Book of Abstracts

GEORGIAN MECHANICAL UNION

saqarTvelos meqanikosTa kavSiri

VIII ANNUAL MEETING Dedicated to the 110th Birthday Anniversary of Ilia Vekua OF THE GEORGIAN MECHANICAL UNION saqarTvelos meqanikosTa kavSiris

VIII yovelwliuri saerTaSoriso ეძღვნება ილია ვეკუას konferencia დაბადებიდან 110 წლისთავს

BOOK OF ABSTRACTS

moxsenebaTa Tezisebi

თბილისის უნივერსიტეტის გამომცემლობა 27.09 – 29.09.2017, TBILISI ISSN 2233‐355X

27.09 – 29.09.2017, Tbilisi

ORGANIZERS: ორგანიზატორები:

• I. Javakhishvili Tbilisi State University • ი. ჯავახიშვილის სახელობის თბილისის სახელმწიფო უნივერსიტეტის − I. Vekua Institute of Applied Mathematics − ი. ვეკუას სახელობის გამოყენებითი მათემატიკის ინსტიტუტი − Faculty of Exact and Natural Sciences − ზუსტ და საბუნებისმეტყველო მეცნიერებათა ფაკულტეტი • Georgian Mechanical Union • საქართველოს მექანიკოსთა კავშირი − Georgian National Committee of Theoretical and Applied Mechanics − საქართველოს ეროვნული კომიტეტი თეორიულ და გამოყენებით მექანიკაში • Tbilisi International Centre of Mathematics and Informatics • თბილისის საერთაშორისო ცენტრი მათემატიკასა და ინფორმატიკაში

SCIENTIFIC COMMITTEE: საერთაშორისო სამეცნიერო კომიტეტი: CHAIR: Jaiani, George (Georgia) თავმჯდომარე: ჯაიანი გიორგი

წევრები: მეგრელიშვილი ზურაბი MEMBERS: Kvitsiani, Tariel (Georgia) აფციაური ამირანი მეუნარგია თენგიზი Apt iauri, Amirani (Georgia) Makhviladze, Nodar (Georgia) s გაბრიჩიძე გურამი მიულერი ვოლფგანგი (გერმანია) Chinchaladze, Natalia (Georgia) Megrelishvili, Zurabi (Georgia) გულუა ბაკური პატარაია დავითი Davitashvili, Teimuraz (Georgia) Meunargia, Tengizi (Georgia) დავითაშვილი თეიმურაზი ფარცხალაძე გაიოზი Gabrichidze, Guram (Georgia) Müller, Wolfgang H. (Germany) ვაშაყმაძე თამაზი ყიფიანი გელა Gilbert, Robert (USA) Pataraia, David (Georgia) კაპანაძე გიორგი შავლაყაძე ნუგზარი Gulua, Bakuri (Georgia) Phartskhaladze, Gaioz (Georgia) კვიციანი ტარიელი შრივასტავა სურეში (კანადა) Kapanadze, George (Georgia) Shavlakadze, Nugzar (Georgia) კიკვიძე ომარი ნატალია ჩინჩალაძე Kikvidze, Omar (Georgia) Shrivastava, Suresh (Canada) Kipiani, Gela (Georgia) Vashakmadze, Tamaz (Georgia) მახვილაძე ნოდარი ჯილბერტი რობერტი (აშშ) საორგანიზაციო კომიტეტი: ORGANIZING COMMITTEE: თავმჯდომარე: ჯაიანი გიორგი CHAIR: Jaiani, George თავმჯდომარის მოადგილეები: ავაზაშვილი ნიკოლოზი, ყიფიანი გელა, VICE-CHAIRS: Nikoloz Avazashvili, Natalia Chinchaladze, Gela Kipiani ჩინჩალაძე ნატალია წევრები: MEMBERS: ბერიაშვილი მარიამი პაპუკაშვილი არჩილი Mariam Beriashvili Archil Papukashvili გაბელაია მირანდა რუხაია მიხეილი Miranda Gabelaia Mikheil Rukhaia გულუა ბაკური შარიქაძე მერი Bakuri Gulua Mary Sharikadze თევდორაძე მანანა ცუცქირიძე ვარდენი Roman Janjgava Manana Tevdoradze ნოზაძე გიორგი ჯანჯღავა რომანი George Nozadze Varden Tsutskiridze

TOPICS OF THE MEETING: კონფერენციის თემატიკა: • Solid Mechanics • მყარ დეფორმად სხეულთა მექანიკა • Fluid Mechanics • ჰიდროაერომექანიკა • Solid-Fluid Interaction Problems • დრეკად მყარ და თხევად გარემოთა ურთიერთქმედების ამოცანები • Applied Mechanics • გამოყენებითი მექანიკა • Technical Mechanics • ტექნიკური მექანიკა • Related Problems of Analysis • ანალიზის მონათესავე საკითხები CONFERENCE WEB-PAGE: კონფერენციის ვებ-გვერდი: http://www.viam.science.tsu.ge/others/gnctam/annual8.htm http://www.viam.science.tsu.ge/others/gnctam/annual8.htm 1 2

LIFE, ACTIVITIES, AND SCIENTIFIC HERITAGE OF ILIA VEKUA

George Jaiani Ilia Vekua Institute of Applied Mathematics & Department of Mathematics of Ivane Javakhishvili Tbilisi State University [email protected] I.Vekua’s parents: Vekua’s Parents house in The present talk is devoted to a con- Liza (Memu) Apshilava and Shesheleti cise survey of scientific, pedagogical, Nestor Vekua and educational activities of the out- Ilia Vekua was born on April 23rd, 1907 in a pic- standing Georgian mathematician and turesque region of Georgia, in Abkhazia, namely in mechanist Ilia Vekua. Biographical data the district Samurzakano in the village Shesheleti. are also given. After finishing a secondary school in the capital of If we look through the Mathematical another ravishing region of Georgia Samegrelo, Ili Vekua Subject Classification, we find Ilia Ve- Zugdidi, in 1925 Ilia Vekua was enrolled at the Fa- kua’s name among the names of a few culty of Physics and Mathematics of the Tbilisi State outstanding mathematicians and mechanists of the world: “30G20 – Ilia Vekua in University (TSU). Generalizations of Bers and Vekua type (pseudoanalytic, p-analytic, 1925 etc, functions). It’s remarkable that only Ilia Vekua has such an honour amidst Georgian scientists. In this connection we quote here

Tbilisi State University

an outstanding American mathematician L. Bers: “While “Vekua’s generalized analytic functions” are basically the same as my “pseudoanalytic functions”, our motivation and our aims were somewhat different”. Ilia Vekua and Nikoloz Muskhelishvili A.N. Krylov

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He graduated from the university with honours in 1930 and, having Sciences. the recommendation of academician Nikoloz Muskhelishvili, left Ilia Vekua was one of the enthusiastic founders of the Razmadze Tbilisi for Leningrad (former and present Saint-Petersburg) and be- Mathematical Institute, Novosibirsk State University and its first came Ph.D. student of the well-known mathematician and mechanist A.N. Krylov. The first works of Ilia Vekua were devoted to the torsion and bending of elastic bars and propagation of electric waves in an infinite layer with parallel plane boundaries. These results subsequently formed his Ph.D thesis “Propagation of Elastic Oscillations in an Infini- te Layer” which he successfully defended at the age of 29. I.Vekua Institute of First Council of the Institute Applied Mathematics of Applied Mathematics

Rector, the founder and the first director of the Institute of Applied Mathematics of the Tbilisi State University until his decease in 1977.

Ilia Vekua Ilia Vekua at the age of 29 at the age of 32 When he was 32 years old, Ilia Vekua received Doctor of Science degree (A Complex Representation of Solutions of Elliptic Equations Computer’s room of the Institute of Applied and its Application to Boundary Value Problems). At the age of 39 he Mathematics (BESM-6/7) became a member of the Georgian Academy of Sciences and a Now it is called I.Vekua Institute of Applied Mathematics of TSU. corresponding member of the Soviet Academy of Sciences. In 1958 At different times he was a Rector of TSU, the second (after he was elected a member (academician) of the Soviet Academy of N. Muskhelishvili) President of the Georgian Academy of Sciences, Professor of Lomonosov Moscow State University, etc.

