №1, 51 (2019) Int. j. math. phys. (Online) N.M. Koshkarbayev, B.T. Torebek 55

Analyzing the above examples, we come to the 6. J. L. Bona, R. Smith, “A model for the two- conclusion that with an increase of the wave way propagation of water waves in a channel,” velocity, the time of wave break-up increases. Math. Proc. Camb. Philos. Soc. 79, (1976): 167-182. Conclusion 7. M. Francius, E. N. Pelinovsky, A. V. Slunyaev. “Wave dynamics in nonlinear media with The present paper is devoted to the Korteweg-de two dispersionless limits for long and short waves.” Vries-Benjamin-Bona-Mahony equation in an Phys. Lett. A. 280:2, (2001): 53-57. infinite interval. This particular problem arises from 8. A.B. Al’shin, M. O. Korpusov, E.V. Yushkov. the phenomenon of long breaking waves with small “Traveling-wave solution to a nonlinear equation in amplitude in fluid. For the Korteweg-de Vries- semiconductors with strong spatial dispersion.” Benjamin-Bona-Mahony equation, we proved the Comput. Math. and Math. Phys., 48, (2008): 764- nonexistence of the singular travelling wave 768. solutions. Moreover, we provide some examples. 9. M.O. Korpusov, E.V. Yushkov. “Local Acknowledgements Solvability and Blow-Up for Benjamin-Bona- Mahony- Burgers, Rosenau-Burgers and Korteweg- The research is financially supported by a grant de Vries-Benjamin-Bona-Mahony Equations.” from the Ministry of Science and Education of the Electronic Journal of Differential Equations 69, Republic of Kazakhstan (Grants №AP05131756). (2014): 1-16. No new data was collected or generated during the 10. S.I. Pokhozhaev. “On the Singular Solutions course of research. of the Korteweg-de Vries Equation.” Mathematical Notes 88:5, (2010): 741-747. References 11. S.I. Pokhozhaev. “On the Nonexistence of Global Solutions for Some Initial-Boundary Value 1. T. Sarpkaya, M. Isaacson. “Mechanics of Problems for the Korteweg-de Vries Equation.” wave forces on o�shore structures. Van Nostrand Differential Equations 47:4, (2011): 488-493. Reinhold.” (1981). 12. S.I. Pokhozhaev. “Blow-Up of Smooth 2. D. Dutykh, E. Pelinovsky. “Numerical simula- Solutions of the Korteweg-de Vries Equation.” tion of a solitonic gas in KdV and KdV-BBM equa- Nonlinear Analysis: Theory, Methods and tions.” Physics Letters A.V. 378, no. 42 (2014): Applications 75:12, (2012): 4688-4698. 3102-3110. 13. S.I. Pokhozhaev. “Essentially nonlinear 3. D. J. Korteweg, G. de Vries, “On the change capacities induced by differential operators.” Dokl. of form of long waves advancing in a rectangular Akad. Nauk 357:5, (1997): 592-594 . canal, and on a new type of long stationary waves.” 14. E. Mitidieri, S.I. Pokhozhaev. “A priori Philos. Mag. 39:5, (1895): 422-443. estimates and blow-up of solutions of nonlinear 4. T.B. Benjamin, J.L. Bona, J.J. Mahony. partial di�erential equations and inequalities.” Proc. “Model equations for long waves in nonlinear Steklov Inst. Math. 234, (2001): 1-362. dispersive systems.” Philos. Trans. R. Soc. Lond. 272, (1972): 47-78. 15. E. Mitidieri, S. I. Pokhozhaev. “Towards a 5. F. Ursell, “The long-wave paradox in the Unified Approach to Nonexistence of Solutions for theory of gravity waves,” Proc. Camb. Philos. Soc. a Class of Differential Inequalities.” Milan Journal 49, (1953): 685-694. of Mathematics 72, (2004): 129-162.

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IRSTI 27.35.14

1A. Issakhov, 2T. Yang, 3A. Baitureyeva 1Faculty of Mechanical Mathematics, Al-Farabi Kazakh National University, Almaty, Kazakhstan, e-mail: [email protected] 3Department of Design Engineering and Mathematics, Middlesex University, London, United Kingdom, e-mail: [email protected] 3Faculty of Mechanical Mathematics, Al-Farabi Kazakh National University, Almaty, Kazakhstan, e-mail: [email protected]

CFD simulation of pollution dispersion from thermal power plants in the atmosphere

Abstract. This paper presents CFD simulation of pollution dispersion from a thermal power plant. Carbon dioxide was chosen as the scattering gas, as it constitutes the main share of emissions from the energy industry. The model was tested using experimental results performed using wind tunnel data available in the literature. A comparative analysis of the results of this article with experimental and numerical data was performed. It showed that the results of this article were closer to the experimental results than the calculations of previous authors. The minimum relative error with the experiment was less by 4.11% at the pipe exit and by 2.52% at a distance x/D=3 from the source, than other results. Based on this verification, the spread of pollution from thermal power plant (TPP) in real physical dimensions was modeled. The k-epsilon turbulence model was used taking into account buoyancy. The calculations were performed using the ANSYS Fluent 18.1 software package. As a result, the distance from the source was calculated, at which pollution will reach the ground surface (~ 2 km). Obtained distance is quite big since this TPP is located in an area which is far from residential settlements and there are no natural or architectural obstacles around. Key words: Navier-Stokes equations, mass transfer, numerical modeling, air pollution, concentration, thermal power plant.

should be paid to the energy sector, since the share Introduction of energy is more than two thirds of total greenhouse gas emissions and more than 80% of Air pollution every year becomes an emissions [2]. Therefore, in this paper, increasingly serious large-scale problem. Plants and was selected as the main test substance of pollution. various energy facilities (such as thermal power ܥܱTheଶ background annual concentration onܥܱ theଶ plants, nuclear power plants, etc.) produce a large Earth is equal to 400.88 ppm=0.0004 (mass amount of pollutants that dispersed in the fraction) [3]. ܥܱଶ atmosphere, damage the flora, fauna, buildings and To determine the extent of air pollution impact harm human health. The European Environment on the environment and people, it is important to Agency (2018) gives the following definition of air take into account the physical principles affecting pollution: “the presence of contaminant or pollutant the movement and dispersion of pollutants [4]. substances in the air that do not disperse properly Due to the rapid growth of computer and that interfere with human health or welfare, or capabilities, in particular, large-scale parallel produce other harmful environmental effects.” [1]. computing, it becomes advisable to use computer According to the final emissions report for 2017, simulation to calculate scientific and technical published in March 2018, global energy-related engineering problems. Nowadays technologies are emissions have increased and reached a historic rapidly developing, as a result of which their maximum. At the same time, special attentionܥܱଶ productivity has increased exponentially over time.

According to the data, over the past 60 years, chosen for the calculation. This method has been computing power has increased in productivity by 1 applied in multiple numerical studies and, when trillion times. [5] compared with experimental data, has shown good Therefore, people use computational fluid agreement. [29]. dynamics to conduct large-scale computer modeling of important scientific and engineering topics [6-7]. Test problem The study of jet behavior in crossflow is A detailed description of the test problem and important for various applications, especially for experiment is given in [30, 31]. The test problem chimneys, because the interaction between the jet domain is a three-dimensional channel with a pipe and crossflow fluids affects the pollution dispersion inside it. The pipe diameter (jet width) was D=12.7 into the atmosphere [8]. [mm], which was used as a characteristic unit of A review and description of the works devoted length. The dimensions of the geometry are shown to the study of the nature of jet motion in a in Figure 1. crossflow is given in [9-14]. Early studies of jets in crossflow were devoted to the derivation of empirical equations for the flight path and the principles of scaling [15-18]. To this end, the authors conducted numerous experimental studies. Recent research in this area is described in [19, 20]. Further, there have been many studies of vortex structures (vortex pairs rotating in opposite directions, horseshoe vortices); stability and destruction of the jet [21-22]. The purpose of this work was to assess the impact of emissions on the environment based on a numerical model of the spread of pollutants from sources. One of the pipes of Ekibastuz Thermal Power Plant-1 (Kazakhstan) was chosen as a real physical object of research [23]. Its height is 330 Figure 1 – Configuration of the computational [m], the pipe diameter is 10 [m]. domain of the test problem

Mathematical model An unstructured grid was constructed, the total Computational fluid dynamics has proven to be number of nodes was 533 697 (See Table 1). The an effective tool for modeling the behavior of jets in ratio of the jet velocity to the velocity of the crossflow. Modeling of such problems is based on crossflow is denoted as R  V jet /V crossflow . In the the resolution of the Navier-Stokes equations (the present work, R=0.5 was considered: the jet velocity equation of continuity and the equation of motion) was 5.5 , the crossflow velocity was 11 [24-25]. It was found that Reynolds-averaged . �� Navier-Stokes equations (RANS) modeling, can �� ] qualitatively predict the behavior of the total flow �� ] and concentration [26]. Past studies have revealed Table 1 – Number of grid points that non-stationary large eddy simulation (LES) models provide good agreement with experimental NI 230 results in pollutant dispersion problems [27]. NJ 100 However, the computational cost of this model is NK 21 about 100 times higher than the cost required for the RANS model [28]. Important observations Total body sizing 0.0025 [m] regarding RANS k-epsilon were noted in [29]. Total nodes 533 697 In present work, RANS k-epsilon model was used. Also, a comparative analysis of the obtained Air was chosen as the main fluid material for the results with the experimental [30] and numerical crossflow and the jet. The Reynolds number has [31] data was carried out. The SIMPLE method was been defined as:

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 56 (2019) 58 CFD simulation of pollution dispersion from thermal power plants in the atmosphere

. (1) other authors in this interval show the values of u/Vjet~0.5 – 0.7. A zero value is more reliable from �� Five types�� of =boundary �� ⁄ conditions= 4700 were used: a physical point of view, as this is a near-wall field. inlet, outlet, periodic, no slip, no flux (see Figure 1). Also, in this region (y/D~0 – 1) the relative errors of According to experimental data, the thickness of the present simulation are smaller, than others (see boundary layer is equal to 2D. The wind velocity Table 2). Based on this data, the solutions obtained profile was defined by a power law with exponent in this work turned out to be more accurate than the 1/7 within the boundary layer and was set as calculations obtained by other authors [30, 31]. The constant 11 above it. Since a smooth surface reason is the quality of the grid: in this work, an was used in the�� experiment, the roughness height unstructured grid was used (the number of nodes was zero. �� ] was 533 697), while in [31] a structured grid was used (the number of nodes was 265 000).

Table 2 – Relative errors of numerical simulations for R=0.5

Min. relative error Max. relative error x/D 0.0 3.0 0.0 3.0 k-epsilon 6.78% 7.54% 87.74% 207.23% k-eps with BC 6.57% 6.2% 85.16% 169.66% SST 6.57% 7.9% 72.9% 170.1% a) Ajersch 9.65% 8.32% 51.37% 279.52% Present paper 2.46% 3.68% 22.58% 89.13%

Ekibastuz thermal power plant-1: full-scale emission distribution modeling

After verification and validation of the numerical algorithm, pollution dispersion from a real power plant (TPP) in full scale dimensions was b) modeled. Ekibastuz Thermal Power Plant-1 (Kazakhstan) was chosen as a real physical object of Figure 2 – Comparison of the obtained results with research (Figure 3). The power plant consists of a experiment data and calculations of other authors. (a) main building and two pipes. The dimensions of the x/D=0.0, (b) x/D=3.0 main building are: length — 500 m, width — 132 m, height — 64 m. The height of the chimneys is 300 meters (built in 1980) and 330 meters (built in Figure 2 shows a comparison of the numerical 1982), the exit pipe diameter for each is 10 meters. results of this study with experimental data and The 3D computational domain has dimensions numerical solutions of other authors at various 8000×2496×3000 m and the distance between the distances from the jet (R=0.5; x/D=0.0 and domain inlet and tallest pipe is 2000 m. Also, the x/D=3.0). At the Figure 2,b the values of the red line tallest pipe is located in the origin of the coordinate (u/Vjet) at y/D=0 approach zero, while the plots of system (Figure 4).

International Journal of Mathematics and Physics 10, №1, 56 (2019) Int. j. math. phys. (Online) A. Issakhov et al. 59

Figure 3 – Ekibastuz Thermal Power Station 1 (Kazakhstan, Ekibastuz city)

Figure 4 – Computational domain and boundary conditions for the Ekibastuz TPP simulation

b) Figure 5 – Computational mesh of Ekibastuz TPP model: (a) cross-section view, (b) scaled view

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 56 (2019) 60 CFD simulation of pollution dispersion from thermal power plants in the atmosphere

The grid was built on the same principle as in exhaust velocity and UH – the horizontal wind the test problem, i.e. refined in the area of the velocity at the reference height (10 m). According to pollution movement trajectory (Figure 5). the data, the following substances are released into Before calculations, mesh sensitivity analysis the atmosphere from the Ekibastuz TPP per year was performed. According to the results, the grid (Table 3). consisting of 5,193,038 triangular elements was chosen as the main grid for further calculations. The distance ZP from the center point P of the Table 3 – Emissions from Ekibastuz TPP (2016) wall-adjacent cells to the ground surface (bottom of Amount of domain) is 0.49 m, there was set inflation with the Type of pollutants Unit growth rate coefficient 1.2 since this is exactly the emissions size of the first cell (height of first cell <1 m) that NOX tons 54,700 was recommended previously for an accurate SO2 tons 132,900 simulation of the atmospheric boundary layer [29]. CO tons 2,800 Since in this case it was not possible to measure Particulate matters tons 28,000 experimentally the profile of velocity , turbulence Persistent organic tons – kinetic energy or dissipation rate , their initial pollutants � Volatile organic profiles were set according to the [33], which tons 115.4 describes the profiles� of these components� that are pollutants most suitable for modeling wind engineering thousand CO2 24 150.7 problems. tons

Thus, the share of in emissions is 99.1% �∗ � + �� and therefore the distribution of this gas was chosen � = �� for calculations. C� � �∗ Results � = �� Figure 8 shows the vertical concentration

� profiles at various distances (x/H=5, 10, 15 and 20, �∗ � = where H=300 m) from the pollution sources for the Where � �� + �� – default Schmidt number value of 0.7. Due to friction velocity, which is calculated as: dissipation, the gas concentration decreases with � � ∗ C =0.09, y =0.2,κ=0.4, u increasing distance from the source. For comparison, the maximum concentration at the distance x/H=20 is almost 13 times less than the κ u�� u∗ = maximum concentration at the distance x/H=5 from Here was setln �toy ��7 +m/s,y� �according to the the source. One can also notice that with increasing meteorological data and the wind rose presented in distance from source the height of the maximum �� Figure 6. � since air measurements as a concentration also increases. Thus, if at x/H=5 the rule are made at this height. According to these data, maximum concentration point was at the height �� the wind �most= often 10 � blows from the southwest y/H=2 (600 m), then at x/H=20 the maximum (more than 288 hours per year), therefore this concentration point is at the height y/H=3 direction was chosen for calculations. (900 m). The temperature of the ground was set to be Figure 9 illustrates the ground-level downstream equal to the maximum average temperature for the concentration profile along the x-axis. According to year: 28 degrees. According to the meteorological the plot, the pollution reaches the earth at a distance data of Ekibastuz city, this temperature is set in July of about x=1.5 km, which is about 1753 m from a (Figure 7). lower stack and 1500 m from a high stack. The The emission temperature was set to 315oС. maximum concentration reaches about 0.04 ppm, Emission rate is 31.5 m/s. Thus, the momentum which is a fairly optimal level of pollution for ratio is M=WS/UH=4.5, where WS is the vertical human health. Figure 10 illustrates the comparison

International Journal of Mathematics and Physics 10, №1, 56 (2019) Int. j. math. phys. (Online) A. Issakhov et al. 61 of ground-level lateral concentration profiles for other cases (Sc=1.4 and 1.0). Figure 11 illustrates different Sc numbers (Sc=0.7, 1.0, 1.2 and 1.4). For the iso-surfaces of the pollution spreading. Due to Sc=0.7, the concentration level is almost 2.5 times dispersion, pollution dissipates with increasing higher than with Sc=1.2 and almost 2.78 than in the distance from the source.

Figure 6 – Wind rose of the Ekibastuz city

Figure 7 – Annual temperature of Ekibastuz city

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 56 (2019) 62 CFD simulation of pollution dispersion from thermal power plants in the atmosphere

Figure 8 – Vertical concentration profiles at various Figure 9. Ground-level (y=0) distances from the pollution sources at central cross downstream concentration profile section (z/H=0).

Figure 10 – Side concentration profiles at ground level (y=0) for different values of the Schmidt number (Sc)

Conclusions dispersion was modeled. The k-epsilon model was used without any additional dispersion The purpose of the study was to investigate models. As a result, it was determined at what the dynamics of the pollution dispersion. The distance from the source the pollution mathematical model and numerical algorithm accumulates on the ground surface. were tested using an experimental test problem. According to the obtained data, with increasing The results were closer to the experimental, distance from the source, the concentration of compared with the data of other authors. Using pollution spreads more widely under the influence the example of a real thermal power plant, of diffusion.