Ilia Vekua and Ilia Vekua with the first Nikoloz Muskhelishvili I. Javakhishvili Tbilisi M. Lomonosov Georgian National graduates of Novosibirsk State University Moscow State Academy of State University University Sciences 5 6

Scientific studies of Ilia Vekua as a rule start with problems of Now, I’d like to quote the following extract from the foreword of Mechanics, continue by fundamental investigations in Complex Ana- Gaetano Fichera to the English edition of I. Vekua’s book in shell lysis, Integral and Partial Differential equations and are crowned with theory: “The present monograph by I.N. Vekua, which is now offered applications to Mechanics. Results of Ilia Vekua, published in leading scientific journals, were summarized in consequent monographs, published in Russian, English, German, Chinese (later also Georgian), and other languages. The most important of his monographs were: New Methods of Solving Elliptic Equations (1948), Generalized Analytic Equations (1948), Generalized Analytic functions (1959), Shell Theory: General Methods of Constructing (first published Russian after his decease in 1982 under the slightly different title), Ricard Courand and Ilia Vekua Ilia Vekua and Gaetano Fichera

to Western applied mathematicians in an English translation, is the natural continuation of the ideological program of the Tbilisi school. It is concerned with shell theory which, as is well known, constitutes an elevated and difficult generalization of plane elasticity. The rele- vant geometric continua of shell theory, although two- dimensional, are no longer planar and their geometric properties deeply influence then in 1985 by Pitman Advanced Publishing Program, and in 2007 by the analysis of the problems that arise in this important field of elas- the Tbilisi University Press in ticity.” Georgian. All these monographs Once again L. Bers. He wrote: “In the theory were awarded by different state of elliptic partial differential equations in the pla- prizes. ne Vekua followed two different paths. One, si- Richard milar to the work of Stefan Bergman, led to Courant explicit formulas for solutions of elliptic equa- and David tions with real analytic coefficients, in terms of Hilbert in arbitrary holomorphic functions of a complex their well- variable. Vekua applied his formulas to very ge- known book “Methods of Mathematical Physics” neral boundary value problems and, in this con- were skeptical about the use of general complex nection, obtained a result which is today recog- representations of solutions of PDEs. Later, Lipman Bers nizable as a precursor of the Atiyah-Singer Courant confessed that the above opinion was index theorem. totally wrong and it was proved by Vekua’s The other main direction of Vekua’s investigations was concerned methods developed on the basis of general com- David Hilbert with properties of systems of two first order elliptic equations, viewed plex representations. as generalized Cauchy- Riemann equations. This time real analyticity of the coefficients is not assumed, and it turns out that the properties

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in question are best understood in terms of generalized complex Moscow function theory. While Vekua’s “generalized analytic functions” are 2. Acting deputy director of the Precise Mechanics and Com- basically the same as my “pseudoanalytic function”, our motivation puter Hardware Institute of the USSR Academy of Sciences, Moscow and our aims were somewhat different. In particular, Vekua applied his 3. Head of the theoretical mechanics chair of the Moscow Physi- theory to elastic shells and to infinitesimal deformations of surfaces in cal Technical Institute 3-space”. 4. Professor of the differential equations chair of the M. Lomono- The life of I. Vekua can be devided in to 8 periods: sov Moscow State University • Shesheleti, Gali 1907-1924 5. Deputy director of the V.A. Steklov Mathematical Institute of • Zugdidi 1925 the USSR Academy of Sciences

• Tbilisi 1925-1930 • Novosibirsk 1958-1964 • Petersburg 1930-1933 1. Member of the Presidium of the Siberian Branch of the USSR • Tbilisi 1933-1951 Academy of Sciences • Moscow 1951-1958 2. Rector of Novosibirsk State University • Novosibirsk 1958-1964 3. Head of the mathematical physics chair of the Novosibirsk • Tbilisi 1964-1977 University 4. Head of the theoretical department of the Hydrodynamics Institute Main positions held: of the Siberian Branch of the USSR Academy of Science

• Tbilisi 1933-1951 • Tbilisi 1964-1977 1. Scientific secretary of the Tbilisi Mathematical Institute 1. Vice-President of the Academy of Sciences of the Georgian SSR 2. Head of the theoretical mechanics chair of the Transcaucasian 2. Head of the mechanics sector of the A. Razmadze Tbilisi Institute of Engineers of Means of Communications Mathematical Institute of the Academy of Sciences of the Georgian 3. Deputy director of the Mathematical Institute of the Geor- SSR gian Branch of USSR Academy of Sciences 3. Rector of the Tbilisi State University 4. Dean of the physical-and-mathematical faculty of the Tbilisi State 4. Scientific supervisor, director of the Institute of Applied Mathe- University matics of the Tbilisi State University 5. Head of the geometry chair of the Tbilisi State University 5. President of the Academy of Sciences of the Georgian SSR. 6. Head of the applied mathematics department at the A. Razmadze Ilia Vekua left an indelible trace everywhere, in every field he had Mathematical Institute of the Academy of Sciences of the Georgian been working. SSR His scientific heritage is immortal and it is being developed in 7. Vice-Rector of Tbilisi State University different countries of the world by his PhD students, PhD students of 8. Chairman of the mathematical and natural sciences department of his PhD students, and his followers. the Academy of Sciences of the Georgian SSR 9. Academician-Secretary of the Academy of Sciences of the Geor- gian SSR 10. Head of the higher mathematics chair of Tbilisi State University

• Moscow 1951-1958 1. Head of a department of the Central Aerohydrodynamics Institute,

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PhD Students ∂u ∂u λ = λ1 + λ2 titt ),()()( z)( =Φ − i , A. Bitsadze (Georgia) P. Dubov (Russia) ∂x ∂y I. Danilyuk (Ukraine) A. Dzhuraev (Tajikistan) problem (1) can be reduced to the problem with an oblique derivati- V. Ivanov (Russia) N. Tovmasyan (Armenia) ve, i.e. , to the Poincaré problem: Define a function harmonic in D L. Mikhailov (Russia) V. Jgenti (Georgia) and satisfying the following boundary condition O. Kharazov (Azerbaijan) G. Gagua (Georgia) ∂u ∂u + λλ = tg ),( ∈ ∂Dt . (2) E. Obolashvili (Georgia) L. Kiknadze (Georgia) 1 ∂x 2 ∂y A. Kalandia (Georgia) Sh. Metskhovrishvili (Georgia) V. Khvedelidze (Georgia) R. Kordzadze (Georgia) In connection with the study of problems (1) and (2) the one- B. Bojarski (Poland) Cher (North Korea) dimensional singular integral equation W. Schmidt (Germany) Sun, Che-shen (China) ϕ ≡ α ϕ + β ϕ + ϕ = tftTtSttttA ),()()()()()()( ∈ ∂Dt , (3) V. Vinogradov (Russia) Y. Zhio Chen (China) is considered, where S is a Cauchy-type singular operator: Y. Krivenkov (Russia) G. Jaiani (Georgia) ϕ τ )(1 dτ ϕ tS )( = , ∈∂Dt , I. Belov (Russia) N. Kaldani (Georgia) πi ∫ τ − t ∂D More than 100 PhD students of PhD Students. and T is the Fredholm integral operator. It is impossible to give an approximate number of disciples. One of the main questions of the theory of singular integral Among them are T. Meunargia, T. Vashakmadze and late D. Gorde- equations of the form (3) is the reduction of this equation to an ziani from I.Vekua Institute of Applied Mathematics of TSU. equivalent second kind Fredholm equation (the problem of equivalent regularization). I. N. Vekua’s solution of this problem is considered Now, we give a brief account of the most typical features of as a brilliant result in the theory of singular integral equations. I. N. Vekua’s rich legacy that have greatly influenced the develop- Developing Carleman’s idea, in the 1940s I. N. Vekua worked out (in ment of the respective problems of mathematics. classical assumptions) a way for constructing a theory of integral The general linear boundary value problem for analytic functions of equations (3) known at present as the Carleman-Vekua method. It one complex variable, studied comprehensively by I. N. Vekua, holds involves three stages: 1) solutions of a characteristic singular equa- the key position in the modern theory of the so-called non- tion (i.e. , eq. (3) for T = 0 ) as well as of its associated equation are Fredholmian problems for elliptic equations. constructed effectively; 2) these solutions are used for an equivalent One of the basic problems of the function theory is associated with regularization of eq. (3) and of its associated equation; 3) Fredholm names of Riemann and Hilbert: Define a function Φ z)( analytic in D integral equations, constructed in this manner, are used for proving and satisfying the boundary condition the Noether theorems for eq. (3). The regularization problem can be solved using, in addition to the λ + ∂∈=Φ Dttgtt ,),()]()(Re[ (1) above-mentioned method, the method of regularization by multiplica- where λ and g are the known functions of the arc abscissa s of ∂D tion of operators. The idea of this method is in the following: it is + required to construct an operator B of type A (see (3)) such that the (i.e. = stt )( ) of D , while Φ t)( is the boundary value of the unknown function for → tz from D . equation ϕ = BfBA be Fredholmian. In this case B is said to be the Introducing the notation left regularizer of A . If the equations ϕ = fA and ϕ = BfBA are

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equivalent, whatever f from the considered class of functions is, theory of normally solvable boundary value problems in the case of B is said to be the left equivalent regularizer of A . It was known that the following second order elliptic differential equation such an operator does not always exist. In this connection a question ∂u ∂u +Δ yxau + yxa + uyxa = ,0),(),(),( (5) arises: is it possible to formulate the problem of constructing a 1 ∂x 2 ∂y 3 Fredholm equation equivalent to eq. (3) in such a way that it should where a , a and a are analytic functions. These problems are an always have a solution? I. N. Vekua answered the question 1 2 3 positively. He showed that there exists an operator B , constructed essential generalization of the Poincaré boundary value problem (2) effectively in quadratures, such that either the equations ϕ = fA and in the case of equation (5). Indeed, the boundary condition is of the form ϕ = BfBA or ϕ = fA and φ = fAB are equivalent in the sense ⎡ ∂ +ki u ⎛ ∂ +ki u ⎞⎤ that either of the equations is solvable, then so is the other, and the ta )( + T ⎜ ⎟ = ),( ∂∈ Dttf , (6) ∑ jk ⎢ ki jk ⎜ ki ⎟⎥ connection ϕ = Bφ exists between their solutions. ≤+ mkj ⎣ ∂∂ yx ⎝ ∂∂ yx ⎠⎦ Using the theory of eq. (3), by means of his own integral where a and f are the known real functions on ∂ , TD are the representations of analytic functions jk jk μ )( dstt Fredholm integral operators. z)( =Φ ∫ In the constructing the theory of this problem I. N. Vekua made ∂D − zt use of the integral representation of all regular solutions of eq. (5) when Φ z)( is a Hölder continuous in D and z = + ϕβϕαdtttzzzzzyxu ],)(),,()(),(Re[),( (7) m−1 ∫ ⎛ z ⎞ ⎛ z ⎞ 0 μ tz ⎜ −=Φ ⎟ ⎜1ln1)()( − ⎟ + μ )( dstds ∫ ∫ where ϕ is an arbitrary analytic function, while α and β are ∂D ⎝ t ⎠ ⎝ t ⎠ ∂D m)( when Φ z)( is Hölder continuous in D , where μ is a real functions constructed by means of the coefficients a1 , a 2 and a 3 . Hölder continuous function of s , I. N. Vekua succeeded in solving Formulas similar to (7) were obtained by T. Carleman, H. Lewy completely the Riemann-Hilbert problem (1) in the following general and S. Bergman. The method of constructing formula (7), known as the Riemann-Vekua method, is considered to be the most simple, formulation: In the domain D whose boundary ∂D is a sufficiently clear and constructive one. Here we recall L. Bers again: “I. Vekua smooth, simple, closed curve; it is required to find an analytic applied his formulas to very general boundary value problems and, in function Φ satisfying the boundary condition m this connection, obtained a result which is today recognizable as a k )( + k )( + precursor of the Atiyan-Singer index theorem”. ∑ λk Ttt k =Φ+Φ tf )(]})[()]()[({Re (4) k =0 I. N. Vekua generalized the integral representation for the elliptic where Φ k )( )( + is a boundary value of the k th order derivative of equation n from D ; and are the functions given on ∂D and T are n −kn Φ λk f k ∑ k uLu =Δ+Δ ,0)( (8) the Fredholm integral operators. k =1 I. N. Vekua’s results obtained in connection with problem (4) where Lk is the k th order differential operator with analytic coeffi- formed the basis of his further research devoted to constructing a cients.

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Employing his formulas, I. N. Vekua investigated the following sults in this direction were obtained by T. Carlemann and M.A. Lav- boundary value problem: Find the regular solution of this equation in rentyev. I. N.Vekua’s intensive research also yielded basic results the simply connected domain D , satisfying the conditions that formed the principal component part of the modern theory of 'ud functions satisfying the equation j njf −== ,1,...,1,0, ∂F ∂F ∂F dν j − q − q FBAF =++ .0 (11) ∂D ∂z 1 ∂z 2 ∂z where ν is the outward normal of ∂D , If qq <+ ,1 then many properties of solutions of system (9) In the theory of general complex representations of solutions of 21 elliptic equations I. N. Vekua discovered a remarkable fact of a remain valid for eq. (11) which is a complex form of a linear elliptic possibility of equivalent reduction of any boundary value problem for system of two equations with respect to the real and imaginary parts eq. (8) to the corresponding boundary value problem for a system of of F . In the constructed theory the known functions 21 ,, Aqq and analytic functions. B are required to be sufficiently smooth. As is well-known, the theory of analytic functions I.N. Vekua made a substantial contribution to the theory of =Φ + yxivyxuz ),(),()( of one complex variable z = x + iy is the metaharmonic functions which are solutions of the Helmholtz theory of the Cauchy-Riemann system equation ∂u ∂v ∂u ∂v ∂ 2u ∂ 2u − = ,0 + = .0 ...++ λ2u =+ ,0 λ = const, p ≥ .2 (12) ∂x ∂y ∂y ∂x 2 0 ∂x1 ∂x p This system is the particular case of the elliptic system He gave the following integral representation of metaharmonic ∂u ∂v − bvau =++ ,0 functions ∂x ∂y 1 q ∂ ∂u ∂v 1 p = 10 p − 10 p ttxtxuxxuxxu 0 λ − dttrI ,)1(),...,,(),...,(),...,( + dvcu =++ ,0 (9) ∫ ∂t ∂y ∂x 0 p − 2 with the real coefficients ,,, dcba which are functions of the real where u is an arbitrary harmonic function, q = , 0 2 variables x y., 2 2 ++= xxr 2 ,... and I is the Bessel function. He also construc- Introducing the notation 1 p 0

∂ ∂ ∂ ∂ ∂ ∂ ted an inverse integral representation of this formula and, moreover, = + ivuzW ,:)( = −i ,:2 = + i ,:2 in the case of a real λ he showed that under the Sommerfeld ∂ ∂xz ∂y ∂ ∂xz ∂y conditions = + + − bcidaA ),(:4 4 = − + + bcidaB ),( ∂u q−− 2/1 q−− 2/1 system (9) can be rewritten as λ =− roui )( and = rOu ),( ∂r ∂W WBAW =++ .0 (10) providing the uniqueness of the solution of the external Dirichlet and ∂z Neumann problems in the case of eq. (12), the second condition is the Back in the 19 th century Beltrami and Picard tried to construct a consequence of the first one. theory of generalized analytic functions = + ivuw . Important re-

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I. N. Vekua showed that his method of constructing solutions of elliptic linear equations could be used in the investigation of the properties of solutions of some nonlinear elliptic equations. Thus, for example, he studied the properties of solutions of the Gaussian equation Δ −= yxvyxKyxv ),,(),(2),(log which enabled him to find a simple proof of the well-known Hilbert theorem on the nonexistence of a regular surface with a negative Figure 2: A sharp cusped prismatic shell with a semicircle projection. Ω∂ is a curvature, conformally homeomorphic to the whole plane. Lipschitz boundary The range of theoretical results obtained by I.N. Vekua is wide. On the basis of his methods of investigation of elliptic equations a consistent theory of elastic shells was constructed. In particular, I.N. Vekua proposed two versions of this theory, one of which is used in the investigation of thin shells, and the other in the construction of a membrane theory of shells. The elastic body is called a prismatic shell if it is bounded above and below by, respectively, the surfaces (so called face surfaces) +)( −)( x3 h 21 xandxx 3 h xx 21 ),,(=),(= Figure 3: A cusped plate with sharp γ 1 and blunt γ 2 edges, γ = γ ∪γ 210 . laterally by a cylindrical surface Γ of generatrix parallel to the x3 - axis and its vertical dimension is sufficiently small compared with ∂Ω is a non-Lipschitz boundary other dimensions of the body (see Fig. 1). In other words, the 3D elastic prismatic shell-like body occupies a In what follows we assume that ±)( 2 bounded region Ω with boundary Ω∂ , which is defined as: h 21 ∩∈ CCxx ωω),()(),( −)( +)( ⎧ 3 ⎫ and Ω ⎨ 321 ∈R xxxxx 21 ∈ω h xxx 321 h xx 21 ⎬,),(<<),(,),(:),,(:= (13) +)( −)( ⎩ ⎭ ⎧ xx 21 ∈ω,),(for0> xxh 21 h xx 21 − h xx 21 ),(),(:=),(2 ⎨ where := ∂∪ ωωω is the so-called projection of the prismatic shell ⎩≥ xx 21 ),(for0 ∂∈ ω := Ω∂∪ΩΩ is the thickness of the prismatic shell Ω at the points xx 21 ),( ∈ ω . h}{2max is essentially less than the characteristic dimensions of ω ~ (see Figures 1-8). Let 3 = xxhx 21 ),( denote the “middle” surface of the prismatic shell, then ~ +)( −)( 2 xxh h xx + h xx ).,(),(:=),( 21 21 21 Figure 1: Prismatic shell with constant thickness In the symmetric case of prismatic shells, i.e., when 17 18