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International Journal of Mathematics and Physics 10, №1, 56 (2019) Int. j. math. phys. (Online) A. Issakhov et al. 63

Figure 11 – Iso-surfaces of the mean concentration C/C0=0.01 (Sc=0.7)

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 56 (2019) 64 CFD simulation of pollution dispersion from thermal power plants in the atmosphere

The farther the distance from the pipe, the lower 12. Hasselbrink, E.F., Mungal, M.G. 2001 the concentration of the substance. The various Sc Transverse jets and jet flames. Part 1. Scaling laws numbers were tested for gas dispersion modelling, for strong transverse jets. J. Fluid Mech. 443, 1–25. Sc=0.7 showed the highest levels of concentration. 13. Muppidi, S., Mahesh, K. 2005 Study of Thus, the obtained numerical data may allow to trajectories of jets in crossflow using direct predict the optimal distance from residential areas numerical simulations. J. Fluid. Mech. 530, 81–100. for the construction of thermal power plants, at 14. Muppidi, S., Mahesh, K. 2008 Direct which the concentration of emissions will remain at numerical simulation of passive scalar transport in a safe level in the future. transverse jets. J. Fluid Mech., 598, pp. 335–360. 15. Callaghan, E.E., Ruggeri, R.S., 1948. References Investigation of the penetration on an air jet directed perpendicularly to an air stream. 1. ECSC. (2018). Glossary: Primary & 16. Margason, R.J., 1968. The Path of a Jet Secondary pollutant. [online] Available at: Directed at Large Angles to a Subsonic Free Stream. http://ec.europa.eu/health/scientific_committees/opi Langley Research Center. nions_layman/en/indoor-air- 17. Golbitz, W.C., 1980. Time Dependent pollution/glossary/pqrs/primary-pollutant- Navier-Stokes Solution of a Turbulent Gas Jet secondary-pollutant.htm (Accessed 5 Jul. 2018). Ejected from a Rectangular Orifice into a High- 2. International Energy Agency (IEA). CO2 Subsonic Crossflow. Faculty of the School of Emissions from Fuel Combustion; 2018 Engineering of the Air Force Institute of 3. CO2 Earth. Annual CO2 Data, [online] Technology. Available at: https://www.co2.earth/annual-co2 18. Cutler, P.R.E., 2002. On the Structure and Accessed: (15 January 2019) Mixing of a Jet in Crossflow. The Universe of 4. Zavila O. Physical Modeling of Gas Adelaide. Pollutant Motion in the Atmosphere, Advances in 19. Erdem E., Kontis K., Saravanan S. Modeling of Fluid Dynamics, Dr. Chaoqun Liu Penetration characteristics of air, carbon dioxide and (Ed.), InTech, 2012 helium transverse sonic jets in mach 5 cross flow 5. Visual Capitalist. (2017). Visualizing the Sensors (Switzerland), 14 (2014), pp. 23462-23489. Trillion-Fold Increase in Computing Power. [online] 20. Cambonie T., Gautier N., Aider J.L. Available at: Experimental study of counter-rotating vortex pair https://www.visualcapitalist.com/visualizing- trajectories induced by a round jet in cross-flow at trillion-fold-increase-computing-power (Accessed: low velocity ratios Exp. Fluids, 54 (2013), pp. 1-13. 15 January 2019) 21. Karagozian, A.R., 2014. The jet in 6. Jiangfei Z. “The new direction of crossflow. Phys. Fluids computational fluid dynamics and its application in 26.http://dx.doi.org/10.1063/1.4895900 industry”, J. Phys.: Conf. Ser. 1064 012060, 2018 22. Fric T.F., Roshko A. Vortical structure in 7. “Computational Fluid Dynamics Past, Pre- the wake of a transverse jet J. Fluid Mech., 279 sent and Future”. [online] Available at: http://aero- (1994), pp. 1-47, comlab.stanford.edu/Papers/NASA_Presentation_20 23. Wikipedia, “Ekibastuz_GRES-1”. [online] 121030.pdf (Accessed: 15 January 2019) Available at: https://en.wikipedia.org/wiki/ 8. Rek Z. et al. “Numerical simulation of gas Ekibastuz_GRES-1 (Accessed: 15 January 2019) jet in liquid crossflow with high mean jet 24. Ferziger J. H., Peric M. Computational tocrossflow velocity ratio “. Chemical Engineering Methods for Fluid Dynamics. Springer; 3rd edition, Science 172 (2017) 667–676 2013, -p. 426 9. Margason, R. J. 1993 Fifty years of jet in 25. Chung T. J. Computational Fluid Dynamics. crossflow research. In AGARD Symp. on a Jet in Cambridge University Press, 2002 – p. 1012. Cross Flow, Winchester, UK. AGARD CP 534. 26. Tominaga, Y., Stathopoulos, T., 2009. 10. Broadwell, J. E. , Breidenthal, R. E. 1984 Numerical simulation of dispersion around an Structure and mixing of a transverse jet in isolated cubic building: comparison of various types incompressible flow. J. Fluid Mech. 148, 405–412. of k–e models. Atmospheric Environment 43, 3200– 11. Karagozian, A. R. 1986 An analytical model 3210. for the vorticity associated with a transverse jet. 27. M. Chavez et al., ”Near-field pollutant AIAA J. 24, 429–436. dispersion in the built environment by CFD and

International Journal of Mathematics and Physics 10, №1, 56 (2019) Int. j. math. phys. (Online) A. Issakhov et al. 65 wind tunnel simulations”. J. Wind Eng. Ind. in a Crossflow: Detailed Measurements and Aerodyn. 99 (2011) 330–339 Numerical Simulations,” International Gas Turbine 28. Cheng, Y., Lien, F.S., Yee, E., Sinclair, R., and Aeroengine Congress and Exposition, ASME 2003. A comparison of large eddy simulations with Paper 95-GT-9, Houston, TX, June 1995, pp. 1–16. a standard k-e RANS model for the prediction of a 31. Keimasi M.R., Taeibi-Rahni M., fully developed turbulent flow over a matrix of “Numerical Simulation of Jets in a Crossflow Using cubes. Journal of Wind Engineering and Industrial Different Turbulence Models”, AIAA journal, Vol. Aerodynamics 91, 1301–1328. 39, No. 12, December 2001 29. Blocken B. et al., “Numerical evaluation of 32. ANSYS Fluent Theory Guide 15, ANSYS pollutant dispersion in the built environment: Ltd., 2013. Comparisons between models and experiments”. J. 33. Richards, P.J., Hoxey, R.P., 1993. Wind Eng. Ind. Aerodyn. 96 (2008) 1817–1831 Appropriate boundary conditions for computational 30. Ajersch, P., Zhou, J. M., Ketler, S., wind engineering models using the k-ε turbulence Salcudean, M., and Gartshore, I. S., “Multiple Jets model. J. wind Eng. ind. aerodyn. 46, 145-153.

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 56 (2019) International Journal of Mathematics and Physics 10, №1, 66 (2019)

IRSTI 28.23.15

1M.N. Kalimoldayev, 1O.Zh. Mamyrbayev, 2A.S. Kydyrbekova, 2N.O. Mekebayev

1Institute of Information and Computational Technology, Almaty, Kazakhstan 2al-Farabi Kazakh National University, Almaty, Kazakhstan e-mail: [email protected]

Voice verification and identification using i-vector representation

Abstract. In the area of voice recognition, many methods have been proposed over time. Automatic speaker recognition technology has reached a good level of performance, but still needs to be improved. Signature verification (SV) is one of the most common methods of identity verification in the banking sector, where for security reasons, it is very important to have an accurate method for automatic signature verification (ASV). ASV is usually solved by comparing a test signature with a registration signature(-s) signed by the person whose identity is declared in two ways: online and offline. In this study, a new i- vector based method is proposed for SV online. In the proposed method, a fixed-length vector, called an i-vector, is extracted from each signature, and then this vector is used to create a template. Several methods, such as the nuisance attribute projection and the within-class covariance normalization, are also being investigated to reduce the intra-class variation in the i-vector space. At the stage of evaluation and decision-making, they also propose to apply the support vector machine with two classes. In this article, a new low-dimensional space, depending on the dynamics and the channel, is determined using a simple factor analysis, also known as i-vector. I-vectors have proven to be the most efficient functions for text independent speaker verification in recent studies. Key words: i-vectors, dimensionality reduction, UBM size, speaker identification

Introduction advantages, signature forgery is more difficult in these methods because they use dynamic As time goes, voice processing technology is characteristics, such as speed and azimuth, which becoming more and more mature. Using advances in are very difficult to simulate. Our focus is on online signal processing and machine learning, ASV is methods. There have been a lot of SV online studies implemented in two ways: online and offline. With that can be grouped into two main categories: offline verification, also called static verification, Methods based on global signature features. we have access only to the signature image [1-3]. In These methods attempt to extract a fixed-length such methods, we usually normalize the image size vector from the entire signature so that the after some preprocessing, and then extract the signatures can be easily compared in vector form. elements from the image using a sliding window. These methods can be further divided into two These functions are then used to compare two subcategories: in the first, we try to extract global signatures. On the other hand, there are online functions from the entire set of signatures. For methods, also called dynamic methods, where example, in [7] Jane uses the number of strokes as a information related to signature dynamics is global feature. The authors use other functions such provided, as well as signature image [4-6]. Dynamic as average speed, average pressure, and the number information includes pressure, velocity, azimuth, of times the pen is lifted during signature in [8]. As etc. In these methods, changes in vertical and a good example of Fierrez-Aguilar, [4] introduced horizontal directions are commonly used as shape- 100 global attributes sorted by their individual related elements. These methods have better discriminatory power. A subset of these functions is performance than autonomous methods and are also used in other studies [5, 9-12]. In the second more reliable because they use more information subcategory, a transformation is applied to the extracted from the signature. In addition to these signature to obtain a fixed-length vector. For © 2019 al-Farabi Kazakh National University Int. j. math. phys. (Online) M.N. Kalimoldayev et al. 67 example, the wavelet transform is used in [13] to assessment, emotion recognition, sound scene extract the feature vector from the entire signature. classification, etc. In the main application of this In another study, the discrete cosine transformation method (i.e. speaker recognition), a vector of fixed (DCT) is used to obtain a fixed length feature vector length, called an i-vector, is extracted from a speech [6]. The proposed method in this paper belongs to signal of arbitrary duration. In this article, we give a this group. description of the i-vector problem and a brief Functional methods. Methods in this category are overview of the initial results. We begin with a very more focused on comparing signatures and brief description of the key components of the i- calculating the distance between two signatures. In vector based on the speaker recognition system. In these methods, each signature is represented using a the following steps, this vector is used for scoring sequence of local features extracted from it. This and recognition. Although i-vector is used mainly in category can also be divided into two subcategories: many speech applications, it is less well known in the methods in the first do not perform any other areas. In this article, we use the i-vector, modeling. In fact, in these methods, a set of which is usually used to recognize the speaker in the references is stored for each individual, and during SV. Despite their different areas, speech biometrics the test, the input signature is compared to the set of and signature biometrics are similar in nature, as references for decision-making. The most common both must extract subject-specific patterns from a method in this subcategory is Dynamic Time- captured signal contaminated by changes from Warping (DTW), which is used in many studies various irrelevant sources. Since the analysis of total [14–17]. variability factors is an embedded i-vector learning The second category includes methods that train step that helps eliminate distractions in biometric a probabilistic model for each person, using analysis and extracts a unique identity signatures in his/her control set. These methods representation vector, we expect the i-vector to be typically use probabilities for evaluation and able to provide a promising solution for the decision making. The most common methods in this signature extraction problem. subcategory are the hidden Markov model (HMM) There are two reasons for using this method for [18–22] and the Gaussian mixture model (GMM) SV. First, online signatures have a variable length, [23–25]. similar to speech signals. Using this method, we can In the area of voice recognition, the use of get a fixed-length vector that facilitates the Gaussian mixture models (GMM) to create following steps in making a decision. Therefore, universal background models (UMM) and after extracting the temporal features from each collaborative factor analysis (JFA), by far the most signature, we extract the i-vector. Since we get a popular i-vector, has increased its accuracy in fixed-length vector for each signature, we can put creating a specific dynamics model. However, this method in the first category above. The second sometimes we do not need to know what language reason is that a person's signatures usually differ the speaker speaks, because in some situations only slightly each time. These differences lead to changes one of them is the most important, while others are within the class, which in turn increase the false relatively less critical. Speaker verification is rejection rate (FRR). In various applications of the i- becoming increasingly important as a solution to vector in speech processing, various ways have been secure biometric keys for industrial, forensic and proposed to reduce intraclass variations, which can government purposes, such as data encryption on also be accepted in this application. Similar to the mobile devices or user verification at contact case of speaker verification, we also suggest using centers. It seems that users are annoyed with the two different methods to reduce the undesirable persistence of multiple PINs and passwords, that is, effects of intra-class changes. Since there are several biometric data that cannot be lost or forgotten, signature samples for each person as a reference set provide significant advantages in terms of usability. at the registration stage, we suggested adding them to the data used to train variation compensation i-vector methods within the class. In addition, we proposed to apply the 2nd class support vector machine The i-vector was first proposed for speaker (SVM) method to distinguish between i-vectors recognition application, and then was applied in extracted from genuine and fake signatures. The other applications, such as language identification, experimental results showed the effectiveness of accent identification, gender recognition, age these ideas on two different databases.

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 66 (2019) 68 Voice verification and identification using i-vector representation

Extract statistics variable. For each observation sequence representing the statement, our i-vector is the point At this stage, for each sequence of attributes, the estimate of the maximum a posteriori (MAP) for the Baum-Welch statistic of zero and first order is hidden variable . Our i-vector extractor learning calculated using UBM [26,27]. procedure is based on the efficient implementation Given that Xi is a complete set of feature vectors proposed in [29].� for learning the i-th signature, the zero and first The contribution of this study is to evaluate the order statistics for the c-th UBM component is result of factors affecting the i-vector, based on the calculated as follows: speaker’s sound identification. We study this in terms of parameters, where we evaluate and analyze (1) how the various parameters of the i-vector extractor, � such as the size of the Universal Background Model �� The variability �of ��the )=speaker∑ � or session is the (UBM) and the dimension of the i-vector, affect the variability manifested by a given speaker from one accuracy of speaker detection. The UBM size refers recording session to another. This type of variability to the Gaussian component, which is the is usually associated with channel effects, although corresponding adapted component in the dynamics this is not strictly accurate, since there are also model. The I-vector dimension is equal to the "rank" changes within the dynamics and phonetic change. of its own matrix. Based on Huang, a greater i- In this approach, the speech segment is represented vector dimension would not give a large by a low-dimensional "identity vector" (ivector – for performance improvement of the classification, but short) extracted by factor analysis. The i-vector significantly increased the computational costs. In approach has become state-of-the-art in speaker [30] the literature discussed by reducing verification, and in this paper we show that it can be computations will allow efficient use of the i-vector successfully applied to speaker identification as in more applications. In this study, the recorded well. The approach provides an elegant way to computation time is to investigate whether both reduce multidimensional sequential input data to a factors affect the computation or not, and low-dimensional vector of features of a fixed length, and the next direction for the next study is while retaining most of the relevant information determined. [28]. The basic idea is that the session-and channel- To record the voice in the present work, a dependent supervectors of the Gaussian mixture complex of technical devices was used, the block model cascade model (GMM) can be modeled as diagram of which is shown in Fig.1. The block diagram of the system includes: microphone 1, low- = + w (2) frequency amplifier 2, analog-to-digital converter 3, software 4, and a computer unit 5, which records the where is the session-and�� channel-independent amplitude-time and calculation of the frequency component of the average supervector obtained dependences of the signal. from UBM,� is the basis matrix covering the To identify an unknown voice recording, a subspace encompassing the important (both for the database of speakers was compiled and registered. dynamics and �the session) in the supervector space, Using the above equipment, voice signals were and is the standard, normally distributed hidden recorded, which formed the database.