−)( +)( ~ unless otherwise stated), bar under one of the repeated indices means xx − 2.,.i),,(=),( xxhexx 0,=),( h 21 h 21 21 that we do not sum. we have to do with plates of variable thickness xxh 21 ),(2 and a mid- By uir , X ijr , eijr , Φ jr we denote the r -th order moments of the plane ω (see Fig. 3). Prismatic shells are called cusped prismatic corresponding quantities ui , X ij , eij , Φ j as defined below: shells if a set γ 0 , consisting of xx 21 ),( ∂∈ ω for which xxh 21 0=),(2 , is not empty (see Fig. 2, Fig. 3). Φ jrijrijrir 21 txxeXu ),,)(,,,( Distinctions between the prismatic shell of a constant thickness +)( ( ,xx ) and the standard shell of a constant thickness are shown in Fig. 4, h 1 2 = Φ − jidxbaxPtxxxeXu .1,3=,,)(),,,)(,,,(: (14) where cross-sections of the prismatic shell of a constant thickness ∫ jijiji 321 r 3 3 with its projection and of the standard shell of a constant thickness −)( h ( 1,xx 2 ) with its middle surface are given in red and green colors, respectively where (common parts are given in blue, see Fig. 4, Fig. 5). ~ 1 xxh 21 ),( x xxa :),( = xxb :),(, = , 3 21 21 xxh 21 ),( xxh 21 ),( and 1 d r τ 2 −1)( r P τ =)( r .0,1,=, r r r L Figure 4: Comparison of cross-sections of prismatic and standard shells r!2 dτ are the r -th order Legendre Polynomials I.Vekua's hierarchical models for elastic prismatic shells are the mathematical models (Vekua 1955, 1985). Their constructing is based on the multiplication of the basic equations of linear elasticity: Motion Equations ∂ 2u Figure 5: Cross-sections of a prismatic (left) and a standard shell with the same j 3 X , Φ+ jiij ρ 321 xxxtxxx 321 ⊂Ω∈ R ,),,(),,,,(= mid-surface ∂t 2 (15) The lateral boundary of the standard shell is orthogonal to the jtt ;1,3=,> "middle surface" of the shell, while the lateral boundary of the 0 prismatic shell is orthogonal to the prismatic shell's projection on Generalized Hooke's law (isotropic case) (Fig. 5). X + μλθδjie θ e ;:=,1,3=,,2= (16) x3 0= ij ij ij ii Kinematic Relations In what follows X ij and eij are the stress and strain tensors, 1 respectively, ui are the displacements, Φi are the volume force ij = + ,, ijji jiuue ,1,3=,),( (17) 2 components, ρ is the density, λ and μ are the Lamé constants, δ ij by Legendre polynomials r 3 −baxP )( and then integration with is the Kronecker delta. Moreover, repeated indices imply summation −)( +)(

(Greek letters run from 1 to 2, and Latin letters run from 1 to 3, respect to x3 within the limits xxh 21 ),( and xxh 21 ),( .

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N By constructing Vekua's hierarchical models in Vekua's first lN r+12 q−1 sr ++ 1 N ⎤ r ∂ h v jr B r r version on upper and lower face surfaces stress-vectors are assumed + ijks h v ks + h X j = ρh 2 jNr L ≡ 0,)(,1,3=,0,=, ∑ ⎥ ∂ t ∑ to be known, while there the values of the displacements ls +1= ⎦ q s where ±)( ±)( ∞ 1 ∞ s +± 1)(21)( s N u i = xxu h (=)),,(,,(: usatxx ±+ =1)() u i 21 21 ∑ is ∑ is N s 0= 2 s 0= 2h u kr v kr 0,=,1,3=,:= Nrk , are calculated from their (displacements') Fourier-Legendre-series hr+1 ∞ 1 r l

ijiji 321 ∑ (:),,,)(,,( += ijrijrir 21 r 3 − baxPtxxeXuratxxxeXu )(),,)(,,)( ail depend only on the thickness, Bijks depend only on the thickness r 0= 2 r −)( +)( and the Lamé parameters, X j depend only on the stresses applied at expansions on the segment x3 ∈ h xx 21 h xx 21 )],(),,([ and vice versa in his second version. the face surfaces, the volume forces, and the thickness. So, we get the equivalent to (15)-(17) infinite system with respect In the static case if N 0= we immediately get the governing to the so called r -th order moments X , e , u . Then, in the system of the N 0= approximation (superscript N 0= is omitted ijr ijr ir below) usual way, we construct an equivalent infinite system with respect to 0 the r -th order moments uir (Vekua 1955). After this, if we suppose − μ hv ,0, αβα+ hv ,0, ααβ− λ hv ,0, βγγX β β 1,2,=,=)(])()[( that the moments whose subscripts, indicating moments' order, are 0 greater than N equal zero and consider for each j=1,2,3 only the − μ 30, αα= Xhv 3 ,),( (19) first N +1 equations Nr )0,=( in the obtained infinite system of and the relations 0 equations with respect to the r -th order moments uir , we obtain the X j0,αα+ X j 0,= N − th order approximation (hierarchical model) governing system h e = ( + vv ), consisting of N + 33 equations with respect to N + 33 unknown αβ 0 2 0, 0,αββα N N h functions uir (roughly speaking uir is an ''approximate value'' of eα 30 = 30,α ev 330 0,=, N 2 uir , since uir are solutions of the derived finite system), i ,1,3= X αj0 = λδ j γγα0 + μee αj0 ,2 0,= Nr . 330 =νXX αα 0 , N N N ⎡⎛ r+12 ⎞ ⎛ r+12 ⎞ ⎤ ⎛ r+12 ⎞ In the static case for the symmetric shell I. Vekua's system in the μ⎢⎜h vα , jr ⎟ + ⎜h v jr,α ⎟ ⎥ + λδ αj ⎜h v r,γγ⎟ ⎣⎝ ⎠,α ⎝ ⎠,α ⎦ ⎝ ⎠,α (18) N 1= approximation has the form − μ hv + hv − λ hv )(])()[( (20) ,0, αβα ,0, ααβ ,0, βγγ N r N r−1 r N N N 0 ⎛ sr ++ 1 ⎞ ⎡ lr ++ 1 lr ++ 1 ⎛ ⎞ + ⎜ Bαjks h v ks ⎟ + a λδ h v l,γγ+ μh ⎜ jil + vv ,, ijl ⎟ − λ hv ,31 β X β β 1,2,=,=)(3 ∑ ∑ il ⎣⎢ ij ⎝ ⎠ rs +1= ⎝ ⎠,α l 0= 0 − μ hv − μ hv X ,=)(3)( (21) ,30, αα ,1 αα 3 21 22

− μ 3 + 3vhvh −λ 3vh ),(])()[( mechanists, since prior to that period equations of mixed type were ,1, αβα ,1, ααβ 1, βγγ applied only to problems of aerodynamics. 1 Because of the depth and importance of I. N. Vekua’s μ 30, ++ ββ1 hvvh X β β 1,2,=,=)3( (22) investigations and the new scientific trends founded by him he ranks 1 3 − μ vh )( ,31, αα hv 0,γγ +++ μλλ31 hhv X 3 .=)23( (23) among the most outstanding scientists of our time. Ilia Vekua was a great public figure and personality, the embodi-

ment of both physical and spiritual greatness. Systems (21), (22) and (20), (23) correspond to bending and tension- That is why he was given everlasting abode at the Mtatsminda compression, correspondingly. Pantheon of Georgian writers and public figures. Note that for the plate of a constant thickness from the ben- 2h ding system (21), (22), under the corresponding assumptions we can derive the classical bending equation q ΔΔu ,= 3 D* where 1 3 = vu 30 2 and He is gone 40 years ago but his scientific results and ideas are still −ν )(1 2 D* = D. important and applied and developed by his former PhD students and − 21 ν followers, in general. Finally, we give in short a survey of develop- Here D is the classical flexural rigidity. So, Vekua's plate bending ment of his results and ideas only in one direction: Lower dimensio- model in the N 1= approximation actually coincides with the nal theories formulated from the 3D theory of elasticity using a classical bending model but by bending Vekua's plate is flexurally displacement ansatz with the truncated (roughly speaking) Fourier more rigid than the classical one. series expansion with respect to the Legendre polynomials. In other In the membrane theory of shells with an alternating Gaussian words in the direction of hierarchical models for prismatic (in curvature the main part is played by an equation of the mixed type, in particular, plates) and standard shells of variable thickness, in particular, by the Holmgren-Gellerstedt equation general. We mostly touch cusped (tapered) prismatic shells*). ∂ 2u ∂ 2u Works of I. Babuška, D. Gordeziani, V. Guliaev, G. Jaiani, y m+12 + = 0 ∂x 2 ∂y 2 I. Khoma, A. Khvoles, T. Meunargia, C. Schwab, T. Tskhadaia, T. Vashakmadze, V. Zhgenti, and others are devoted to further ( m ≥ 0 is an integer) and Lavrentyev-Bitsadze equation analysis of I.Vekua's models (rigorous estimation of the modelling ∂ 2u ∂ 2u + signy = 0 error, numerical solutions, etc.) and their generalizations (to non- ∂x 2 ∂y 2 shallow shells, to the anisotropic case, etc.). for which the Tricomi problem and its various generalizations acquire In 1955 and 1965 I.Vekua pointed out the importance of investti- clearly defined mechanical sense. This fact, discovered by I.N. Vekua *) in the mid-1950s evoked great interest among mathematicians and For references see G. Jaiani, Cusped Shell-like Structures, Springer, Heidelberg-Dordrecht-London-New York, 2011 23 24

gation of cusped prismatic shells, which is connected with degenerate Dirichlet, weighted Neumann, and mixed BCs when the thickness (singular) equations and systems, but no one tackled this problem. In satisfies the unilateral condition 1968 he, as my PhD advisor, suggested to concentrate my interest on ≥ xhxxh k ,),(2 = consth > ,0 = constk ≥ ,0 x ≥ .0 (25) this problem. Until the early nineties of the last century only my 2021 0 2 The bending [see system (21), (22)] vibration problem can be publications were devoted to this problem preceded with a work of investigated in an analogous manner. A. Khvoles, dealing with Vekua’s representations of general solu- The method of investigation of hierarchical models based on the tions of some equations connected with the cusped prismatic shells. idea to get Korn’s type inequality for 2D models from the 3D Korn’s Later, I managed to involve in studies in this direction my PhD inequality for non-cusped domains and then to use Lax-Milgram students and some colleagues within the framework of international theorem belongs to D. Gordeziani (1974) which (the method) found projects. its complete realization in works of D. Gordeziani, G. and M. Applying the function-analytic method, developed by Fichera Avalishvili (2003). By means of the solutions of these 2D BVPs, a (1956, 1960), the particular case λ = μ of Vekua’s system (19) of sequence of approximate solutions in the corresponding 3D region is the N = 0 approximation for general form cusped prismatic shells constructed. This sequence converges in the Sobolev space H 1 to the have been investigated by Jaiani (1988). The main conclusion says solution of the corresponding original 3D BVP. The analogous ⎛ ∂h ⎞ approach is developed by Schwab (1996). This idea with the that at a blunt cusped edge ⎜ +∞= ⎟ the displacement vector ⎝ ∂n ⎠ corresponding modifications was successfully used by G. Jaiani, components can be prescribed, while the sharp cusped edge S. Kharibegashvili, D. Natroshvili, and W. Wendland (2003, 2004) in the case when the cusped prismatic shell occupies a Lipschitz 3D ⎛ ∂h ⎞ ⎜0 ≤ ∞< ⎟ should be freed from BCs (Keldysh type BVP for domain, on the face surfaces stress vectors, while on the non-cusped ⎝ ∂n ⎠ edge weighted moments of displacement vector components are displacements). The last result concerning sharp cusped edges is true given. With the help of variational methods, the existence and uni- for the Nth approximation as well (Jaiani, 2001). In the queness theorems for the corresponding 2D BVPs are proved in the N = 1 approximation for the symmetric prismatic shell in the case appropriate weighted function spaces. The systems of differential = xhxxh k ,),(2 = consth > ,0 constk ≥= ,0 x ≥ ,0 (24) equations corresponding to the 2D variational hierarchical models are 2021 0 2 explicitly constructed for a general system of functions and for the the tension-compression system (20), (23) is investigated by G. Legendre polynomials, in particular, i.e., in I.Vekua’s case. Investiga- Devdariani, G. Jaiani, S. Kharibegashvili, D. Natroshvili (2000). The tions for plates, prismatic and general standard shells whose thick- existence and uniqueness of generalized solutions of BVPs with ness may vanish on their boundaries but occupy Lipschitz 3D do- Dirichlet [for the weighted zero moments when k <1 and for mains are carried out by D. Gordeziani G. Avalishvili, M. Avalish- weighted first moments when k < 3/1 ] and Keldysh type (for the vili, B.Miara (2004-2010). Their works (2004, 2005) are also devoted weighted zero-moments when k ≥1 and for the weighted first to the design of a hierarchy of 2D models for dynamical problems moments when k ≥ 3/1 ) BCs are proved in weighted Sobolev within the theory of multicomponent linearly elastic mixtures in the spaces. case of prismatic shells with thickness which may vanish on some Jaiani and Schulze (2007) studied the vibration tension-compre- parts of its boundary, provided that the 3D domain occupied by the ssion system [(vanishing of the vibration frequency corresponds to prismatic shell is the Lipschitz one. The above method does not the static system) (20), (23)] under all reasonable nonhomogeneous allow to consider BVPs when on the cusped edge either displace-

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ments or loads (the loads in this case are concentrated along the cus- r ped edges) are prescribed. where bis depend only on the thickness. Coerciveness of the corres- By N. Chinchaladze, R. P. Gilbert, G. Jaiani, S. Kharibegashvili, ponding bilinear form is shown and uniqueness and existence results D. Natroshvili (2008) the well-posedness of BVPs for elastic cusped for the variational problem are proved. The structure of the spaces k plates (i.e., symmetric prismatic shells) in the N th approximation X N is described in detail and their connection with weighted Sobo- N ≥ 0 of I.Vekua’s hierarchical models [see system (18) in the static lev spaces is established. Moreover, some sufficient conditions for a case] under all the reasonable BCs at the cusped edge and given linear functional arising in the right hand side of the variational displacements at the non-cusped edge are studied. The approach is equation to be bounded are given. N = 2,1,0 approximations are applicable in the same way also for non-symmetric prismatic shells. considered in detail. Peculiarities characterizing these concrete Special attention is drawn to the N = 2,1,0 approximations as to models are exposed. Note that for the r th order moments Dirichlet important cases from the practical point of view. For example, and Keldysh BCs are correct when N = 0 and N = 1 models, roughly speaking, coincide with the plane 1 1 k < and k ≥ , = Nr ,,0 deformation and Kirchhoff-Love model, respectively. It is assumed r +12 r +12 that the cusped plate projection ω has a Lipschitz boundary respectively. The works of G. Jaiani (1973,2008) deals with a system consis- ω γ ∪=∂ γ10 , where γ 0 is a segment of the x1 -axis and γ1 lies in ting of singular partial differential equations of the first and second the upper half-plane x > ;0 moreover, in some neighborhood of an 2 order arising in the zero approximation of I.Vekua’s hierarchical edge of the plate, which may be cusped, the plate thickness has the models of prismatic shells, when the thickness of the prismatic shell form (24). Then γ 0 will be a cusped edge for k > 0 . Note that in the varies as a power function of one argument and vanishes at the last case, on the one hand, a 3D domain Ω occupied by the plate is cusped edge of the shell [see (24)]. For this system of special type a non-Lipschitz for k >1; on the other hand, the governing system nonlocal BVP in the half-plane is solved in the explicit form. The consisting of N + 33 simultaneous equations [see (18) in the static BVP under consideration corresponds to the stress-strain state of the cusped prismatic shell under the action of concentrated forces and case] is elliptic in Ω and has an order degeneration on γ for any 0 concentrated couples. k > 0 . The classical and weak setting of the BVPs in the case of the Surveys of the works by G. Giorgadze, G. Avalishvili, N. Chin- N th approximation is considered. For arbitrary k ≥ 0 appropriate chaladze and R.P. Gilbert, T. Meunargia, T. Vashakmadze and some k other talks of the program of the GeoMech8 can be considered as weighted function spaces X N ω)( which are crucial in analysis of the k continuation of the present survey in the sense of presentation more problem are introduced. X N ω)( is the completion of the space in detail the development of the heritage of Ilia Vekua which (the D ω)]([ N +33 with the help of the norm: development) should be treated as his heritage in a wide sense. n 3 N 3 2 ⎛ 1 ⎞ 2 dω 1 ⎛ 1 ⎞ k ννν),( k ⎜ +== ⎟ er ijr ()ν = ⎜r + ⎟ X N X N ∑ ∫ ∑ ∑∑ rj==0 ⎝ 2 ⎠ω i 1. h 4 rj==0 i 1, ⎝ 2 ⎠ 2 N r r ⎡ r+1 s+1 ⎛ ⎞⎤ dω × h νν)( ++ js + bbh is νν , ∫ ⎢ ,, ijrjir ∑ ⎜ is js ⎟⎥ ω ⎣ rs += 1 ⎝ ⎠⎦ h

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ON DEVELOPMENT OF I. VEKUA METHOD FOR frameworks of classical linear theory of elasticity, thermoelasticity, CONSTRUCTION OF HIERARCHICAL MODELS OF theory of mixtures, thermoelasticity with microtemperatures, and

ELASTIC STRUCTURES non-classical theories of thermoelasticity. On the basis of the results

Gia Avalishvili*, Mariam Avalishvili** of investigation of dimensional reduction algorithms for plates, shells *I. Javakhishvili Tbilisi State University, Faculty of Exact and Natural and bars pluri-dimensional hierarchical models were constructed and Sciences, Tbilisi, Georgia, [email protected] studied for elastic multistructures consisting of several parts with **University of Georgia, Tbilisi, Georgia, [email protected] different geometric shapes.