�

4 1 2 3 5

Figure1 – A block diagram of the hardware of the voice identification system

International Journal of Mathematics and Physics 10, №1, 66 (2019) Int. j. math. phys. (Online) M.N. Kalimoldayev et al. 69

When scaling, or weighing, all experimental data where Xik is the i-th variable of the k -th sample, are reduced to the same scale. This procedure is – is the i-th scaled variable of the k -th necessary to reduce the impact on the analysis of sample,�� Sdev – is the standard deviation of the strongly pronounced variables. There are various sample.�� ways of scaling [28], in this work standardization Figure 2 shows the block diagram of the method has been applied, since it is the most studied and used in this paper. The i-vector system consists of tested. Standardization uses standard deviation – two main parts: the front part and the rear part. The Sdev, which is one of the most commonly used first consists of the extraction of cepstral features weighting factors. In addition, each element of the and UBM learning, while the latter includes matrix X is multiplied by the value 1/Sdev: sufficient statistical calculation, T-matrix training, i- vector extraction, dimension reduction and , (3) evaluation. � �� =��

Receiving voice Preprocessing Function extraction recordings

Microphone Calculation of conversion Signal normalization recording factors VAD method, (MFCC)

Figure 2 – Block diagram of the experiment with the speaker's identification system

Function extraction (4) First, simple energy speech activity detection �� (VAD) is performed to discard unnecessary part of where is the measured� amplitude,= is the the voices. Energy-based VAD is used when energy maximum amplitude, normalized amplitude,�� i = 0,1, values are first calculated at the frame level, ..., k. �� followed by data normalization and, finally, voice In order to remove the second factor (speech activity detection. A class with a higher average is rate), time normalization was performed. The considered to be the speaker's sound, and therefore second factor was taken into account the corresponding segments of the sound are programmatically by using the same number of preserved until smooth. Second, the characteristics samples. The amplitude-frequency characteristics in of MFCC and log energy, together with their first the form of the recorded audio signal spectrum were and second order derivatives, are computed in 20 ms analyzed directly. The frequency spectra had the of Hamming window frames every 10 ms. Then the form of the dependence of the amplitude A from the determination of the activity of the speaker’s sound frequency f. is applied, and the speaker’s sound is normalized in When recording voice signals in real conditions, accordance with the standard normal distribution. it is possible to impose random factors, including To solve the problem of identification of the both external mechanical noise and hardware noise. individual, the analysis of the individual frequency Median filtering [31] was used for their suppression, spectrum of voice signals is the main one. In this which consisted in exclusion from the initial setting, the first two factors (the amplitude and emission signal. duration of the signal) are random and need to get The current lack of a clear systematization of rid of them. To do this, all signals were reduced to voice features, as well as the existence of a large one amplitude, that is, the amplitude normalization number of voice characteristics of various levels, was performed: such as the basic tone [32], formant frequencies

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 66 (2019) 70 Voice verification and identification using i-vector representation

[33,34] and others [35,36], is a certain difficulty in variability has become a modern approach to voice choosing most informative features and recognition. characteristics for a specific identification method This method, which was introduced after its and requires a separate study. This section provides predecessor, the joint factor analysis, can be qualitative and quantitative estimates for the considered as a method of extracting a compact selection of informative voice characteristics. The representation with a fixed length in the presence of difference in voice timbres is described by different an arbitrary length signal. The extracted compact frequency spectra of voice signals. Fourier unit vector can then be used either to measure decomposition is a natural mathematical apparatus similarity based on vector distance or as input for for frequency spectrum analysis. Processing of data any further feature transformation or modeling. representing numerical amplitude-time dependences There are certain steps to extract the i-vector from can be carried out using discrete Fourier transform: the signal. First the features should be extracted from the input signal, then the Baum – Welch statistics should be extracted from the features, and π �� finally the i-vector is calculated using these � � �� statistics. We will explain these steps in detail �(�) =��(�)� , below. �� (5) For each statement, the corresponding feature sequence is eventually transformed into an i-vector � = �,1,�, � , � � 1 using a GMM-based i-vector extractor with three π �� different UBM-sized components trained from the � � 1 �� combined features from all the samples included in �(�) = �� (�)� , our training data (Fig.3). Three UBM sizes that � �� (6) make up 32, 64 and 128 components. Assuming that we have calculated the zero and where A(k), a(n)� = are �,1,�, direct � , � and � 1 inverse discrete first order statistics using (7) and (8), we can Fourier transforms, respectively, k and n are sample calculate the a posteriori covariance matrix numbers, and N is the number of samples. [i.е. ], average (i.e. and second Coefficients A (k) can be used precisely as the moment� (i.e.� for wi, using� the following elements of the matrix X, forming rows in this relations: �� ,� � � �� ] matrix. �� ] (7) I-vector extraction � �� ,��=(��∑� ��(��)�� ∑� ��) I-Vector based systems. As explained earlier, at (8) � �� present, the i-vector in the space of complete � � � � � � � � �� ] = �� ,�� ∑ � ∑ � (� )� (9) � � � � � � � �� ,� �=��,��]��]

Classification

scaling signal separation (weighing) main component calculation (based on GMM)

Figure 3 – Using projection methods

International Journal of Mathematics and Physics 10, №1, 66 (2019) Int. j. math. phys. (Online) M.N. Kalimoldayev et al. 71

Post processing of dimension reduction is to project the i-vectors into a space where the variability between accents is Since the simulation of i-vectors contains maximal and the variability within the accent is information about the dynamics and channel minimized. In this study, three different variability in the same space at the same time, the measurements were experimented with: 100, 200, channel compensation technique in the common and 400. Thus, we optimized the parameters of the factor space is required to eliminate undesirable i-vector to experiment and evaluate the result. effects. Channel-compensated approaches play a major role in I-vector speaker recognition systems. Scoring Therefore, channel compensation is necessary to ensure that test data obtained from different Finally, the identification result from the system channels can be properly evaluated by loudspeaker is given by calculating the similarity score. The models. For channel compensation to be possible, simplest and fastest counting function, that is, the channel variability should be modeled explicitly. cosine distance, is calculated between the i-vectors Before calculating the verification estimates, from the dynamics model and the i-vector from the channel bleaching, linear discriminant analysis test segment. The decision-making and evaluation (LDA) and within-class covariance normalization process is then computed, and the system (WCCN) were performed to compensate for the performance is then represented using accuracy channel. 91,11%, CMC curves, and detection error tradeoff We used the same dataset to train the total (DET). variability matrix to evaluate the LDA and WCCN t-SNE for individual signatures using raw i- matrices. Since the extracted i-vectors contain vectors (i.e., without applying any transformations) variations both within and between accents, the goal (Figure 4)

Figure 4 – Two-dimensional data representation

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 66 (2019) 72 Voice verification and identification using i-vector representation

Figure 5 – Three-dimensional data representation

As shown in Fig. 5, for both projection methods, The speech database of voice data intended for all points are combined into compact areas not tasks of voice identification and differing in intersecting with each other, each area corresponds considerable number of repetitions of phrases by the to the records of one speaker. This indicates that same speakers is created. The use of this database both methods have provided a reliable separation of by increasing the number of repetitions provides a speakers by voice data. more accurate assessment of the identification Note that the direct comparison of the above result. graphs of accounts for the methods of main Comparison of various informative parameters components (Fig. 4) and projections on latent of voice signals used as a feature vector in structures (Fig. 5) does not allow to quantify these projection analysis methods has been carried out. calculation options. The advantage of one method The residual dispersions were calculated that over another can be estimated from the residual showed the preference for the use of voice dispersion graph. identification of the Mel-cepstral decomposition coefficients, which improve the separation of the Result and discussion source signals according to their features and reduce the contribution of random distortions. A series of experiments was conducted to study We found that the accuracy increases as the the effect of the number of UBM components, dimension of the i-vector increases. In addition, our vector dimension, and post-processing techniques. results showed that the UBM with smaller size These experiments were conducted using a set of outperforms larger UBM. In addition, the result of voices – a set of open source toolkit and extensible the time calculation shows that the processing takes tools for recognition of the modern level. 50 votes longer when the dimension of the i-vector increases taken from the database were used for the and the size of UBM is larger. evaluation. The adaptation of projection methods of the main Conclusion components and projections on latent structures in relation to the analysis of acoustic signals in In this article, we studied how the i-vector technologies of personal identification by voice has extractor parameter, such as the UBM size and i- been carried out. vector dimension, affects the accuracy of voice

International Journal of Mathematics and Physics 10, №1, 66 (2019) Int. j. math. phys. (Online) M.N. Kalimoldayev et al. 73 identification. As for the parameters, the highest 8. Lee.L.L., Berger.T., Aviczer.E. “Reliable accuracy was achieved when using UBM with Online Human Signature Verification Systems.” Gaussians and an i-vector dimension. They are IEEE Trans. Pattern Anal. Max Intell. 18, (6) similar to those reported in the general voice (1996): 643–647. recognition literature. As for the data, we found that 9. Nanni, L., Lumini, A. “Parsen window the selection of UBM training data is the most classifier ensemble for online signature important part, followed by the dimension of the i- verification.” Neurocomputing 68 (2005): 217–224. vector. This is understandable because the earlier 10. Lei, H., Govindaraju, W . “Comparative components of the system affect the quality of the study of the consistency of functions in verifying remaining steps. signatures online.” Pattern Recognit. Lett. 26, (15) For further research, we propose to study the (2005): 2483-2489. effect for a larger i-vector dimension and a larger 11. Richiardi, J., Ketabdar, H., Drygajlo, A . UBM size. For this study, we do not do this because “Selecting local and global functions for online of the long computation time, because we use the signature verification.” 2005 Proc. Eighth int. Conf. small size of the speaker’s database. In the Analysis and recognition of documents (2005): 625– following studies, we can reduce the computation 629. time by exploring other factors that influence this, 12. Nanni, L. “Experimental comparison of and add additional data to further study this effect of single-class classifiers for online signature the experiment. verification,” Neurocomputing 69, (7) (2006): 869– This work carried out in the framework of the 873. project “Development of technologies for 13. Leytman DZ, George S.Е. “Verification of a multilingual automatic speech recognition using handwritten signature online using wavelets and deep neural networks”. neural back distribution networks”. 2001 Proc. Sixth Int. Conf. Analysis and recognition of documents References (2001): 992–996. 14. Kholmatov A., Yanikoglu B. “Identity 1. Kalera, M.K., Srihari, S., Xu, A . “Verifying Authentication Using the Improved Signature and Identifying Signatures Offline Using Distance Verification Method on the Internet”, Pattern Statistics” Int. J. Pattern Recognit. Artif. Intel. 18, Recognit. Lett. 26, (15) (2005): 2400–2408. (07) (2004): 1339–1360. 15. Vivaracho-Pascual, C., Faundez-Zanuy, M., 2. Singh, J., Sharma, M . “Verifying signatures Pascual, JM “An effective low-cost approach for offline using neural networks.” J. Inf. I-manager. recognizing signatures online based on the Technol.1, (4) (2012): 35. normalization of length and fractional distances.” 3. Daramola, S.A., Ibiyemi, TS.S . Pattern Recognit. 42, (1) (2009): 183–193. “Autonomous signature recognition using the 16. Sato, Y., Kogure, K. “Online signature hidden Markov model (HMM).” Int. J. Comput. verification based on the form, movement, and Appl. 10, (2) (2010): 17–22. pressure of the letter.” Proc. Sixth Int. Conf. Pattern 4. Firres-Aguilar, J., Nanni, L., Lopez, Recognition.( 1982): 823–826. Penalba, J., et al . “An online signature verification 17. R. Martens, Klezen L. “Optimization of system based on a combination of local and global dynamic programming for online signature information.” Audio and video biometric verification”. 1997 Proc. Fourth Int. Conf. authentication of identity (2005): 523–532. Document Analysis and Recognition 2 (1997): 53– 5. Nanni, L . “Advanced Improved multi- 656. match method for online signature verification with 18. Firres J., Ortega-Garcia J., Ramos D. global functions and tokenized random numbers.” “Online Signature Verification Based on HMM: Neurocomputing, 69, (16) (2006): 2402-2406. Character Extraction and Signature Modeling.” 6. Liu Y., Yang, Z., Yang, L . “Online Pattern Recognit. Lett., 28, (16) (2007): 2325–2334. Signature Verification Based on DCT and Sparse 19. Dolfing, J., Aarts, E., Van Oosterhout, J. Representation,” IEEE Trans. Cybern. 45, (11), “Online Signature Verification Using Hidden (2015): 2498-2511. Markov Models.” 1998 Proc. Fourteenth Int. Conf. 7. Jane AK, Griss FD, Connell S.D . “Online Pattern Recognition 2 (1998): 1309–1312. Signature Verification.” Pattern Recognit. 35, (12) 20. Van BL, Garcia-Salichetti S., Dorizzi B. B. (2002): 2963-2972. “On the use of the Viterbi path along with

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International Journal of Mathematics and Physics 10, №1, 66 (2019) Int. j. math. phys. (Online)

International Journal of Mathematics and Physics 10, №1, 75 (2019)

IRSTI 27.35.14

1,2*A.S. Zhumali, 1B.A. Satenova, 1,2O.L. Karuna

1Al-Farabi Kazakh National University, Almaty, Kazakhstan 2National Engineering Academy, Almaty, Kazakhstan *e-mail: [email protected]

Lattice Boltzmann method simulation of thermal flow dynamics in a channel

Abstract. The objective of this paper is the simulation of thermal flow dynamics in a channel. The mathematical model in a two-dimensional formulation is described by Navier-Stokes equations, continuity and temperature equations. For the numerical simulation of the problem the Lattice Boltzmann method applying the D2Q9 model is used. The validity of this method is tested by comparing the numerical solution to the analytical solution of the planar channel flow and error rates are calculated for various sizes of the computational grid. The test problem of thermal Poiseuille flow in the channel was solved to deactivate the correctness of the developed algorithm. Very good agreement between the exact and numerical solution of this problem is shown. Key words: The lattice Boltzmann method, thermal flow dynamics, Poiseuille flow.

Introduction The goal of this paper is the numerical implementation of thermal flow dynamics in a Thermal flows play an important role in the flow channel in a two-dimensional case. With the help of dynamics. Recently, there has been an effort to LBE method the profiles of velocities and increase the capability of the lattice Boltzmann temperature at different values of parameters in the method in order to solve for fluid flows including system of differential equations and at different time heat transfer [1, 2]. A detailed analysis can be found instants are investigated. in [3]. Generally, the thermal lattice Boltzmann model Statement of the problem (TLBE) can be divided into several categories [4]. The first is the multispeed scheme, the second is the In this paper we considered 2-D thermal flow in double distribution function (DDF) scheme and the planar channel. The flow driven by a body force. last is the hybrid thermal lattice Boltzmann equation We set cold temperature at the bottom wall and hot (HTLBE) scheme [3]. The multi-speed scheme is a temperature at the top wall of the channel plain extension of the Boltzmann isothermal models (Figure 1). with a lattice, in which only the velocity distribution function is affected. In double distribution function scheme, different distribution functions are used, one for the velocity field and the other for the temperature field or internal energy. The main advantage of the DDF scheme compared to the multi-speed scheme is to increase the numerical stability, and therefore it is widely used. The hybrid computational scheme combines the LBE and Finite difference (FD) or Finite volume (FV) methods [5]. In this paper we use DDF scheme. Figure 1 – The considered area

76 Lattice Boltzmann method simulation of thermal flow dynamics in a channel

The governing equations can be written as [6]: Each cell contains a fixed number of distribution functions, which represent the number of fluid

2 particles moving in these discrete directions. up1  u i uu  i F , Depending on the dimension and the number of ij x t x  x  xx directions of velocity, there are various models that jij j can be used. In the present study, a two-dimensional u flow and a two-dimensional square lattice with nine i  0, discrete velocities (D2Q9 model) are examined. For x each velocity vector, the value of the distribution i function is stored. In the D2Q9 model (Figure 2), 2 TT  the velocities are calculated using the formulas uT k . j       t xjj xxj eee01(0,0), (1,0), 2 (0,1),

ee34 ( 1,0), (0, 1),  Here, is the linear function of the temperature ee56 (1,1), ( 1,1),

T and Fx is the body force ee78 ( 1, 1), (1, 1),

     0   (  TT 00 ), where xc / t and k – lattice velocity direction. 2 x  0 uF max /8 H , where 0 is the average density, T0 is the average temperature,  is the coefficient of thermal expansion. We assume that at the initial time, the velocity and temperature in the channel are zero. Periodic boundary conditions are used at the channel inlet and outlet for u 21 ,, Tu , p . And the following boundary conditions are applied on channel walls

u(, x x 0,) t u (, x x Ht ,) 112 112 Figure 2 – D2Q9-model  uxx212( , 0,) t uxx212 ( , Ht ,) 0,

Tx(12 , x 0, t ) Tcold , Distribution functions are calculated by solving the lattice Boltzmann equation, which is a special Tx(12 , x Ht ,) Thot . discretization of the Boltzmann kinetic equation. Numerical method After introducing the Bhatnagar – Gross – Crook approximation, we can formulate the Boltzmann The lattice Boltzmann equation (LBE) method is equation in the form [10] a discrete model of a continuous medium. Currently, Δt the LBE method may well compete with traditional f (x  e Δt,t  Δt) f (x,t) f eq(x,t) f (x,t) , methods of computational hydrodynamics, and in ii i τ i i some areas (flows in a porous medium, multiphase and multicomponent flows) it has significant where Δt denotes the lattice time step, ei is the advantages [7-9]. By this method an intermediate discrete lattice velocity in the direction i, τ denotes scale model is used to simulate fluid flow. It applies eq simulation of the motion of fluid particles in order the lattice relaxation time, fi is the equilibrium to capture the macroscopic parameters of the fluid. distribution function. The equilibrium distribution The area is discretized by uniform cartesian cells. functions are calculated by the formula

International Journal of Mathematics and Physics 10, №1, 75 (2019) Int. j. math. phys. (Online)

A.S. Zhumali et al. 77

 2 2  where xB is the coordinates of the boundary node, eq e u 1 e u 1 u f   1 i  i   f is the pre-streaming distribution function with i i  2 4 2  i cs 2 cs 2 cs   discrete velocity ei , which points into the boundary.

fi is the post-streaming distribution function in the where the values of the weight coefficients i are direction opposite to e . For the constant as follows: i temperature boundary, the distribution function can 4/ 9, i  0,  be obtained as i  1/9, i 1,2,3,4,        1/36, i  5,6,7,8, gi (xB ,t t) gi (xB ,t) 2 iTB ,

where T is the temperature of the top or bottom and cs  c / 3 is the lattice speed of sound. B The macroscopic variables for the density and wall and i is the weight coefficients. velocity of a fluid are calculated as the first two Algorithm for applying the lattice Boltzmann moments of the distribution functions for each cell: method: 1. Discretization of the physical domain and 8 1 8 nondimensionalization of the related parameters;   f , u  f e . 2. Choice of simulation parameters;  i   i i i0 i1 3. Domain initialization; 4. Collision step; For the temperature field, the distribution g is 5. Application of the boundary conditions; 6. Streaming step; 7. Calculation of the macroscopic parameters; g (x e Δt,t  Δt)  ii 8. Verification of convergence criteria. If Δt  eq  criterion is met, then end of routine, else go to 4.  gi(x,t)  g ii (x,t) g (x,t). τg The equilibrium distribution functions for the Analytical solution and numerical results temperature field are determined by the formulas: In order to check the developed algorithm for  2 2  solving the problem of thermal flow, the problem of eq e u 1 e u 1 u two-dimensional Poiseuille flow in channel is g   T 1 i  i   i i  2 2 4 2 2  solved.  cs cs cs  Table 1 sets the parameters for calculating the test problem. The simulation was performed with The temperature field is calculated by the different sizes of the computational domain: formula Nx  N y 10050, 200100 . The maximum 8 velocity umax in the channel and the sound speed cs T   gi. are 0.1 and 0.5773, respectively. Kinematic i0 viscosity v = 9.021 · 10–3. The height of the channel As the no-slip boundary condition in fixed walls H = 1. The Reynolds number Re = umax · H/v ≈ 10. of channel the mid-link bounce back scheme is used In the present study the Prandtl number [6,11]. According to the scheme, the wall boundary  is located half lattice away from the boundaries of Pr   0.7 . Relaxation parameters are defined as the fluid nodes. At the boundary node, the outward k post-streaming distribution is equal to the inward  k [12]    0.5 and  g   0.5 . pre-streaming distribution: c2 t 2c2t s s Along the X axis the constant pressure f (x ,t  t)  f (x ,t), i B i B difference is maintained:

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78 Lattice Boltzmann method simulation of thermal flow dynamics in a channel

p 8u The comparison of the exact solution with the  max ,  2 results of the numerical solution is observed in xx inout (  yy bottop ) Figures 3 – 5. L1 and L2 norms of error were calculated by where p is the pressure difference, the following formulas:  p  pp inout , pout and pin are the pressure at the outlet and at the inlet of the channel,  q n ( x , x 21 ,  qt b () x , x 21 , t ) respectively,  is the dynamic viscosity, x and ,xx 21 out  L t)(  , 1 ( xq x ,, t ) xin are the outlet and the inlet boundaries,  b 21 ,xx 21 respectively, ytop and ybot are the top and the bottom walls boundaries, respectively. 2  q n x x 21  qt b x x 21 ,,(),,( t )

,xx 21 Table 1 – Simulation parameters  t)(  . L2 2  b ( xq x 21 ,, t )

Parameters ,xx 21 scaling factor, scale scale  1:2

number of points along the x axis Nx N x  100  scale where the index n means numerical solution and the index b means analytical solution. number of points along the y axis, Ny N 50  scale y Increasing the grid resolution helps to reduce the relaxation parameter,    3 /16 0.5 error norms. If we assume that the error norm is

maximum velocity in a channel, umax umax  0.1/ scale known for different grid sizes and their ratio of the kinematic viscosity, v   (2  1) / 6 sizes of each grid to the initial one is m, then we can determine the order of accuracy using the following Reynolds number, Re Re 10 formula: Prandtl number, Pr Pr 0.7

channel outlet pressure, pout pout  1  0 t)(  tn  log)( m   .   m t)(  The analytical solutions for the velocity and the temperature fields are calculated as [12,13]: In the test problem, the error norms were calculated at time  Lt 2  1/ . The accuracy orders  y2  of the numerical algorithm depending on the grid )( yu u 1)   , exact max  2  size are presented in table 2.  L  Figure 3 shows the predicted cross-sectional profile in the force-driven channel flow for velocity ()TT and comparison between LBM simulation and T() yT  top bot y exact bot H analytical solution. Solid line is the analytical  4  . solution and the symbol is the numerical result. 1 2  2y   Prumax  1   1  3   H  

2 Table 2 – The accuracy orders of L1 and L2 of the velocity, depending on the grid size at t ν⋅/ L =1

Grid size uL()1 Order of accuracy, n uL()2 Order of accuracy, n 100 50 4.6600 104 3.8978 6.8264 102 1.9489 200 100 1.6972 104 3.8175 4.1198 103 1.9087

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A.S. Zhumali et al. 79

Also, Figure 4 shows the temperature cross- numbers. Here, solid lines are the analytical sectional variation at different time instants in solutions and the symbols are the numerical results. comparison with the analytic solution. Solid lines As can be seen from the figures, the numerical are the numerical results and the symbol is the results agree well with the analytical solutions. The analytical solution. And Figure 5 demonstrates the general results in terms of streamwise temperature temperature cross-sectional profiles in comparison and velocity for time instants t  0.1 and t 1 are with the analytic solution for different Prandtl shown in Figures 6 and 7, respectively.

Figure 3 – Velocity profile of a 2D Poiseuille flow. Comparison of the exact solution with the result of the numerical solution at

TTtop  1,bot  0, Pr 0.7, umax  0.1.

Figure 4 – Temperature variation of a 2D Poiseuille flow. Comparison of the exact solution with the result of the numerical solution

at TTtop  1,bot  0, Pr 0.7, umax  0.1

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80 Lattice Boltzmann method simulation of thermal flow dynamics in a channel

Figure 5 – Temperature profiles of a 2D Poiseuille flow. Comparison of the exact solution with the result of the numerical solution at

TTtop  1,bot  1, Pr 0.7and Pr 1.5, umax  0.1

(a) (b) Figure 6 – Streamwise temperature at Pr 0.7 , Re 10 and computational time t  0.1(a) and t 1(b)

(a) (b)

Figure 7 – Streamwise velocity at Pr 0.7 , Re 10 and computational time t  0.1(a) and t 1(b)

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A.S. Zhumali et al. 81

Conclusion 4. Zhaoli G., Chaguang Z., Baochang S. "Thermal lattice Boltzmann equation for low Mach The basic aim of this paper is the development of number flow," Physical Review E 75 (2007): mathematical model for thermal flow in a channel 036704 and the implementation of numerical simulation of 5. J. Wang, L.-P. Wang, Z. Xiao, Y. Shi, and the problem by the Lattice Boltzmann method S. Chen, “A hybrid numerical simulation of applying the D2Q9 model. The validity of this isotropic compressible turbulence,” J. Comput. method is tested by comparing the numerical Phys. 229, 5257 (2010). solution to the analytical solution of the planar 6. Chao Li, Lian-Ping Wang. “An immersed channel flow. The comparison of the exact solution boundary-discrete unified gas kinetic scheme for with the numerical solution for test problem of simulating natural convection involving curved thermal Poiseuille flow given in Figures 3–5 shows surfaces” Int. J. of Heat and Mass Transfer. – a very good agreement and relationship. It is (2018): 1059-1070. determined that the numerical method has a second 7. McNamara G.R., Zanetti G. “Use of the order of accuracy in time. This means that the Boltzmann equation to simulate lattice-gas developed algorithm may well be applied to solving automata” Phys. Rev. Lett. vol. 61, no. 20 (1988): the problem of the dynamic thermal flow in a three- 2332–2335. dimensional region. This result will be obtained and 8. Chen S., Doolen G.D. “Lattice Boltzmann shown in a future research. method for fluid flow” Annu. Rev. Fluid Mech. –. V. 30. P. 329–364. References 9. Nourgaliev R.R., Dinh T.N., Theofanous T.G., Joseph D. “The lattice Boltzmann equation 1. Y. Peng, C. Shu, Y.T. Chew, “Simplified method: theoretical interpretation, numerics and thermal lattice Boltzmann model for incompressible implications” Int. J. of Multiphase Flow vol. 29, no. thermal flows”, Phys. Rev. E 68 (2) (2003) 1 (2003): 117–169. 026701.1–026701.8. 10. Kruger, T., Kusumaatmaja, H., Kuzmin, A., 2. A.D’Orazio, M. Corcione, G.P. Celata, Shardt, O., Silva, G., Viggen, E. M. “The Lattice “Application to natural convection enclosed flows Boltzmann Method.” Springer (2017): 694. of a lattice Boltzmann BGK model coupled with a 11. Portinari M. “2D and 3D verification and general validation of the lattice boltzmann method” Diss purpose thermal boundary condition”, Int. J. Therm. École Polytechnique de Montréal (2015): 129. Sci. 43 (6) (2004): 575–586. 12. C.-H. Liu et al. “Thermal boundary 3. P. Lallemand, L.-S. Luo, “Theory of the conditions for thermal lattice Boltzmann lattice Boltzmann method: Acoustic and thermal simulations”, Computers and Mathematics with properties in two and three dimensions”, Phys. Rev. Applications 59 (2010): 2178–2193. E 68 (3) (2003) 036706.1–036706.25. 13. F.M. White, “Viscous Fluid Flow,” 3rd ed., McGraw-Hill, New York (2006).

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 75 (2019)

International Journal of Mathematics and Physics 10, №1, 82 (2019)

IRSTI 44.39.29

1R.K. Manatbayev, N.Kalasov1,*, B. Bektibai1, Ye.Nurymov1, K.Baktybekov2, A. Syzdykov2, E.M. Zulbukharova1 1 al-Farabi Kazakh national university, Faculty of physics and technology, Dept. of Thermal physics and Technical Physics, Almaty, Kazakhstan 2 Kazakhstan Gharysh Sapary, Astana, Kazakhstan *e-mail: [email protected] Definition of the basic geometric parameters of a carousel type turbine its technological solution

Abstract. In the last 10-15 years, the use of wind energy has been developing rapidly. To date, more than 20,000 wind power units have been installed in the world, the total capacity of which exceeds several MW. Kazakhstan has significant resources of wind energy. The most known in this respect are the resources of the Dzungar Gate and Shelek Complex, located in the Almaty region (Kazakhstan). Their possibilities for use in the generation of electric power of air flows are unique. In this paper, the main calculations for determining the influence of the design characteristics of the Darrieus wind turbine on its energy efficiency are presented. The dependence of the maximum coefficient of wind energy use of vertical axle wind wheels on the number of blades at a constant filling factor σ, on the number of blades at their constant width, on the lengthening of the blade λ is studied. Based on these results, design characteristics for a rotor with a power of 1 kW are determined, and a diagram of the wind turbine which can provide thermal protection of the wind turbine by using natural ventilation of warm air inside the rotating elements of the windmill arising from centrifugal forces is also given. And another way of solving the technological problem is proposed. Renounce a long rotating shaft and replace it with a short axis, on which two bearings will rotate with a wind wheel attached to them. Application of the proposed design of the wind farm will simplify the manufacturing technology. Key words: wind turbine, Darrieus, aero dynamical characteristics, natural ventilation, geometrical parameters

It is known that when designing a wind turbine, Introduction the designer needs to know the effect of the main parameters such as blade lengthening, the number of The development of a reliable method of blades, blade thickness and filling factor on rotor pro- protecting a working wind turbine from adverse ductivity. weather conditions is relevant. In this connection, Currently, the main source of information for in this paper, we propose the development of a wind designing wind turbines with the Darrieus rotor is turbine with an anti-icing system. This uses natural an experiment. The most complete and comprehen- ventilation of the flow elements of the wind turbine sive results of experimental studies were published with warm air, which does not allow the sticking of in [2-4]. wet snowflakes on the surface of the apparatus and In order to develop a semi-industrial version of the formation of icing.�� [�1�]� ��The present work is devot- the wind turbine, studies were conducted to determine ed to the study of the effect of geometric parameters the geometric parameters of the turbine. Thus, the of a turbine on its performance and some technologi- calculations of the influence of design parameters cal solution. on the energy characteristics of the turbine being

© 2019 al-Farabi Kazakh National University Int. j. math. phys. (Online) R.K. Manatbayev et al. 83 developed with an anti-icing system have been decrease. This is especially pronounced when the performed. number of blades is increased, their chord is reduced to maintain the constancy of the filling factor σ. It is Determination of geometric parameters of the more effective with increasing the number of blades rotor influencing its energy characteristics to maintain the length of the chord. Fig. 2 shows the results of calculations for the 1) Effect of lengthening the blade. The study of the influence of the number of blades at a lengthening of the blade is one of the main constant filling factor. It can be seen that as the number parameters of the design of the Darrieus rotor, which of blades increases, the wind energy utilization factor determines its aerodynamic characteristics. The decreases. levels of aerodynamic loads, including the torque are depending from the magnitude of elongation. For a single blade, the nature of this dependence is almost the same as for the wing. For elongations λ<1, the magnitude of aerodynamic loads varies linearly from λ. With increasing elongation, these loads approach their asymptotic values for λ> 5. It follows from Fig. 1 that it is necessary to use blades with an elongation greater than 6 to provide an acceptable coefficient of wind energy utilization.

Figure 2 – Dependence of Cp on the number of blades at a constant filling factor σ=const, l=3.3 m

Fig. 3 shows that with the blade width unchanged,

the efficiency of Cp falls less significantly than with constant filling with increasing number of blades.

Figure 1 – Dependence of Cp on the elongation of blades n=2, b/D=0.167, R=1.65 m, l= variable

2) The influence of the number of blades. Another important parameter of the design is the number of blades of the Darrieus rotor. To assess the influence of the number of blades on the energy characteristics of the rotor, special studies are carried out on rotor models with different number of blades. The study shows that the single-blade rotor has the highest energy characteristics. But in this case, the torque is experiencing large pulsations in time, which gives rise to a bunch of dynamic problems. Figure 3 – Dependence of Cp on the number To smoother torque of the rotor we may increase of blades with a constant width the number of blades, but energy efficiency will b=const, D=3.3m, l=3.3m

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 82 (2019) 84 Definition of the basic geometric parameters of a carousel type turbine its technological solution

3) Effect of blade thickness. The effect of the b is approximately 2.1. The cross-sectional area 2 relative thickness of the blade profile on the value of f1=0.0154 m . the maximum value of the coefficient Cp for different Thus, geometric parameters influencing the ener- Reynolds numbers Re is shown in [5]. gy characteristics of a turbine with thermal protection The greatest effect is achieved for blades with a were determined, and internal hydraulics and warm relative thickness of 0.15

Cp for thin blades. It should be noted that the same character of the dependence on the relative thickness The limitation of carousel type wind turbine of the profile is observed for the traction force created and its technical solution by the flapping wing [2]. 4) Influence factor effect. The filling factor is The main disadvantage in the vertical position: associated with two parameters of the rotor design: the vertical axis of rotation and its vertical mounting the number of blades nb and the ratio of the chord of in the bearing supports. The manufacture of supports the blade to the diameter of the rotor b/D. It should is also a complex technological task; it is required to be noted that with an increase in the filling factor σ, ensure the alignment of the supports and their rigid- the value of the rapidity of z decreases, at which Cp ity. Also, a disadvantage of the design is the large reaches its maximum. mass of the shaft, which presses on the lower rota- In order to have 1 kW Darrieus rotor with direct tional support, creates a large friction torque, which blades with thermal protection, with an average wind reduces the efficiency of the installation [8]. speed U of 7 m / s, we find the streamlined surface of It is clear that the task of making such a shaft the rotor according to the formula: and its installation in a hull is a complex task. As a solution to this problem, a modular system is used. It (1) is made from several parts-modules, and then assembled in place in one unit. where Nb – power, W; Cp – wind utilization coefficient In this paper, we propose another method for solv- is equal to 0.4; ρ – 1.29 kg/m3 – air density. ing this complex technological problem. Renounce a The streamlined area of the rotor with a power of long rotating shaft and replace it with a short axis, on 1 kW should be 11.3 m2. If we assume that the rotor which two bearings will rotate with a wind wheel at- diameter D and the blade height l are equal, then D = tached to them. Application of the proposed design of l≈3.36 m. With an experimentally valid lengthening the wind farm will simplify the manufacturing tech- of the blade λ = l / b = 6-8, the length of the chord of nology. In the old design, it was necessary to have a the blade can be b =3.36/(6-8)=0.56-0.42 m, and the long rotating vertical shaft, which must be fixed in chord, on average, will be b = 0.55 m. two places below and above. For a 1 kW wind tur-

Since the maximum value of the coefficient p C bine, a shaft of about 8 m in length is required. The for different Re values is achieved for blades with length of the shaft is almost equal to the length of a relative thickness of 0.15

International Journal of Mathematics and Physics 10, №1, 82 (2019) Int. j. math. phys. (Online) R.K. Manatbayev et al. 85 tively short body 1.0 m long. Bearing misalignment bled on the mounting housing assembly on the key- can also have a sufficiently large value of 1-2 mm. way and fixed with a special nut to the rotating shaft This error will be compensated by the elasticity and 1. All assemblies can be mounted on supports of vari- flexibility of the blade design itself. ous designs and heights.