In the present talk results on application of I. Vekua's dimensio- Acknowledgement. This work was supported by Shota Rustaveli nal reduction method, its extensions and generalizations for construc- National Science Foundation (SRNSF) [217596, Construction and tion of hierarchical models of elastic structures are presented. In [1] investigation of hierarchical models for thermoelastic piezoelectric Ilia Vekua constructed a hierarchy of two-dimensional models for structures]. linearly elastic homogeneous plates with variable thickness. In this paper, multiplying equations, corresponding to three-dimensional References model by Legendre polynomials with respect to the variable x of 1. Vekua, I.N.: On a way of calculating of prismatic shells, Proc. A. 3 Razmadze Inst. Math. Georgian Acad. Sci., 21 (1955), 191-259 plate thickness, integrating them and expanding components of the (Russian). displacement vector-function, stress and strain tensors into orthogo- 2. Vekua, I.N.: Shell theory: General methods of construction, Pitman nal Fourier-Legendre series with respect to x3 and considering par- Advanced Publishing Program, Boston, 1985. tial sums of the series a hierarchy of two-dimensional models for pla- 3. Gordeziani, D.G.: On the solvability of some boundary value problems for a variant of the theory of thin shells, Dokl. Akad. Nauk SSSR, 215 te was constructed. Note that the classical Kirchhoff-Love and (1974), 6, 1289-1292 (Russian). Reissner-Mindlin models can be incorporated into the hierarchy, 4. Gordeziani, D.G.: To the exactness of one variant of the theory of thin obtained by I. Vekua so that it can be considered as an improvement shells, Dokl. Akad. Nauk SSSR, 216 (1974), 4, 751-754 (Russian). of the frequently used engineering plate models. Mathematical 5. G. Jaiani, On a mathematical model of bars with variable rectangular models of plates and shells, constructed by I. Vekua, are presented in cross-sections, ZAMM, 81 (2001), 3, 147-173. his monograph [2]. First results on the investigation of mathematical models constructed by I. Vekua were obtained by D. Gordeziani in [3, 4], SOLITARY WAVES IN FLUIDS WITH VARIABLE where the static two-dimensional models for general thin shallow DISPERSION shells in Sobolev spaces were investigated and the relationship Vasily Yu. Belashov*, Elena S. Belashova**, Oleg A. Kharshiladze*** between the static three-dimensional model and two-dimensional * Kazan Federal University, Kazan, Russia, [email protected] ones in the spaces of classical smooth enough functions in the case of ** Kazan National Research Technical University named after A.N. homogeneous isotropic linearly elastic plate with constant thickness Tupolev – KAI, Kazan, Russia was studied. Later on, applying variational approach and an idea of I. ***I. Javakhishvili Tbilisi State University, Tbilisi, Georgia, Vekua and its generalization [5] static and dynamical hierarchical [email protected] two-dimensional models for plates and shells and one-dimensional models of bars were constructed and investigated within the We study the problem of dynamics the 2D and 3D solitary waves in fluids with the varying in time and space dispersive 29 30

parameter =β β t r),( . For example, that have place on studying of The equations’ set d = ∂ H d, ye/Bxe = −∂ H/B , i xitiyiti i the evolution of the 3D FMS waves in magnetized plasma, which is tρ∂ + v ⋅ ∇ρ = 0 , = −(zv ˆ × ∇ψ /) B , Δψ = −ρ describing the conti- described by the KP equation [1], when β is a function of the Alfvén nuum or quasi-particles with Coulomb interaction models [1], where velocity vА = f [ B (t, r), n (t, r)] (n is the plasma density) and the angle ∧ 2 22 ρ is a vorticity or charge density and ψ is a stream function or θ=θ Bk )( : =β A cv 0i )(cot2/( −θω mm ie )/ . Similar situation potential for inviscid fluid and guiding-centre plasma, respectively, takes place for the ion-acoustic (IA) waves in collisional dusty and H is the Hamiltonian, was considered. For numerical simulation plasma when in the absence of dissipation the dispersion law are the CD method specially modified was used. The results showed that =ω kVs where = 00 + /)/)(/( mTnnmTV iieiies is the IA speed for some conditions the interaction is nontrivial and can lead to in dissipationless plasma with constant-charge dust. It is clear that the formation of complex forms of vorticity regions, such as the vorticity dispersion will be variable with variation of ratio of plasma filaments and sheets, and also can ended to formation of the turbulent components. Similar situation can also take place for solitary waves field. The theoretical explanation of the effects is given on the basis on shallow water with variable depth [1]. We present here the results of the generalized critical parameter which determines qualitative of numerical simulation of the solitary waves in the KP model character of interaction. We investigated the applications to dynamics distracting from a specific type of medium for different model of vortex structures in the atmosphere, hydrosphere and plasma, functions β. As a result we have obtained the different types of stable namely: evolution of the cyclonic type synoptic and ocean vortices, and unstable solutions including the solutions of the mixed "soliton − and interactions in the vortex-dust particles system, and also non-soliton" type for different character of dispersion variations. dynamics of streams of charged particles in a uniform magnetic field. Our approach may be useful for the interpretation of effects Acknowledgement. The work is performed according to the Russian associated with turbulent processes in fluids and plasmas. Government Program of Competitive Growth of Kazan Federal University. Acknowledgement. The work is performed according to the Russian References Government Program of Competitive Growth of Kazan Federal 1. Belashov, V.Yu., Belashova, E.S.: Solitons: Theory, simulation, University. applications. Publishing Center “School”, Kazan, (2016). References 1. Belashov V. Yu. Modeling of dynamics of vortex structures in continuous media. Astrophys. Aerospace Technol. 4(3) (2016), 28. NUMERICAL MODELING OF INTERACTION OF VORTEX STRUCTURES IN FLUIDS AND PLASMAS

Vasily Yu. Belashov*, Oleg A. Kharshiladze** ON SOME SOLUTIONS OF ELASTIC MATERIALS WITH VOIDS *Kazan Federal University, Kazan, Russia, [email protected] **I. Javakhishvili Tbilisi State University, Tbilisi, Georgia, Lamara Bitsadze [email protected] I.Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University, Tbilisi, Georgia, [email protected] The results of numerical modeling of interaction of the vortex structures in a continuum, and, specifically, in fluids and plasmas in In this talk the 3D quasi-static theory of elasticity for materials 2D approach, when the Euler-type equations are valid, are presented. with voids is considered. The representation of regular solution of the system of equations in the considered theory is obtained. There the 31 32

fundamental and some other matrixes of singular solutions are and torque on a roller are defined. Therefore we define the power constructed in terms of elementary functions. characteristic of the technological process.

References References 1. Iesan D. A Theory of Thermoelastic Materials with voids, Acta 1. Малинин Н.Н. Ползучесть в обработке металлов. М.: Машинос- Mechanica, 60 (1986), 67-89. троение, (1986). 216 с. 2. Cowin Stephen C., Nunziato Jace W. Linear theory of elastic materials 2. Киквидзе О.Г., Булекбаева Г.Ж., Кипиани П.Н. Пластическое with voids. Journal of Elasticity, 13 (1983), 125–147. деформирование наплавленного слоя на плоской поверхности. //Georgian Engineering News, №1 (Vol. 77). (2016), 64-66.