Figure 4 – Fixed rigid support

The shaft does not require high structural requirements, since the shaft does not have original surfaces. All these factors greatly simplify the technology of manufacturing a wind power plant, by about 25- 30%. It should be noted that it is the problem of a long vertical shaft that is the main one, which con- strains the wide application of rotor wind turbines 1 – rotation shaft; 2 – rigid case; 3 – hubs’ in practice. Also, the mass of the shortened structure 4 – wind turbine mounting unit; 5 – wind turbine blade; with the hollow internal shaft is much smaller than 6 – generator; 7 – electric drive of the vertical movement the mass of the long shaft; therefore, the force of their of the working body; 8.9 – special nut; 10 – washer; weight will create a much smaller friction torque in 11 , 12 – bearings. the lower rotational support, which undoubtedly will Figure 5 – Assembly design wind turbine increase the efficiency of the installation. This solution can be used in any scheme of a wind farm with a vertical axis of rotation of the rotor, for example, Blades 5 windmill is made of aluminium and the scheme of Savonius, Darrieus, Evans, Musgrove, welded to the guiding body 4 by argon welding. carousel, etc. [8, 9]. The guide body assembly is welded to the hub 3 Wind installation with a shortened construction according to the working drawing of the rotor assem- from the point of evaluation for manufacturability bly. The shortened rigid body 2 is made of a standard has a number of advantages, since the design is ori- thick-walled tube of steel CT 3. Two standard bear- ented to manufacturing in small shops, which gives a ings 11 and 12 are used in this design. For its intend- reduction in the technological cost. This design also ed purpose, the thrust bearing 11 is thrust, since it has the advantage of being transported to remote ar- is affected by the weight of the rotor assembly. The eas for the use of farms and in small businesses. lower radial single row bearing 12 operates on the The assembly structure consists of 12 parts (see rotational movement of the torque transmission from Fig.5.). A rigid shortened body is assembled on the the pulley 6 to the generators generated electrical en- mounting housing 7. The rotor assembly is assem- ergy.

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 82 (2019) 86 Definition of the basic geometric parameters of a carousel type turbine its technological solution

A 2 m long chord and a 0.5 m chord will be made productivity, time spent on technological preparation of aluminium AD0 according to GOST 4784-97 or of production, manufacturing, maintenance and aluminium grade 1050 according to EN 573-3 with repair of the product. Therefore, the design of the a thickness of 2 mm. Using tabular data with x val- technological process for the manufacture of parts ues, the distance from the profile’s bottom (in relative must be preceded by an analysis of the technological units, from 0 to 1, or percentages), yt is the coordinate nature of its design and, if necessary, processing for of the top point and yb is the coordinate of the bottom manufacturability. Technological design of the details point of the profile (also in relative units or percent- of the wind power plant is estimated at two levels – ages). For the symmetrical wing profile NASA-0021, qualitative and quantitative. A qualitative assessment the volume of the blade of the installation was de- and quantitative demonstration of the conformity of termined. The density of aluminium is ρ = 2712 kg / the design of parts during the analysing the designs m3. Thus, the mass of one blade of the symmetrical of a wind power plant should met the following wing profile NASA-0021 is m = 11.366 kg. Since we requirements: have 2 blades, the total mass of the blades will be m – the design of the wind turbine shall be standard = 22.732 kg. or consist of standard and unified structural elements; Swing length of 1 m and chord 0.5 m, as well as – standard and unified blanks are used for the the blade, will be made of aluminum AD0 according manufacturing of details of the wind turbines; to GOST 4784-97 or aluminum grade 1050 accord- – dimensional accuracy and surface roughness ing to EN 573-3 with a thickness of 2 mm. Using the of wind turbine components are optimal, reasonably previous method, the mass of the fly was also deter- structurally and economically; mined, which is equal to m=11.366kg. – in determining the rigidity, shape and size, The shaft of rotation of the installation with a as well as the mechanical and physicochemical length of 1.334 m and an inner diameter dв=24mm properties of its material, the components of the will be made of structural alloy steel grade 30ХГСА, windmill were taken into account the capabilities density of which is equal to ρ=7850kg/m3. With the of the manufacturing technology, storage and help of the drawing, the volume of the rotation shaft transportation conditions; was calculated. Multiplying the volume of the shaft – accuracy and roughness of the surfaces of by the density of the material, we determined the the components of the windmill ensure the required mass of the shaft m = 1.688 kg. accuracy of installation, processing and control; Using the drawings, the mass of the outer casing – the procurement of the components of the was determined, which will also be made of struc- wind turbine must be obtained in a rational way tural alloy steel grade 30ХГСА, density of which is (taking into account the volume of output and the equal to ρ = 7850 kg/m3. The mass of the outer tube type of production); is m = 17.34 kg. – in all designs of wind turbine parts, access As a result, having combined the masses of all to the surfaces to be treated and the possibility of the parts of the installation, the mass of the actual simultaneous processing of several blanks; installation was determined to be m = 53.126 kg [10]. Based on the analysis of the initial information of the wind turbine (the assembly drawing of the unit, Conclusions the drawing of the part, the program and the annual volume of output, the type of production, the service The dependence of the maximum coefficient of purpose of the unit and the part), one can draw a wind energy use of vertical axle wind wheels on the conclusion on the expediency of a fundamental number of blades at a constant filling factor σ, on change in the method of obtaining the initial billet. the number of blades at their constant width, on the The correspondence of the surface, which lengthening of the blade λ is studied. The geometric will be used as technological bases, is revealed, characteristics of the blade and fly are obtained, and compliance with their requirements to the which are the main parts of the rotor, which affect its technological bases of the billet is checked. The total energy efficiency. mass of a real installation is determined using the Technological design of the components of the drawings. Also, for each part, a selection of materials wind power plant has a direct relationship with labor was made from which the plant will be manufactured.

International Journal of Mathematics and Physics 10, №1, 82 (2019) Int. j. math. phys. (Online) R.K. Manatbayev et al. 87

Nomenclature a vertical axis of rotation, Thermal physics and aero λ – Elongation; mechanics, 8, 329-334 (2001)

Cp – wind energy utilization coefficient; 4. A.K.Ershina, R.K.Manatbaev, Organization n – Number of blades; of natural ventilation inside the working Darrieus b – Chord of the blade, m; turbine, Small energetics, 1-2, 63-66 (2013) D – Diameter, m; 5. A.K.Ershina, R.K.Manatbaev, Determination R – Radius, m; of the hydraulic resistance of the symmetrical wing l – Blade’s height, m; profile NASA-0021, KazNU bulletin: mathematics, σ – Filling factor; mechanics and informatics section, 51, 56-58 (2006) c – Relative thickness; 6. K.J.Thuryan, D.Strickland, D.E.Berg, Power Re – Reynolds number; of wind turbines with a vertical axis of rotation, Aero U – average wind speed, m/s cosmic technics, 1988 Ф – perimeter of the wing, m; 7. D.N.Gorelov, Aerodynamics of vertical axed 2 f1 – cross sectional area, m . wind turbines, KAN Polygraph center, 2012 8. D.DeRenzo, I.I.Shefter, Wind energy, References Energoatomizdat, 1982 9. Manatbayev R.K., Yershina Sh. A., Yershina 1. R.K. Manatbayev, A. Georgiev, R. Popov, A.K.Method for Thermally Protecting Operating D. Dzhonova-Atanasova, А.А. Kuikabayeva, E.M. Revolving Type Wind-Power Installation, Involves Zulbukharova. The effect of design parameters Causing Internal Natural Ventilation in Wind on energy characteristics of Darrieus rotor. Unit by Centrifugal Force to Heat All Elements International journal of mathematics and physics, of Wind-Power Installation by Warm Air. Patent 74, 94-98 (2016). Number(s):RU2008137251-A; RU2447318- 2. D.N.Gorelov. The analogy between the flap- C2International Patent Classification:F03D-003/00 ping wing and the wind wheel with the vertical axis Derwent Class Code(s): Q54 (Starting, ignition); of rotation. Applied mechanics and technical phys- X15 (Non-Fossil Fuel Power Generating Systems) ics, 50, 152-155 (2009). Derwent Manual Code(s): X15-B01A6 3. D.N.Gorelov, I.N.Kuzmenko, Experimental 10. Askarov E.S. Wind power plant with vertical evaluation of the limiting power of a wind wheel with axis of rotation of the wind wheel, KZ Patent№2219

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 82 (2019) International Journal of Mathematics and Physics 10, №1, 88 (2019)

IRSTI 29.03.77

1F.F. Komarov, 2*T.A. Shmygaleva, 2N. Akanbay, 2S.A. Shafii, 3A.A. Kuatbayeva 1Belarusian state university, Minsk, Belarus 2al-Farabi Kazakh National university, Almaty, Kazakhstan 3 Sh.Ualikhanov Kokshetau State university, Kokshetau, Kazakhstan *e-mail: [email protected] Optimization calculation algorithms on cascade and probabilistic functions and radiation defects concentration at the ionic radiation

Abstract. Work is performed within a cascade and probabilistic method which essence consists in receiving and further usage of the cascade and probabilistic functions (CPF) for various particles. CPF make sense to probability that the particle generated at some depth of h’ will reach a certain depth of h after n number of impacts. In work optimization calculation algorithms of the cascade and probabilistic functions (CPF) depending on interactions number and particles penetration depth is offered, concentration of radiation defects at ionic radiation for the purpose of reduction calculation time and quality. For calculation CPF and concentration radiation defects there is an area of result, border of this area and a calculation step. Borders and a calculation step selection automation is executed. Key words: Optimization, algorithm, calculation, cascade and probabilistic, ion, defect formation, binary search, ternary search, function.

Introduction of ions with substance problems, for example [5-8]. Losses of energy on ionization and electron shells In recent years much attention is paid to excitement of the environment atoms weren't questions of various processes mathematical considered, therefore the elementary CPF was used. modeling. Mathematical models development, At charged particles to substance interaction on their calculation algorithms, research objects allows to movement way there are continuous energy losses. describe many phenomena [1-3]. Modeling on the These losses result in strong dependence for both computer radiation defect formation processes in power the flying particles spectrum, and the primary solid bodies at radiation is considered by us their beaten-out atoms (PBOA) from penetration depth. various charged particles and computer modeling The interaction run on PBOA formation features on cascade and probabilistic functions and significantly depends on energy in this connection radiation defects for ions. Such works necessity is there was a need of receiving the physical and connected with a solid body defects generation and mathematical models considering real dependences evolution management problem, for receiving, of the elementary act various parameters on energy, eventually, materials with the set properties [4]. For depth. metals radiation by ions is in an efficient manner Earlier in most cases at concrete calculations the changes of such properties as the metal durability, elementary cascade and probabilistic function (CPF) corrosion resistance, fatigue, depreciation etc. In this was generally used, it isn't always justified as the direction the solid body radiation physics would be run of interaction depends on energy [9,10]. It is left rather academic occupation which isn't interest necessary to investigate the received CPF behavior to practical applications without researches. It is taking into account energy losses for ions, prove necessary to notice that a large number of works is properties which they have to possess both with devoted to radiation defect formation at interaction physical, and with mathematical points of view,

© 2019 al-Farabi Kazakh National University Int. j. math. phys. (Online) F.F. Komarov et al. 89 develop calculations algorithms and make CPF losses for each particles grade depending on energy calculations depending on interactions number and are known and described by analytical expressions, particles penetration depth, spectrums of primary in particular, by the Bethe-Bloch formula) [10]. beaten-out atoms and radiation defects Impacts happen to atoms, cores discretely. After concentration. crashes primary particles keep the movement Nowadays well-known methods are next: the direction. At the movement charged particles Monte Carlo method, the Boltzmann kinetic through substance their run depends on energy equation, Fokker-Planck’s equation, Lindhard’s through the interaction section which is calculated methods, Vineyard's methods, etc. The choice of for ions by Rutherford's formula [10]. Observations this or that method applicability is very complex depths are according to the tables of the ion- challenge as, on the one hand, because of similar implanted impurity spatial distribution parameters tasks complexity often it is necessary to apply too [12]. The calculated interaction section is many approximations which significantly worsen approximated by the following expression: the calculations accuracy to obtaining final result, and on the other hand – for obtaining qualitative 11 result it is necessary to overcome very great ()h  1 , (1) computing difficulties. Eventually, everything is 00a() E kh defined by a problem specifics and the researcher ability to apply this or that method to the specific 0, а, Е0, k – approximation parameters, 0=1/ objective solution. The cascade and probabilistic 0. method is one of options in theoretical methods From a recurrence relation for transition numerous calculation for the spatial and power probabilities distributions of the falling and secondary particles in the environment [9–11]. From this point of view it’s usage in scientific research is necessary.  h', hE ,   n 0 h 11  ,(2) Main results      n1(h ', h '', E 00 ) ( h '', h , E 0 )  1dh '' h' 00 a( E ' kh '')  The following physical model is given. Charged particle on the movement way continuously loses we receive expression for CPF taking into account the energy on ionization and excitement (energy energy losses for ions in the following look:

n    E0  kh  l ln   1  E0  kh  hh    E0  kh    n (h ,, hE0 )=   exp  *  (h  h) , (3) n! n  E kh      ak  0 0 0     where n – interactions number, h', h –ion generation functions taking into account energy losses for ions and registration depths, l=1/(0.ak). depending on interactions number and particles Calculations of cascade and probabilistic penetration depth were carried out on a formula:

(E kh ') ln 0 ()E kh 0 n (hh ')  E kh'  (4) ak hh '1 0 n(h ', hE . )   *exp  ln  . 0 i1i ak  E kh  0 00  0    

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 88 (2019) 90 Optimization calculation algorithms on cascade and probabilistic functions ...

For optimization CPF calculation algorithms E2max  depending on interactions number and particles Ck (E0 , h) W (E0 , E2 , h)dE2 , (7) penetration depth, vacancy clusters concentration Ec are used Stirling formulas [13]: 4mc2 mc2 n n 1 2 n! n e 2n . (5) E2 max  E1 ; 2  22 ()mc1 mc2 1 1 ln n! (n  )ln n  n  ln(2 ) . (6) E2max – the greatest possible energy acquired by 2 2 2 atom, – an ion rest energy. Ck(E0,h) is defined Ions spend the main energy part for ionization m1c and excitement the environment atoms (to 99%) and taking into account that a particle energy at h depth only 1% goes for atomic structure defects formation. is E1(h). As E1(h)=E0-E(h), that setting energy At charged particles with material interaction dot losses on ionization and excitement E(h), we defects, Frenkel's couples, big congestions the receive the corresponding observations h depths vacancy and interstitial atoms can be formed. from a Bethe-Bloch formula. The primary beaten- Radiation defects at ionic radiation concentration out atoms spectrum is defined by the following calculation is carried out on a formula [11]: ratio:

h n1  h  h  wE , E , hdh      1 2 W E0 , E2 , h   n (h ) exp  , (8) nn   (h) 0 hk2  2  1 2

where n0, n1 – initial and final value of interactions definition range. CPF  n ()h , entering expression number from a cascade and probabilistic function (8), has next appearance:

n   E   1  0  ln  0ak   1  E0   h   E0 kh    n (h)    exp   h , (9) n        n!0  E0 kh   0  ak     

1 24 1 24 1 (h)  10 (cm), 2  10 (cm).  1   n  n  1 2 0 0 0     aE0  kh 

 Section 2 is calculated by Rutherford's formula, d( E12 , E )/ dE 2   (,EE12 ) . (10) 1, 2 – run on an ion - atomic and atom - atomic  impact respectively; k– integer bigger units; E1

wE1, E2 ,h – PBOA spectrum in the elementary Substituting expression (10) in formulas (7), (8) act, E2 – primary beaten-out atom energy. The PBOA spectrum in the elementary act is we receive: calculated on a formula:

E h EE 2 max dE n1  h h'' dh  d 2max 2    CEhk (,)0  2   n (h ')exp EE E nn   (h ') 2max d Ec 2 0 hk 2 21 2

International Journal of Mathematics and Physics 10, №1, 88 (2019) Int. j. math. phys. (Online) F.F. Komarov et al. 91

Carrying out transformations, we come to the following expression:

h E (E  E ) n1  h  h  dh  d 2 max c     Ck E0 ,h   n (h )exp  , (11) E (E  E ) nn   (h) c 2 max d 0 hk2  2  1 2

where Еd – average energy of shift, Е0 – initial formula (11) with usage (9) it is impossible as in energy of a particle, Ес – threshold energy. each member of CPF there is an overflow.

To calculate radiation defects concentration on a Expression for  n ()h it is presented in the form:

   E     ln 0       1  E0  h'   E0  kh'  n (h',h, E0 )  exp lnn!nln0  ln    nln  h') (12)       0ak  E0 kh' 0  ak        energy of primary particle and penetration depth. When calculating CPF for ions there are All CPF calculations for a formula (4) have been difficulties consisting in approximating coefficients made on a С#, as the DBMS for dataful operation selection and in finding real area of result both MS SQL Server 2014 environment was used. CPF depending on interactions number and from calculations results for ions depending on penetration depth. The area of result is influenced by interactions number and particles penetration depth the flying particle and target atomic number, initial are given in figures 1,2.

0 4 8 12 16 32 33 34 35 36 3 n*10 -2

-4

-6

-8

-10 1 2 3 lg 

Figure 1 – CPF dependence on interactions number for the titan in iron at h=0,0001; 0,0002; 0,0003 (cm); E=1000 keV (1-3)

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 88 (2019) 92 Optimization calculation algorithms on cascade and probabilistic functions ...

1234567-4 0 h*10 ,cm

-2

-4

-6

-8

lg-10 1 2 3 -12

Figure 2 – Dependency ψ n (h’, h, E0), from h for aluminum in the titan at E0 =800 keV for n= 732; 2702; 5697 (1-3)

Due carrying out CPF calculations depending on Let's give the regularities arising due finding a impacts number and particles penetration depth of real range defined for CPF calculated depending on result area behavior and a calculation step penetration depth: regularities are revealed. Let's note some of them. 1. Calculations shows that a small atomic weight Result area behavior regularities for CPF calculated of the flying particle and small depths area of result depending on interactions number consist in the CPF depending on h is close to h, which following: corresponds h/. With increase in observation depth 1. With initial energy (the flying particle and a the area of result is displaced to the right and target same) reduction with the same depth the area narrowed. of result is narrowed and displaced to the left. 2. With initial energy of a particle (the flying 2. With the flying particle increase in atomic particle and a target same) reduction with the same weight the area of result finding is displaced to the observation depth the area of result is displaced to left relatively h/ and is narrowed. the right and narrowed. 3. With the flying particle big atomic weight the 3. With increase observation in depth for any CPF maximum value is displaced to the left flying particle and any target the area of result is relatively h/ already with small depths, and with displaced to the right. big depths the result is in narrow area. 4. Depending on the flying particle atomic 4. The narrowest area of result turns out with the number at the same value of depth h the area of flying particle big atomic weight and small target on result is displaced to the right. the end of a run. 5. At the flying particle atomic number great At the step choice following regularities take value the area of result is displaced to the right place: relatively h, corresponding h/ already with small 1. For the flying particle small atomic weight and depths and the area of result is considerably small depths the step is small (about 10-20), with narrowed. increase in observation depth it begins to increase. Step behavior regularities due CPF calculating 2. With the flying particle atomic weight gain in depending on particles penetration depth are the step respectively increases, reaching several revealed. hundred and even thousands. 1. For the flying particle small atomic weight the 3. With the flying particle big atomic weight and step is small, with increase in observation depth it small targets the step considerably increases. increases, and on the end of a run very strongly.

International Journal of Mathematics and Physics 10, №1, 88 (2019) Int. j. math. phys. (Online) F.F. Komarov et al. 93

2. With initial energy of a particle with the same Ternary search defines that the minimum or a observation depth (the flying particle and a target maximum can't lie either in the first, or in the last same) reduction the step also increases. third of area, and then repeats search on the 3. With the flying particle atomic weight increase remained two thirds. for the same observation depth the step increases at The concentration values calculated by a formula first gradually, then is very sharp. (11) have the following behavior: for the easy flying 4. Step on atomic number dependence tends particles curves increase, reaching a maximum, then increase. decreases to zero. With initial energy of a particle For automation and optimization finding CPF increase curves are displaced to the right. With area of result depending on interactions number, increase in threshold energy Ес concentration values penetration depths have been realized algorithms decreases and curves pass much below, transition Ternary [14] and Binary [15] searches. The Ternary through a maximum is carried out more smoothly. search algorithm has been modified taking into While energy Е0 =100 keV the curve decreases. Due account CPF specifics: it exists in limited area. In the flying particle increase in atomic weight the the existing algorithms the division coefficient equal value of function in a maximum point increases and, to 3 is used (ternary search). In the developed therefore, curves pass above while values of depths program complex the coefficient can vary. Binary decrease. Calculation algorithms optimization with (binary) search (it is also known as a halving and a formulas (5), (6) usage is performed. After carrying dichotomy method) - the classical element search out optimization in formulas (4), (11) it is visible algorithm in the sorted array (vector) uses array what a counting duration was considerably reduced, crushing on half. It is used in computer science, for example, for germanium in aluminum at E0 = calculus mathematics and mathematical 1000 keV, E1 = 120 keV calculation time was 1 hour programming. Ternary search is a method in 44 minutes. After optimization calculation time has computer science for finding the minimum or less than 1 minute. Calculations comparison results maximum of function, which at first strictly before optimization and after it is given in the increases, then strictly decreases, or on the contrary. table 1.

Table 1 – Definition range borders of radiation defects concentration for germanium in silicon at Ес=50 keV и Е0=1000 keV

4 h*10 , cm Ск, cm Е0,keV n0 n1 1 2 0,1 10476 1000 219 560 5' 1'' 5,3 17598 800 25146 27958 10' 2'' 10,6 29380 600 69624 74258 25' 3'' 15,8 51189 400 147578 154312 1h 7'' 18,9 77629 300 227841 236220 3h29' 15'' 19,9 90354 260 264188 273220 4h12' 20'' 20,9 107041 220 308961 318741 5h30' 25'' 21,8 124137 180 359803 368257 7h06' 35'' 22,3 123290 140 394307 403204 10h01' 1' 23,2 118373 100 474116 486299 12h41' 2' 23,9 50357 70 563193 575375 15h26' 7' 24,1 -20064 60 596160 608342 17h19' 10'

Here 1, 2 – calculation time before carrying out Calculations results of are given in figures 3-5 optimization and after it. and in tables 2-5 [16-21].

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 88 (2019) 94 Optimization calculation algorithms on cascade and probabilistic functions ...

1 18

15 2 Ck*10 , cm 12

6 2

3 3 0 0.0 0.5 1.0 1.5 2.0 -3 h*10 , cm

Figure 3 – Radiation defects concentration dependence on depth for the titan nitrogen ions radiation at Е0=1000 keV, Ес=50 keV(1), 100 keV (2), 200 keV (3)

14 5 3 2 4 1 12 2 Ck*10 , СМ 10

4 7 8 6 9 2 12 11 10 0 01234567 h*10-4, СМ

Figure 4 – Radiation defects concentration dependence on depth in the ionic radiation for nitrogen in silicon at Ес=50 keV; Е0= 1000, 800, 500, 200, 100 keV (1-5); Ес=100 keV; Е0= 1000, 800, 500, 200 keV (6-9); Ес=200 keV; Е0= 1000, 800, 500 keV (10-12)

International Journal of Mathematics and Physics 10, №1, 88 (2019) Int. j. math. phys. (Online) F.F. Komarov et al. 95

2,5 C *103 , cm k 2 2,0

1,5 1

1,0

0,5

0,0 01234567 h*10-4, cm

Figure 5 – Radiation defects concentration dependence on depth in the ionic radiation for nitrogen in silicon (1) and nitrogen in germanium (2) at Е0= 1000 keV; Ес=50 keV

Finding the radiation defects concentration area 3. With increase in primary particle initial energy of result at ionic radiation has allowed to find this at the same value of threshold energy and area’s behavior regularities. Let's note some of penetration depth, value of radiation defects them. concentration decreases. 1. With increase in threshold energy with the 4. The radiation defects concentration area of result same penetration depth value of radiation defects borders depending on penetration depth increases, the concentration considerably decreases, area of result borders change range fluctuates from 0 to 5000. borders don't change. 5. Depending on threshold energy at the same 2. Depending on penetration depth the radiation energy and the same penetration depth the border defects concentration value increases. don't change.

Table 2 – Radiation defects concentration definition range borders for nitrogen in the titan at Ес=50 keV, Е0= 1000 keV

4 h*10 , cm Ск, cm Е0, keV n0 n1 τ 0,1 453,93 1000 0 27 1″ 1,7 504,21 900 61 224 3″ 3,5 569,57 800 196 439 4″60 5,4 650,76 700 376 681 6″ 7,3 747,10 600 596 970 7″90 9,4 878,12 500 894 1341 9″ 11,6 1050,64 400 1286 1840 13″ 12,8 1165,35 350 1545 2142 14″ 14 1294,26 300 1846 2474 15″70 14,5 1352,72 280 1987 2648 17″ 15 1412,73 260 2138 2820 18″ 15,5 1473,16 240 2301 2995 19″ 16,1 1556,9 220 2514 3247 21″ 16,6 1612,65 200 2709 3461 22″ 17,2 1688,03 180 2967 3795 24″50 17,8 1746,76 160 3258 4105 26″ 18,4 1765,86 140 3588 4455 29″ 19 1695,9 120 3971 4885 31″45 19,6 1422,67 100 4422 5397 35″ 20,3 677,95 80 5071 6110 41″ 20,6 0 70 5406 6452 42″

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 88 (2019) 96 Optimization calculation algorithms on cascade and probabilistic functions ...

Table 3 – Radiation defects concentration definition range borders for nitrogen in the aluminum at Ес=50 keV и Е0=1000 keV

4 h*10 , cm Ск, cm Е0, keV n0 n1 0,1 296 1000 0 32 3,5 335 900 84 310 7,1 381,5 800 250 583 10,9 439 700 469 899 14,9 511 600 751 1277 19,1 605 500 1116 1742 23,6 734 400 1607 2344 26 817 350 1923 2723 28,5 919 300 2304 3175 29,6 970 280 2493 3396 30,6 1017,6 260 2677 3611 31,7 1074 240 2895 3864 32,9 1142 220 3154 4163 34 1205 200 3414 4463 35,2 1278 180 3728 4821 36,5 1361 160 4108 5253 37,7 1422 140 4505 5703 39,1 1487,2 120 5041 6300 40,4 1460 100 5630 6965 41,9 1270 80 6465 7894 42,6 976 70 6934 8414 43,4 413 60 7558 9102 44,1 -745 50 8207 9816

Table 4 – Radiation defects concentration definition range borders for nitrogen in the silicon at Ес=200 keV и Е0= 800 keV

4 h*10 , cm Ск, cm Е0, keV n0 n1 τ 0,1 66,71 800 0 32 01″15 6,1 73,10 700 598 972 02″30 12,5 79,67 600 1499 2060 05″42 19,3 85,05 500 2688 3428 10″49 26,5 84,91 400 4271 5228 46″56 30,3 79,16 350 5282 6307 1′ 01″10 34,2 64,25 300 6483 7634 1′ 17″50 35,9 54,08 280 7070 8249 1′ 24″09 37,6 39,79 260 7701 8926 1′ 31″41 39,3 19,77 240 8384 9700 1′ 44″40 41,0 0 220 9124 10461 1′ 52″80

Table 5 – Radiation defects concentration definition range borders for carbon in the titan at Eс=100 keV, Е0= 500 keV

4 h*10 , cm Ск, cm Е0, keV n0 n1 τ 0,1 213,66 500 0 30 1″ 2,1 238,34 400 132 337 4″ 3,3 251,38 350 269 541 7″ 4,5 255,62 300 439 769 9″ 5 253,24 280 520 881 13″ 5,5 246,35 260 610 994 17″ 6 232,94 240 705 1110 24″ 6,5 210,03 220 811 1252 26″

International Journal of Mathematics and Physics 10, №1, 88 (2019) Int. j. math. phys. (Online) F.F. Komarov et al. 97

7,1 176,16 200 951 1423 33″ 7,7 119,25 180 1109 1610 36″ 8,2 24,45 160 1256 1789 11″ 8,8 0 140 1456 2044 12″

Conclusions 2. Czarnacka, K., Komarov, F.F., Romanov, I.A., Zukowski, P., Parkhomenko, I.N. AC In work, analytical expressions of cascade and measurements and dielectric properties of nitrogen- probabilistic functions taking into account energy rich silicon nitride thin films. // Proceedings of the losses for ions from recurrence relations for 2017 IEEE 7th International Conference on transition probabilities are received. The Nanomaterials: Applications and Properties, NAP approximating expression entering a recurrence 2017. - No. 5. P. 657-669. relation is picked up and approximation coefficients 3. Komarov, F.F., Milchanin, O.V., are found, so that the theoretical correlation relation Parfimovich, I.D., Tkachev, A.G., Bychanok, D.S. was rather high [22-24]. Algorithms are developed Absorption and Reflectance Spectra of Microwave and optimization of calculation cascade and Radiation by an Epoxy Resin Composite with probabilistic functions taking into account energy Multi-Walled Carbon Nanotubes. //Journal of losses depending on interactions number and Applied Spectroscopy.-2017. – Vol. 84. – No. 4. –P. particles penetration depth, radiation defects 570-577. concentration is made at ionic radiation, calculations 4. Komarov F.F. Nano- and microstructuring are carried out. Behavior regularities on CPF area of of solids by swift heavy ions//Physics-Uspekhi. - result and calculation step depending on interactions 2017. – Vol. 187. – No. 5. –P. 465-504. number and particles penetration depth are revealed. 5. Komarov, F.F., Romanov, I.A., Vlasukova, It is shown that the area of result is influenced L.A., Mudryi, A.V., Wendler, E. Optical and significantly by the primary particle initial energy, Structural Properties of Silicon with Ion-Beam penetration depth, the flying particle atomic number Synthesized InSb Nanocrystals.// Spectroscopy.- and target. Automation and optimization area of 2017. – Vol. 120. – No. 1. –P. 204-207. result borders selection and calculation step with 6. Komarov, F.F., Konstantinov, S.V., ternary and binary search algorithms usage is Strel’nitskij, V.E., Pilko, V.V. Effect of Helium ion executed. Algorithms optimization on calculation irradiation on the structure, the phase stability, and the cascade and probable functions taking into the microhardness of TiN, TiAlN, and TiAlYN account energy losses, depending on interactions nanostructured coatings. //Technical Physics.-2016. number and particles penetration depth, – Vol. 61. – No. 5. –P. 696-702. concentration the vacancy clusters is performed in 7. Al'zhanova, A., Dauletbekova, A., the ionic radiation case with Stirling formula usage. Komarov, F., Akilbekov, A., Zdorovets, M. Results comparison of calculation for time before Peculiarities of latent track etching in SiO2/Si carrying out optimization and after it is executed. structures irradiated with Ar, Kr and Xe ions.-2016. The program complex is developed in the Microsoft – Vol. 374. – No. 1. –P. 121-124.// Nuclear Visual Studio 2015 environment in the C# Instruments and Methods in Physics Research, programming language. The database is created in Section B: Beam Interactions with Materials and the Microsoft Server 2014 environment. Atoms.-2016. – Vol. 426. – No. 1. –P. 1-56. 8. Komarov, F.F., Zayats, G.M., Komarov, References A.F.,Michailov, V.V., Komsta, H.Simulation of radiation effects in SiO2/Si Structures.// Acta 1. E.G. Boos, A.A. Kupchishin, A.I. Physica Polonica A.-2015. . – Vol. 3. – No. 1. –P. Kupchishin, E.V. Shmygalev, T.A. Shmygaleva. 854-860. Cascade and probabilistic method, solution of 9. Komarov, F., Uglov, V., Vlasukova, L., radiation and physical tasks, Boltzmann's equations. Zlotski, S., Yuvchenko, V.Drastic structure changes Communication with Markov's chains. -Monograph. in pre-damaged GaAs crystals irradiated with swift Almaty: KazNPU after Abay, Scientific Research heavy ions.// Nuclear Instruments and Methods in Institute for New Chemical Technologies and Physics Research, Section B: Beam Interactions Materials under al-Farabi KazNU, 2015. – 388 p. with Materials and Atoms.-2015. – Vol. 360. – No. 4. –P. 150-155.

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 88 (2019) 98 Optimization calculation algorithms on cascade and probabilistic functions ...