DETERMINATION OF STRESS – STRAIN STATE OF WELDED LAYER AT ROLLING BY CREEP THEORY CUSPED PRISMATIC SHELL-FLUID INTERACTION PROBLEMS Gulmira Bulyekbaeva*, Omar Kikvidze** *Caspian State University of Technologies and Engineering named Natalia Chinchaladze*, Robert Gilbert** after Sh. Yessenov [email protected] * Ilia Vekua Institute of Applied Mathematics & Department of Mathematics **Akaki Tsereteli State University [email protected] of Ivane Javakhishvili Tbilisi State University, Tbilisi, Georgia **University of Delaware, DE, USA y Technological process of a weld carries out restoration of worn- out surfaces of details. For the purpose of improvement of mechani- The talk is devoted to the updated survey of problems with the cal characteristics and quality of a surface of the built-up layer plastic some elastic cusped structure-incompressible fluids interaction deformation is necessary that is often carried out by a rolling. At a problems, when in the solid part either the Kirchhoff-Love plate or rolling plastic deformation is made by a rigid cylindrical roller which Vekua’s prismatic shell in the lower order approximations are makes the flat parallel movement with a constant velocity. considered (see, e.g. [1-6] and references therein). We consider a two-dimensional, established, viscous – plastic Application of I. Vekua’s dimensional reduction method to the flow of a material on the basis of the theory of hardening [1]: viscous Newtonian fluid occupying thin prismatic domains will be nm also presented [7]. = a ee κξσ References where = ξκdt is the Odqvist parameter, σ is the equivalent ∫ e e 1. N. Chinchaladze, On some nonclassical problems for differential stress, ξ is the equivalent speed of deformation, ,, nma are equations and their applications to the theory of cusped prismatic shells. e Lect. Notes TICMI, 9 (2008) constants of material. 2. N. Chinchaladze, G. Jaiani, On a cylindrical bending of a plate with two The nonlinear differential equations of equilibrium are written cusped edges under action of an ideal fluid. Bull. TICMI 2, 30–34 down concerning components of flow velocity whom include also (1998) average stress [2]. 3. N. Chinchaladze, G. Jaiani, On a cusped elastic solid–fluid interaction Boundary conditions for numerical integration of the equations problem. Appl. Math.Inform. 6(2), 25–64 (2001) are written down. After determination of velocity of a flow we calcu- 4. N. Chinchaladze, R. Gilbert, Cylindrical vibration of an elastic cusped late components of a deviator of stress by the equations of Saint- plate under the action of an incompressible fluid in case of N = 0 approximation of I. Vekua’s hierarchical models. Complex Var. Theor. Venant–Levy–Mises. The force necessary for deformation of a layer Appl. 50(7–11), 479–496 (2005) 33 34

5. N. Chinchaladze, R. Gilbert, Vibration of an elastic plate under action ning the gas non-stationary flow in the inclined and branched of an incompressible fluid in case of N = 0 approximation of I. Vekua’s pipeline is obtained. Some results of numerical calculations of gas hierarchical models. Appl. Anal. 85(9), 1177–1187 (2006) flow in the inclined and branched pipelines are presented. 6. N. Chinchaladze, R.P. Gilbert, G.V. Jaiani, S. Kharibegashvili, D. Natroshvili, Initial-boundary value problems for solid–fluid composite structures, ZAMP, 63 (4), 625-653 (2012) CONFLICTS AND CATASTROPHES 7. N. Chinchaladze, G. Jaiani, Hierarchical Mathematical Models for Solid–Fluid Interaction Problems (Georgian). Materials of the Guram Gabrichidze International Conference on Non-Classic Problems of Mechanics, Georgian National Academy of Sciences, Tbilisi, Georgi, Kutaisi, Georgia, 25–27 October, Kutaisi, 2, 59–64 (2007) [email protected]

The purpose of the work is to study conditions of conflict-free TRANSIENT GAS FLOW MODELING IN INCLINED AND and sustainable development of processes as well as its regulation BRANCHED PIPELINE possibilities.

Teimuraz Davitashvili Extrapolation and generalization of the results obtained in the I. Javakhishvili Tbilisi State University, I.Vekua Institute of Applied mechanics give the basis to state that any process or situation may Mathematics, Tbilisi, Georgia, [email protected] also be characterized by two global parameters - the change of any common features that unite the multiplicity and the attitude of the Natural gas distribution networks are complex systems with objects or subjects involved in this union towards this change. hundreds or thousands of kilometers of pipes, compression stations In any process, any kind of distinction is regarded as the cause and many other devices for the natural gas transportation and of the conflict. Representation of the new vector form of matrix distribution service. Achievement in the power consumption points shows it clearly. with the required conditions is the main practical aspect and the most Conditions that provide the conflict-free and sustainable difficult issue in the gas transmission pipeline system. Determination development of the processes are formulated. of gas pressure and flow rate distribution along the pipelines is a ne- The received results are compared with the rule of relationship, cessary step for solving the above mentioned question. Searching of with so-called Golden rule and other models which are considered in gas flow pressure and flow rate along the inclined and branched modern Game theory. pipeline network has much more practical value but represents more difficult issue. For this purpose development of the mathematical models describing gas non-stationary flow in the branched and inclined pipeline systems are actual. The purpose of this study is determination of gas pressure and flow rate special and temporally distribution along the pipeline based on simplified one-dimensional partial differential equations governing the gas non-stationary flow in the inclined and branched pipeline. The simplification is established on the hypothesis that the boundary conditions do not change quickly and the capacity of gas duct is relatively large. Analytical solution of the simplified one-dimensional partial differential equations gover-

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I. VEKUA'S THEORY OF THE ANALYTIC FUNCTIONS: THE SOLUTION OF ONE PROBLEM OF THEORY OF THE SPACE OF GENERALIZED ANALYTIC FUNCTIONS SHELLS BY METHOD OF I. VEKUA FOR

Grigori Giorgadze APPROXIMATION N=3

I.Vekua Institute of Applied Mathematics, Tbilisi, Georgia, Bakur Gulua [email protected] I. Javakhishvili Tbilisi State University, I.Vekua Institute of Applied Mathematics, Tbilisi, Georgia The idea of using the methods of complex analysis for a wider Sokhumi State University, Tbilisi, Georgia class of functions then the space of analytic functions takes its origin [email protected] from the end of 19th century and is actually connected with the period of conception of complex analysis as the independent branch We consider the geometrically nonlinear and non-shallow of science. First attempts for the extension of the complex methods spherical shells for I. N. Vekua N=3 approximation. The concrete were connected with the generalization of the Cauchy-Riemann problems, using complex variable functions and the method of the system and the investigation of the properties of the space of small parameter has ve been solved. solutions for such a system. After the appearance of the monographs of I.Vekua and L.Bers, Acknowledgement. The designated project has been fulfilled by a generalized analytic functions due to the terminology of Vekua and financial support of Shota Rustaveli National Science Foundation pseudo-analytic functions due to Bers these problems are the subject (Grant SRNSF/FR/358/5-109/14). of investigation by many scientists. We will consider also application of Bers-Vekua theory in the conformal fields theory, in the theory of deformation of complex SOME THREE-DIMENSIONAL BOUNDARY structures and the theory of two dimensional exactly solvable models VALUE PROBLEMS FOR AN ELASTIC BINARY MIXTURES of quantum mechanics (see [1]). WITH DOUBLE POROSITY

Roman Janjgava References I.Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State 1. Akhalaia G., Giorgadze G., Kaldani N., Jikia V., Manjavidze University, Tbilisi, Georgia, [email protected] N., Makatsaria G. Elliptic systems on Riemann surfaces. Lecture Notes of TICMI, vol. 13, pp.3-154, 2012. The talk deals with a linear system of equilibrium equations for elastic bodies with double porosity, where the solid body skeleton is a mixture of two isotropic materials. The general solution of this IMPROVEMENT WAYS OF CRANE-TRANSPORT AND system of equations is represented by means of harmonic functions ROAD-TRANSPORT MACHINES’ WORKING APPLIANCES and a metaharmonic function. On the basis of the general solution

Vazha Gogadze and using the method of separation of variables, the class of boun- A.Tsereteli state university, Georgia dary value problems for a rectangular parallelepiped is analytically [email protected], [email protected] solved.