10. Kodanova, S.K., Ramazanov, T.S., Issano- 17. Nowak, S.H., Bjeoumikhov, A., von va, M.K., Golyatina, R.I., Maiorov, S.A Calculation Borany, J., Wedell, R., Ziegenrücker, R. Examples of ion stopping in dense plasma by the Monte-Carlo of XRF and PIXE imaging with few microns method.// Journal of Physics: Conference Series.- resolution using SLcam® a color X-ray camera //X- 2018. – Vol. 946. – No. 1. – P. 148-154. Ray Spectrometry. – 2015. – Vol. 44. – No. 3. – P. 11. Smyrnova, K.V., Pogrebnjak, A.D., 135-140. Beresnev, V.M., Kravchenko, Y.O., Bondar, O.V. 18. Bjeoumikhov, A., Buzanich, G., Langhoff, Microstructure and Physical–Mechanical Properties N., Soltau, H., (...), Wedell, R. The SLcam: A full- of (TiAlSiY)N Nanostructured Coatings Under field energy dispersive X-ray camera.// Journal of Different Energy Conditions.// Metals and Materials Instrumentation.-2012. - Vol. 7. – No. 5. – P. 248- International.-2018. – Vol. 24. – No. 2. –P. 241-439. 252. 12. Kozhamkulov, B.A., Kupchishin, A.I., 19. Bock, M., Skibina, J., Fischer, D., (...), Bitibaeva, Z.M., Tamuzs, V.P. Radiation-Induced Beloglazov, V., Steinmeyer, G.Nanostructured Defect Formation in Composite Materials and Their fibers for sub-10 fs optical pulse delivery.// Laser Destruction Under Electron Irradiation // Mechanics and Photonics Reviews.- 2013. - Vol. 7. – No. 3. – of Composite Materials. -2017. – Vol. 53. – No. 1. – P. 566-570. P. 59-64. 20. Dubovichenko, S., Burtebayev, N., 13. Kupchishin, A.I. On positrons irradiation in Dzhazairov-Kakhramanov, A., (...), Kliczewski, S., the defects of vacancy type.// IOP Conference Sadykov, T. New measurements and phase shift Series: Materials Science and Engineering.-2017. – analysis of p16O elastic scattering at astrophysical Vol. 184. – No. 1. –P. 234-248. energies.// Chinese Physics C.-2017. - Vol. 41. – 14. Kupchishin, A.I., Niyazov, M.N., Vorono- No. 1. –P. 485-495. va, N.A., Kirdiashkin, V.I., Abdukhairova, A.T.The 21. Kupchishin, A.I., Shmygalev, E.V., Shmy- effect of temperature, static load and electron beam galeva, T.A., Jorabayev, A.B.Relationship between irradiation on the deformation of linear polymers.// markov chains and radiation defect formation IOP Conference Series: Materials Science and Engi- processes by ion irradiation.// Modern Applied neering.-2017. – Vol. 168. – No. 1. –P. 229-274. Science.- 2015. – Vol. 9. – No. 3. – P. 59-70. 15. Kupchishin, A.I., Kupchishin, A.A., 22. Kupchishin, A.I., Kupchishin, A.A., Voronova, N.A. Cascade-probabilistic methods and Shmygalev, E.V., Shmygaleva, T.A., Tlebaev, K.B. solution of Boltzman type of equations for particle Calculation of the spatial distribution of defects and flow// IOP Conference Series: Materials Science cascade- Probability functions in the materials.// and Engineering. – 2016. Vol. 110. – No. 1. – P. Journal of Physics: Conference Series. – 2014. – 123-129. Vol. 552. – No. 2. – P. 89-96. 16. Kupchishin, A.I., Kupchishin, A.A., 23. Voronova N. A., Kupchishin, A.I., Voronova, N.A., Kirdyashkin, V.I.Positron sensing Kupchishin, A.A., Kuatbayeva A.A., Shmygaleva, of distribution of defects in depth materials. // IOP T.A., "Computer Modeling of Depth Distribution of Conference Series: Materials Science and Vacancy Nanoclusters in Ion-Irradiated Materials".// Engineering. -2016. - Vol. 110. – No. 4. – P. 248- Key Engineering Materials. – 2018. – Vol. 769. – P. 258. 358-363.

International Journal of Mathematics and Physics 10, №1, 88 (2019) Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 99 (2019)

IRSTI 44.31.31

1,*Z.Kh. Gabitova, 2M. Gorokhovski, 1A. Septemirova, 1A. Kalybekov, 1G. Bulysheva 1 Al-Farabi Kazakh National University, Almaty, Kazakhstan 2 Ecole Centrale de Lyon, Lyon, France *e-mail: [email protected] Numerical simulation of heat and mass transfer processes in combustion chamber of pk-39 boiler

Abstract. Over the past decades, the consumption rates of fossil energy resources has not declined, despite fundamental changes in the political, economic and social structures of society. According to experts, the share of coal in the global fuel and energy balance is more than 27%, thanks to which almost 45% of the world’s electricity is generated. For each country in the world, the structure of the national energy balance is determined by the availability of its own sources of fuel and energy resources. In the Republic of Kazakhstan, coal reserves are estimated at 30 billion tons, which is 3.4% of the world’s coal reserves. However, Kazakhstan’s coal is characterized by low calorific value and high ash content, its combustion leads to the formation and delivery of large quantities of pollutants in air, soil and water. Pollution of the atmosphere is one of the global problems of mankind, whose solution is to optimize the combustion process and realization stringent environmental requirements for specific emissions of harmful substances with waste gases. To solve this difficult task, there is a need to improve equipment, introduce new technologies and use alternative methods for organizing a combustion process, the basis of which is to study the processes of heat and mass transfer in the presence of combustion. Irreplaceable powerful method of theoretical research of currents at availability of burning is numerical modelling. The results of a three-dimensional numerical simulation of aerodynamics, temperature flow and carbon oxides are presented. Studies have been carried out for a pulverized-angle flame of various dispersity. A comparative analysis of the obtained results with the results of the field experiment is carried out. The obtained results will allow choosing the optimal variant of the combustion process organization in order to increase its efficiency and reduce the negative impact on the environment. Key words: simulation, coal combustion, aerodynamics, monodisperse flame, polydisperse flame, concentration fields.

Introduction systems, optimization of combustion processes and other) [1,2]. Numerical modeling is sufficiently accurate and At the present stage of development of the energy inexpensive way to analyze complex processes that industry, immediate consideration and resolution of occur during combustion of the fuel in the combustion environmental issues are required. Due to the fact, chambers of real power plants, and it allows to that for most countries the main sources of pollutant simultaneously consider the complex of processes emissions into the atmosphere are companies that are almost impossible to do, conducting in situ operating in the burning of low-quality raw materials experiments. Only the numerical modeling and as well as with poorly equipped with flue gas cleaning carrying out computational experiments optimally systems, the problem of pollution of the Earth’s solve scientific and project engineering tasks in this atmosphere is an urgent. Environmentally hazardous area (improvement, design of new boilers; burners emissions, which are products of coal combustion [3- upgrade; development of multistage fuel combustion 5] reactions cause enormous damage to the earth’s

© 2019 al-Farabi Kazakh National University Int. j. math. phys. (Online) 100 Numerical simulation of heat and mass transfer processes in combustion chamber of pk-39 boiler ecosystem. It is therefore necessary to carry out a are also divided by a pier. In the center of burners is detailed study of physical and chemical processes located oil-fired nozzle for lighting and lighting of that occur during combustion of energy fuels and to the flame. The performance of each of the eight coal- solve the problem of environmentally “pure” making fired burner fuels is 4 t/h. use of coal [6,7]. Computational experiments on research heat and mass transfer processes have been carried out Methodology of investigation by the starting FLOREAN [8] software package, the geometry of the combustion chamber was created For carrying out computational experiment the by a computer program «PREPROZ» (Fig. 1b). The combustion chamber of the real power boiler BKZ- software package FLOREAN was created to solve 160 Almaty TPP-3 (Kazakhstan) was selected. The problems in the field of burning solid fuel and was boiler BKZ-160 of drum-type furnace with dry slag repeatedly tested in many modern studies [1-12]. removal has a calculated steam generating capacity During the numerical simulation of heat and mass 160 t/h at a pressure of 9,8 MPa and a temperature transfer process, the control volume method has of the superheated steam 540 0C. The boiler has a been applied. Combustion chamber of a power boiler U-shaped profile with a rectangular prism furnace. BKZ-160 has been divided into control volumes; it is Combustion chamber volume is 790 m3. On the sides possible to obtain 217 536 computational areas. of the combustion chamber located four blocks direct Numerical simulation was carried out on the basis flow slot burners (two burners in the block) which of solutions of the Navier-Stokes equations, equations directed at a tangent to the circle with a diameter of of heat diffusion and diffusion of components of the one meter. Each burner has a fuel mixture channel reacting mixture and the reaction products based on and two secondary air channel, they are located from thermal radiation and multiphase media, equations above and from below the channel of air-fuel mixture, of state, and chemical kinetics equations defining the and divided lined piers. The top and bottom burners intensity of nonlinear energy and matter [8, 13-15].

A) B)

Figure 1 – A) Scheme of the furnace, B) General view of the camera, broken down into control volumes

power plant) in the present work a numerical For a qualitative description of combustion calculation of a turbulent pulverized coal flame was processes in a real three-dimensional physical and carried out taking into account the dispersion of chemical system (combustion chamber of Thermal coal. The percentage distribution of carbon particles

International Journal of Mathematics and Physics 10, №1, 99 (2019) Int. j. math. phys. (Online) Z.Kh. Gabitova et al. 101 in size: dp=10 μm– 10%; dp=30 μm– 20%; dp=60 heat shields and heat losses, which prolongs the life μm– 40%; dp=100 μm– 20%; dp=120 μm– 10% of individual elements of the boiler plant, and also corresponds to a polydisperse flame, dp=60 μm– increases the heat removal surface, which speaks of 100% – is the averaged diameter, which corresponds the advantages of the furnaces with the tangential to a monodisperse flame. Numerical calculation in the arrangement of the burners. The aerodynamics work was carried out for the two cases listed above. of flow in the combustion of monodispersed and polydispersed flames has some differences; however, Results of numerical simulation if it is necessary to make quick estimates, in numerical simulation of the aerodynamic characteristics of the Let us consider the profiles of aerodynamics coal combustion process, one can use the model of combustion of a turbulent pulverized flame in burning a particle of averaged size, which in turn different sections along the length of the flame. Fig. reduces the expenditure of computer time [1, 16-21]. 2 shows the distribution of the full-velocity vector Being the UNFCCC framework (the United in the longitudinal section of the furnace during Nations Framework Convention on Climate Change) combustion of a monodisperse and polydisperse since 1995 and the Kyoto Protocol since 2009, flames. Obtained velocity fields allow us to visually Kazakhstan has a principled position and pursues analyze the aerodynamics of reacting flows in the a consistent policy in the field of preventing global combustion chamber. The fields of the full-velocity climate change, in the field of reducing the carbon vector show the value of the flow velocity of the intensity of the economy and in the field increasing medium and its direction at each point. energy efficiency, creating conditions for the In the Fig. 2 the area of fuel and oxidizer is clearly transition to technologies for environmentally “pure” visible: counter dust and gas streams from opposing burning of energy fuel [22]. In this connection, tangential burners create a vortex in the central part the study of the concentration characteristics of on the location of burners and level of active burning greenhouse gases is an urgent task. zone. Clearly visible is the recirculation zone with Fig. 4,5,7 show a comparative analysis of carbon reverse gas currents [15]. Part of the flow is directed oxide concentration distributions for the case of a down to the funnel, forming two symmetrical polydisperse and monodisperse flare. Analyzing vortex in the area below the burner arrangement, it the Fig. 4 it can be argued, the nature and pattern is typical both for burning of a monodisperse flame of carbon monoxide СО and carbon dioxide СО2 and for burning of a polydisperse flame. However, are different from each other. Concentration of in a longitudinal section of the combustion chamber carbon oxide reaches area of the maximum values symmetry is broken relative to the vertical axis of the in a zone of active burning, unlike carbon dioxide chamber when burning polydisperse flame (Fig. 2b). which concentration increases as it moves out of the It means that burning of dust and gas streams with combustion chamber. different particle sizes affects to the character of the Concentrations of poly- and monodisperse flow stream. flames in the field of an arrangement of burners do In cross-section chamber at a level between not differ. The average value of the concentration of the lower and upper tiers of burners there is a clear carbon monoxide for polydisperse flame in the first picture of the current (Fig. 3). The pulverized coal tier of burners (z= 4,81m) is 0,184•10-2 kg/kg, for streams flowing into the chamber deviate from the monodisperse is 0,185•10-2 kg/kg, in the second tier direction of the burner axes (located tangentially) (z = 5,79m) is 0,279•10-2 kg/kg both for poly- and towards the adjacent walls, with which they make up for monodisperse flames (Fig. 4A). In the area of a smaller angle. Fusing into the total flow, the jets active burning the concentration of carbon monoxide create a volumetric vortex with a vertical axis of CO reaches the maximum value, chemical processes rotation, which, as it rises, untwists and then moves of formation of carbon monoxide CO fade to output along the axis, as can be seen clearly in Fig. 2. from the combustion chamber, for polydisperse flame The central vortical motion of the pulverized coal at the exit of the combustion chamber the mean value stream leads to uniform heating of the combustion is 1,35•10-4 kg/kg, for monodisperse is 0,61 10-4 kg/ chamber walls, to a decrease in the slagging of the kg (Fig. 4A).

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 99 (2019) 102 Numerical simulation of heat and mass transfer processes in combustion chamber of pk-39 boiler

A) B)

Figure 2 – Field of a vector of full velocity in the longitudinal section of the combustion chamber (x = 3,16 m) for A) monodisperse flame; B) polydisperse flame

А) B)

Figure 3 – Field of a vector of full velocity in the cross-section of the combustion chamber (z = 5,3m) for А) monodisperse flame; B) polydisperse flame

International Journal of Mathematics and Physics 10, №1, 99 (2019) Int. j. math. phys. (Online) Z.Kh. Gabitova et al. 103

0,005 0,20 A Polydisperse B Monodisperse , kg/kg 2 CO

CO, kg/kgCO, 0,004 0,15

0,003

0,10 0,002

0,05 0,001

Polydisperse Monodisperse 0,000 0,00 0 2 4 6 8 10 12 14 16 18 20 22 0 2 4 6 8 10 12 14 16 18 20 22 Height, m Height, m

Figure 4 – Comparison of the average values of concentration А) СО, В) СО2 for poly- and monodisperse flame on height of the combustion chamber

А) B)

Figure 5 – Distribution of the carbon oxide concentration in the longitudinal section of the furnace combustion chamber (y = 3,7m) for А) monodisperse flame; B) polydisperse flame

C 1450 o 1400 A B 1400 1200

Temperature, Temperature, 1350 1000 C� 800 o 1300

600 1250 Temperature � Temperature 400 1200 Polydisperse Polydisperse Monodisperse Monodisperse 200 1150 Experiment Experiment

0 1100 0 2 4 6 8 10 12 14 16 18 20 22 4,0 4,5 5,0 5,5 6,0 6,5 7,0 7,5 8,0 8,5 9,0 9,5 10,0 Height, m Height �m�

Figure 6 – Comparison of average temperature values for poly- and monodisperse flames and a comparison with the field experiment [23]: А) on height of the combustion chamber В) zone of active burning Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 99 (2019) 104 Numerical simulation of heat and mass transfer processes in combustion chamber of pk-39 boiler

А) B)

Figure 7 – Distribution of carbon dioxide concentration in a longitudinal section of the combustion chamber of the combustion (y = 3,7m) for А) monodisperse flame; B) polydisperse flame

Analyzing the distribution of CO concentration in possible to determine value of concentration in any the longitudinal sections of the combustion chamber point of furnace by a color scale of the received (Fig. 5), it can be said that in the active combustion figures which is not always possible to obtain during zone, there is a clear difference in the formation of the field experiments on the thermal power plant.

CO for a mono- and polydisperse flames, which So the average values of carbon dioxide СО2 in the indicates that the particle size has a significant effect longitudinal section of the combustion chamber (y on the formation of reaction products. The maximum = 3,7m) for polydisperse flame is 0,155 kg/kg, for values of carbon monoxide CO are explained by the monodisperse flame is 0,158 kg/kg (Fig. 7). At the exit intensive physical and chemical interaction between of the combustion chamber average concentration of the fuel carbon and air oxygen, and with increased carbon dioxide for polydisperse flame is 0,1876 kg/ temperatures in this region (Fig. 6). kg, for monodisperse flame is 0,1895 kg/kg. Fig. 6 shows the experimental points obtained directly from measurements at the thermal power Conclusion plant [23]. It is confirmed that the numerical simulation results are in good agreement with the In the present work, the calculation of results of a natural experiment. It is leading to the aerodynamics, thermal and concentration conclusion of the applicability of the proposed characteristics of the combustion of mono- and physical-mathematical model of combustion polydisperse flames is performed; the results of the processes, used in the present work. It should also study can formulate the following conclusions: be noted that the experimental data obtained directly – A detailed picture of the structure of the from TPP-3 lie closer to the temperature curve of the flame is obtained, which includes a developed polydisperse flame, from which it can be argued that recirculation zone with return currents of the the polydisperse flame model is more sensitive and combustion products; reflects a more real process of burning pulverized – It is noted that the character of formation of the coal at Almaty TPP‑3. concentration fields CO and CO2 is different. The Analyzing Fig. 4B and Fig. 7 it can be said that maximum concentration of carbon monoxide reaches as flow moves out of the combustion chamber CO2 in the zone of active combustion, and the formation is restored from CO, this regularity is fair both for of carbon dioxide CO2 increases as it moves towards monodisperse, and for polydisperse flames. It is the outlet from the furnace; International Journal of Mathematics and Physics 10, №1, 99 (2019) Int. j. math. phys. (Online) Z.Kh. Gabitova et al. 105

– The results of computer simulation of conduction using parametric identification.” temperature T were compared with the results of Measurement Techniques 54 (2011): 318-323. field experiments, the analysis of which confirms 4. Pilipenko, N. V. “The systematic errors the correctness of the chosen model of numerical in determining the nonstationary heat-exchange experiment. conditions with parametric identification.” In conclusion, we note that the nature of combustion Measurement Techniques 50 (2007): 880-887. of mono- and polydisperse dust has differences, i.e. 5. Askarova A. S., Bekmuhamet A., Maximov V. the influence of fineness of grinding has a significant Yu., et al. “Mathematical simulation of pulverized influence on the processes of heat and mass transfer coal in combustion chamber.” Procedia Engineering in the combustion chamber of CHPP boilers. The 42 (2012):1259-1265. combustion model of polydisperse dust more 6. Bolegenova, S. A., Maximov, V. Yu., A., accurately reflects the actual combustion process, Gabitova, Z. Kh., et al. “Computational method for which confirms the comparison with the full-scale investigation of solid fuel combustion in combustion experiment. However, the application of this model chambers of a heat power plant.” High Temperature requires large computer, time resources. The results 53, 5 (2015): 792-798. obtained in this study will give recommendations for 7. Beketayeva M., Ospanova Sh., Gabitova Z. K. optimizing the burning process of pulverized coal in “Investigation of turbulence characteristics of burning order to reduce pollutant emissions and creations of process of the solid fuel in BKZ 420 combustion power stations on ‘pure’ and an effective utilization chamber.” WSEAS Transactions on Heat and Mass of coal. Transfer 9 (2014): 39-50. Also in the future, the authors of this work 8. Müller, H. Numerische Berechnung are planing to conduct a numerical experiment to dreidimensionaler turbulenter Strömungen determine other concentration characteristics of in Dampferzeugern mit Wärmeübergang the turbulent combustion process (NOx, SOx, NH3, und chemischen Reaktionen am Beispiel des NCN), taking into account the fineness of grinding SNCR-Verfahrens und der Kohleverbrennung: fuel, which mutually complement each other and Fortschritt–Berichte VDI–Verlag, 1992 (in ensure the integrity of the entire study. German). 9. Bolegenova, S., Bolegenova, S., Bekmukhamet, Acknowledgment A., et al. “Numerical experimenting of combustion in the real boiler of CHP.” International Journal of The work was carried out within the framework Mechanics 7, 3 (2013): 343-352. of the financing of the Ministry of Education and 10. Askarova, A., Beketeyeva, M., Safarik, Science of the Republic of Kazakhstan. Grant No. P. et al. “Numerical modeling of pulverized coal 0115РК00599 combustion at thermal power plant boilers.” Journal of Thermal Science 24, 3 (2015): 275 282. References 11. Lavrichsheva, Ye. I., Leithner, R., Müller, H., Magda, A., et al. “Combustion of low-rank coals 1. Bekmukhamet A., Bolegenova S.A., in furnaces of Kazakhstan coal-firing power plants.” Beketayeva M.T, et al. “Numerical modeling of VDI Berichte 1988 (2007): 497-502. turbulence characteristics of burning process of the 12. Karpenko, Yu. E, Messerle, V. E., Ustimenko solid fuel in BKZ 420 140-7c combustion chamber.” A.B, et al. “Mathematical modelling of the processes International Journal of Mechanics 8 (2014): 112- of solid fuel ignition and combustion at combustors 122. of the power boilers.” 7th International Fall Seminar 2. Askarova, Maximov, V., Ergalieva, A., et on Propellants, Explosives and Pyrotechnics, Xian 7 al. “3-D modeling of heat and mass transfer during (2007): 672-683. combustion of solid fuel in Bkz-420-140-7C 13. Leithner, R., Numerical Simulation. combustion chamber of Kazakhstan.” Journal of Computational Fluid Dynamics CFD: Course of Applied Fluid Mechanics 9, 2 (2016):699-709. Lecture, Braunschweig. – 2006. – 52 p. 3. Pilipenko, N.V., Sivakov, I.A. “A method 14. Vockrodt, S., Leithner, R, Schiller, А., et al. of determining nonstationary heat flux and heat “Firing technique measures for increased efficiency

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 99 (2019) 106 Numerical simulation of heat and mass transfer processes in combustion chamber of pk-39 boiler and minimization of toxic emissions in Kasakh coal 19. Askarova, A. S., Messerle, V. E., Ustimenko, firing.” 19th German Conf. on Flames, Germany A. B., et al. “Numerical simulation of pulverized 1492 (1999): 93-97. coal combustion in a power boiler furnace.” High 15. Askarova, A. S., Bolegenova, S.A., Temperature 53, 3 (2015): 445 452. Maximov, V.Y., et al. “Numerical research of 20. 2Gorokhovski, M., Chtab-Desportes, A., aerodynamic characteristics of combustion chamber Voloshina, I., Askarova, A. “Stochastic simulation of BKZ-75 mining thermal power station.” Procedia the spray formation assisted by a high pressure.” AIP Engineering 42 (2012): 1250-12-59. Conference Proceedings, Xian 1207 (2010): 66-73. 16. Buchmann M. A., Askarowa A. “Structure 21. E. I. Heierle, S. A. Bolegenova, V. Ju. of the flame of fluidized-bed burners and combustion Maximov, S. A. Bolegenova, R. Manatbayev, M. T. processes of high-ash coal.” Gesellschaft Beketaeva, A. B. Ergalieva. “CFD Study of harmful Energietechnik, Combustion and Incineration – substances production in coal-fired power plant of Eighteenth Dutch-German Conference on Flames, Kazakhstan.” Bulgarian Chemical Communications VDI Вerichte 1313 (1997): 241-244. Special Issue E (2016): 260-265. 17. Karpenko, E. I., Messerle V. E., Ustimenko, 22. Pilipenko, N. V., Gladskikh, D. A. A. B. “Plasma enhancement of combustion of solid “Determination of the Heat Losses of Buildings fuel.” Journal of High Energy Chemistry 40, 2 and Structures by Solving Inverse Heat Conduction (2006): 111-118. Problems.” Measurement Techniques 57 (2014): 18. Messerle, V. E., Ustimenko, A. B., 181-186. Maksimov, V. Yu., et al. “Numerical simulation of 23. Alijarov, B. K., Alijarova, M. B. Combustion the coal combustion processinitiated by a plasma of Kazakh Coals in Thermal Power Stations and source.” Thermophysics and Aeromechanics 21, 6 Large-Power Boiler Houses, Almaty, 2011 (in (2014): 747 754. Russian).

International Journal of Mathematics and Physics 10, №1, 99 (2019) Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 107 (2019)

IRSTI 29.03.85

1,2B.A. Iskakov, 1,2Y.M. Tautayev, 1,2T.Х. Sadykov, 3,4A.L. Shepetov, 3,5N.M. Salikhov 1Satbayev University, Institute of Physics and Technology, Almaty, Kazakhstan 2Al-Farabi KazNU, Almaty, Kazakhstan. 3Tien Shan Scientific Station, Almaty, Kazakhstan 4P.N. Lebedev Institute of Physics of RAS, Moscow, Russia 5Institute of Ionosphere, Almaty, Kazakhstan Development and creation of a software system for the monitoring system mac1

Abstract. A special system of acoustic detectors was created to search for possible correlations between wide air showers and the signal of elastic oscillations from the depths of the earth’s crust. They are intended for joint synchronous work with storm installation. Key words: microphone, signals, microcircuit, program, acoustics.

A preliminary search for short-term acoustic an ungrounded secondary winding and does not emission signals in events related to the group have direct electrical contact with the rest of the passage of high-energy muons was carried out in a electronic circuit or the power lines of the external special experiment at the Tien-Shan high-altitude electrical network. From the power source to the station in 2012. Upon completion of the modification microphone, this voltage is applied through the of the Tien-Shan stormwater installation and the second pair of twisted wires. Thus, the microphone transition to regular recording of wide atmospheric node of the measuring system is electrically isolated showers, experiments of this kind it is expected to from all common ground lines and power supply to continue in full. the electronic circuits. A highly sensitive microphone with a sensitivity Acoustic detector signals are recorded in a special of 20 mV/Pa in the acoustic frequency range of room, which is located directly at the upper edge of 500-1000 Hz is located at a depth of 50 m from the the well. In the room are the other nodes forming the ground surface inside a well drilled in the rocky signal equipment. The differential amplifier (element soil. The distance between the well and the storm D1 in Figure 1) provides 100 times the amplification detector system is approximately 200 m. The circuit of the useful signal while simultaneously suppressing of electronic equipment which provides registration common mode noise that occurs on a long link. The is shown in Figure 1. The transmission of electrical common mode rejection ratio of this type of amplifier signals from a microphone from the depth of a is ~70 dB. At the output of the differential amplifier, well is made over a cable line formed by a twisted bipolar sinusoidal signals are formed, belonging to pair of wires through a transformer junction. The the acoustic range (~102 – 104Hz) and ready for microphone and the signal-carrying intermediate digitization by means of an ADC system. The low- small-sized transformer are a single building block, frequency selector built on the operational amplifiers which is completely lowered into the well. A constant D2 and D3 serves to highlight the modulating voltage of ± 3V for powering the microphone is amplitude of the acoustic signal of the low-frequency produced by an independent power source, which envelope, which is provided for registration through is built on the basis of a separate transformer with a separate ADC channel.

A special small-scale low-power ADC system Analog Devices. The clock sequences C (clock) and has been developed for recording acoustic detector CS required for the operation of the microcircuits are signals. It is placed together with the formation generated by a computer microprocessor running a schemes directly at the upper edge of the well. special driver program. The same program accepts The ADC system is based on the Raspberry PiB+ the results of converting the input signals that come single-board computer on a Broadcom BCM2835 from the DO outputs of the ADC chips in the form of a type microprocessor with a clock frequency of 700 serial binary code and converts them into binary data MHz. This computer, through the lines of a general- bytes. A schematic diagram of the connection of the purpose digital I/O port embedded in it, controls two ADC elements to the Raspberry PiB+ microcomputer ADC elements with AD7887 chips manufactured by is shown in Figure 2.

Figure 1 – Connection diagram of a microphone in an acoustic detector

The signals of the ADC microcircuits are The block diagram of the software package that connected to the pins of the GPIO port of the is used to record the data of the acoustic detector is microcomputer through the buffer elements of the shown in Figure 3. As can be seen in this diagram, TTL logic. They are part of the K155LN1 chip the program interacts directly with the hardware and through resistive dividers, which are necessary registers built into the Raspberry PiB microcomputer for matching the level of logic signals from the parallel I/O port, to the lines of which the control microprocessor with the level of the AD7887 chip. signals of the ADC chip are connected k09raspi.c The input analog signals are recorded over two driver. This program is written in the compiled information channels of the small-sized ADC system programming language C and is responsible for the continuously with a period of 2 ms. Measured data formation, in accordance with the corresponding is collected for subsequent on-line analysis in a file temporal characteristics, of two clock sequences C on a local disk that connects to the Raspberry PiB+ and CS. They provide the necessary synchronization microcomputer via its built-in USB interface. of the ADC chips. In response to each clock cycle of

International Journal of Mathematics and Physics 10, №1, 107 (2019) Int. j. math. phys. (Online) B.A. Iskakov et al 109 the synchronization sequence C, the next bits of the are processed by the driver software, and parallel data code are received from the microcircuits, which bytes of the 12-bit ADC code are formed from them, are registered by the k09raspi.c program. on the two which are placed by this driver into a segment of input lines of the parallel port. The received data bits shared memory that is readable by other programs.

Figure 2 – Compact ADC system

Responsible for sampling the data generated packed in an irregular way, before they are sent for by the driver from shared memory and further permanent storage they are processed once again by processing this information is the measurement a special program – the utility k09006. It converts management program k09001. This program them into regular archive files .dat.gz, which can be performs the standard functions of supporting the read by standard archivers such as zip, gzip and the measurement process, reads, immediately after like. After the conversion, the data archive files are launch, a set of configuration parameters from the automatically sent to the general processing center input file k09in. It adjusts the algorithm of its work via wireless communication lines. to the specifics of a specific measurement setup, Measurement management program k09001 is tracks the appearance of new information packages written in C ++ compiled programming language, in the shared memory segment, generates an array which allows to satisfy rather stringent requirements of binary data with measurement results from these for its speed. In addition, the required processing packages, and writes this array to a file with the .zdat speed of incoming information is achieved due to extension on the local disk. Before writing to the disk, double buffering and parallelization of operations the array is compressed by the archiver integrated in this program. When it starts, it creates two buffer into the program k09001, which reduces the total arrays in the computer’s memory to write data, at amount of output data and shortens the information each moment of time one of these arrays is filled exchange with the disk. Since the files recorded with current data. Information from the second at the in this way directly during the measurements are same time is being written to the computer disk in

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 107 (2019) 110 Development and creation of a software system for the monitoring system mac1 a separate stream of execution. The Raspberry PiB the normal operation of the entire measurement + microcomputer, which provides the process of process: automatic start of the necessary utility digitizing the signals of an acoustic detector, runs on programs, monitoring and automatic correction of a specially adapted version of the Linux operating the system clock, the ability to remotely connect via system. The programs included in this system ensure communication lines to monitor the system operation the performance of normal operations to support on-line and etc.

Figure 3 – Block diagram of the software system for the monitoring system

One of the specialized programs designed to work which implement various data processing algorithms, with the archived data files of the acoustic detector search for short peaks of intensity, etc. In particular, is the visualization program k09007, which, when one of these modules is used to calculate the averaged launched in graphical mode of operation, allows parameters of the recorded signals: the rms values of you to view the measurement results stored in these the ADC codes, which are calculated both for the files in the form of interactive graphs. The program analog signal removed from the microphone and for k09007 is written in Python, and its graphical window its low-frequency envelope, as well as the number interface is implemented using the Tkinter library, of events recorded per minute with ADC codes which is included in the standard distribution of this exceeding a number of threshold values. language. Graphs are drawn in the program window by means of the Python graphical library Matplotlib. Conclusion In addition to the interactive user mode of direct interaction with the user, the k09007 program also As shown in the block diagram (Figure 3), the supports a non-interactive mode, in which information averaged parameters of the acoustic signal calculated from archive files with measurement results, instead in this way are loaded by the program k09007 into the of directly displayed in the graphics window, is special table for the acou database of the same name. transmitted to external plug-in modules k09007exe, The information stored in this table can be requested

International Journal of Mathematics and Physics 10, №1, 107 (2019) Int. j. math. phys. (Online) B.A. Iskakov et al 111 by the web server to be displayed in text or graphic of the Republic of Kazakhstan – Series of geology form on the Tian-Shan station web page. Processing and technical sciences 3, 429 (2018): 47-56. requests from the web server and preparing for it the 5. Ambrosio M. et al. (MACRO Collab.). necessary information is made through an auxiliary “Vertical muon intensity measured with macro at the program “CGE-general-purpose script k14003. Ggran Sasso laboratory. “ Physical Review D 52, 7 (1995): 3793-3802. References 6. Gusev G. A., Zhukov V. V., Merzon G. I. et al. “Cosmic rays as a new instrument of seismological 1. Berger Ch. et al. (Frejus Collab.). “Experimen- studies.” Bull. Lebedev Phys. Inst. 38, 12 (2011): tal study of muon bundles observed in the Fréjus de- 374–379. tector. “ Physical Review D, 40, 7 (1989): 2163. 7. Tribedy P. Journal of Physics: Conference Se- 2. Beringer, J., Arguin, J. F., Barnett, R. M., ries Ser. 832 (2017) 012013. Copic, K., Dahl, O., Groom, D. E., ... & Yao, W. M. 8. Kroemer O. 2009 Proceedings of the 31st “Review of Particle Physics. “ Physical Review D 86 ICRC.- p. 25 (2012): 1-22. 9. Adamson P. et al. 2007 Phys. Rev. –Vol. D76. 3. Young, J.B., Presgrave, B.W., Aichele, H., –P. 052003. Wiens, D.A. and Flinn, E.A. “The Flinn-Engdahl 10. Beisembayev U., Dalkarov O.D., Sadykov Regionalisation Scheme: the 1995 revision. “ Physics T.Kh. et.al. Nuclear Instruments and Methods in of the Earth and Planetary Interiors 96, 4 (1996): Physics Research A. V.832 2016 – p.158–178. 223-297. 11. G. V. Kulikov and G. B. Khristiansen. 1959 4. Shepetov, A. L., Sadykov, T. K., Mukashev, K. Soviet Physics JETP 35(8):441-444. M., Zhukov, V. V., Vil’danova, L., Salikhov, N. M., 12. J. Abraham, Pierre Auger Collaboration 2010 ... & Argynova, A. K. “Seismic signal Registration Physics Letters B – V685. – p. 239–246. with an acoustic detector at the Tian Shan Mountain 13. M. Witek 2017 EPJ Web of Conferences 141, Station. “ News of the National Academy of Sciences 01007.

Int. j. math. phys. (Online) International Journal of Mathematics and Physics 10, №1, 107 (2019) Contents

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