In above work is described kinematic and structural researches Acknowledgement. The designated project has been fulfilled by a of crane-transport and road-machines’ working appliances. financial support of Shota Rustaveli National Science Foundation

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(Grant SRNSF/FR/358/5-109/14). MODELING OF NONELASTIC INTERACTIONS OF OPTICAL SOLITONS

References Oleg Kharshiladze*,****,Vasily Belashov**, 1. Natroshvili D. G., Jagmaidze A. Ya., Svanadze M. Zh.: Some Jemal Rogava***, Khatuna Chargazia***,**** problems of the linear theory of elastic mixtures, TSU press, 1986 (in *Dept. of Physics, I. Javakhishvili Tbilisi State University, Tbilisi, Georgia, Russian). **Kazan Federal University, Kazan, Russia, 2. Khomasuridze N., Janjgava R. Solution of some boundary value ***I. Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi thermoelasticity problems for a rectangular parallelepiped taking into State University, Tbilisi, Georgia, account microthermal effects, Meccanica, 51 (2016), 211-221.\ **** M. Nodia Institute of Geophysics, I. Javakhishvili Tbilisi State University, Tbilisi, Georgia, [email protected] ON ONE PROBLEM OF THE PLANE THEORY OF ELASTICITY FOR THE REGION WITH A PARTIALLY Investigation of laser soliton propagation and interaction in UNKNOWN BOUNDARY optical fibber for the information transmission is a very actual

Giorgi Kapanadze problem. This interaction sufficiently changes the characteristics of I. Vekua Institute of Applied Mathematics of I. Javakhishvili the light field and distorts transmitted information. For the control of Tbilisi State University, Tbilisi, Georgia the soliton shape, stability and dynamics, it is necessary to study an A. Razmadze Mathematical Institute of I. Javakhishvili influence of fibber defects, dispersive and nonlinear inhomogeneities, Tbilisi State University, Tbilisi, Georgia and nonstationary parameters of medium on the character of soliton [email protected] propagation. The problem reduces to the nonlinear Schrodinger

(NLS) equation for the amplitude of the light field with coefficient The problem of the plane theory of elasticity with a partially functions having spatial and temporal inhomogeneities. unknown boundary (the problem of finding an equally strong Fourier splitting method for the NLS equation was used at nu- contour) for a rectangular plate weakened by an equally strong merical modeling, and the inhomogeneities of coefficient functions contour (the unknown part of the boundary) is considered. It is were taken into account. The NLS equation is divided into linear and assumed that the linear segments of the boundary are under the action nonlinear parts, dispersive and nonlinear effects are considered sepa- of normal contractive forces with the given principal vectors and the rately, corresponding operators are assumed commutative. Implicit unknown part of the boundary is free from external forces. The scheme of finite-difference method is used for investigation of soli- condition for the unknown contour to be equally strong is that the ton propagation in non-uniform and nonstationary environment. tangential normal stresses are stable on it. Numerical modeling shows that inhomogeneity of medium For solving the problem, the methods of complex analysis are changes the amplitudes of solitons and other light impulses, their ve- used; the sought complex potentials and equations of an unknown locities of propagation, their quantity that is caused by their nonelas- contour are constructed effectively (in the analytical form). tic interaction in inhomogeneous fibber. Nonstationary medium chan-

ges a form of impulse and affects its spectral features. Changes of Acknowledgement. This work was supported by a financial support modulation of the parameters of medium make possible variation of of Shota Rustaveli National Science Foundation (Grant character of nonelastic interaction at solitons SRNSF/FR/358/5-109/14). attraction-repulsion.

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References References 1. Agrawal G.P. Nonlinear Fiber Optics. – Academic Press, – 2013. 1. Varamashvili N., Chelidze T., Devidze M., Chelidze Z., Chikhladze V., 2. Belashov V.Yu., Belashova E.S. Solitons: theory, simulation, Surmava A., Chargazia Kh., Tefnadze D. Mass-movement and seismic applications. Kazan, 2016. processes study using Burridge-Knopoff laboratory and mathematical 3. Belashov V.Yu., Vladimirov S.V. Solitary Waves in Dispersiv Complex models. Journal of Georgian Geophysical Society, Issue (A), Physics of Media. Theory, Simulation, Applications. Springer-Verlag GmbH&Co, Solid Earth, (2015), 19-25. 2005. 2. Dieterich J.H. Modeling of rock friction 1. Experimental results and constitutive equations. Journal of Geophysical Research, 84, B5, (1979), 2161-2168. DYNAMICAL ANALYSIS OF FRICTIONAL 3. Burridge R., Knopoff L. Model and theoretical seismicity. Bulletin of the AUTO-OSCILLATIONS Seismological Society of America, 57 (3): (1967), 341-371.

, , 4. Ruina A. Slip instability and state variable friction laws. Journal of Oleg Kharshiladze* ***, Khatuna Chargazia** ***, Geophysical Research, 88, (1983), 10359-10370. Nodar Varamashvili***, Dimitri Amilaxvari*, Levan Dvali* 5. Chelidze T., Varamashvili N., Devidze M., Chelidze Z., Chikhladze V., *Dept. of Physics, I. Javakhishvili Tbilisi State University, Tbilisi, Georgia Matcharashvili T. Laboratory study of electromagnetic initiation of slip. **I. Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State Annals of Geophysics, 45 (2002), 587-599. University, Tbilisi, Georgia *** M. NodiaInstitute of Geophysics, I. Javakhishvili Tbilisi State University, Tbilisi, Georgia [email protected] CALCULATION OF THE RIBBED SHELL BY A FINITE ELEMENT METHOD

An earthquake is commonly described as a stick-slip frictional Gela Kipiani, Seit Bliadze instability occurring along preexisting crustal faults. The seismic Aviation University of Georgia, Tbilisi, Georgia cycle of earthquake recurrence is characterized by long periods of [email protected], [email protected] quasi-static evolution (tectonic processes) which precede sudden slip events accompanied by elastic wave radiation: the earthquake. This In this article, the development of the stages of calculating plates succession of processes recalls the behavior of nonlinear relaxation and shells with stiffeners is described. It is noted that the theory of oscillations. ribbed shells is one of the most controversial and incomeplete In the present work the nonlinear dynamics of three-block sections of the general theory of shells. The calculations of shells systems due to Burridge-Knopoff model of dry frictions without with longitudinal and transverse stiffeners by the finite element velocity restrictions is also studied using numerical methods and method are considered. In the first variant, the stiffener is considered possibility of „stick-slip“ motion and triggering of instabilities by as a beam end element, and the sheath is modeled as a finite element recording of acoustic emission, accompanying the slip events is by a thin plate. In the second case, both the ribs and the shell are revealed. Observation data analysis is carried out. Spectral features, modeled as a finite element by a thin plate. In the third case, the recurrent quantitative and qualitative characters, wavelet diagrams of edges and shells are modeled as a single three-dimensional finite obtained signals are studied. Stick–slip occurs only within a narrow element. In the fourth case, the edges are modeled as a three-dimen- range around these critical speeds of a system. External damping can sional volume element, and the shell as a finite element of the plate. prevent stick–slip motion. Experimental studies of the indicated shells were carried out in the laboratory of strength at the testing base of the JSC "Tbilaviamshe- ni", certified according to BS EN ISO 9001 and EN 9100. A compa- 41 42

rative analysis of the theoretical and experimental results (Accuracy between the components of stress tensor with the angle of internal ≈ 5 %). friction, and between the speed deformation tensor components, which show the coincidence of the direction of the maximum speed References of motion deformation with one of the directions of the family of 1. Strang G., Fix G. J. An Analysis of the Finite Element Method. sliding lines (Ishlinsky-Geniev Condition), plus considering the Prentice-Hall, inc. (1973). plasticity caused by snow humidity [1]. A body of finite sizes is taken as a model of the avalanche body. Its geometric sizes and difference between the center of masses and MATEHMATICAL MODELING OF THE DYNAMICAL speeds of moving the frontal areas are considered. The obtained PROCESSES OF AVALANCHE-LIKE CURRENTS results play the role of such auxiliary factors, without which no

Tariel Kvitsiani correct analysis or interpretation is possible. This concerns such Georgian Technical University, Tbilisi, Georgia, [email protected] parameters, as the height and length of an avalanche body, coordinate of its longitudinal profile, front speed as the function of distance and Aiming at studying the dynamic processes of avalanche-like time, power parameters influencing the resistances, impact of cold currents, the work gives the data about the conditions of avalanche winds on the snow cover, etc. playing an essential role in examining hazard, in particular, the dependence of avalanching on surface the validity of the obtained results [2,3]. inclination giving a general view of the processes of origination and distribution of avalanches and being quite important in theoretical Reference and quantitative evaluation. 1. Kvitsiani T. Slope Stability and Avalanche-Like Currents. Publishing A continuous deformed body model is taken as a main physical- House “Technical University”. Tbilisi, (2010), p. 140. mechanical model of snow avalanches thoroughly showing the nature 2. Losov K.S. Some Questions of Avalanche Motion and Other Similar of plasticity and loosening. The dynamic processes of snow avalan- Phenomena. The Works of All-Union Meeting on Avalanches. Leningrad: Hydrometeodizdat, (1978), 64-69. ches are described by means of a system of equations of a composite 3. Moskalev I.R. Origination and Motion of Avalanches. Leningrad: environment (landslide-slide) presented as hydrodynamic equations Hydrometeodizdat, (1976), p. 272. and its one-dimensional option [1]. These systems consider loose and plastic properties and plasticity as well, which is typical of bin- gamma liquids at the expense of considering motion limit strain in the dynamic equations. When the parameters of the grain loose environment equals zero, the given system is automatically transfor- med into the Hencky-Ilyushin equation, which describes so called plastic-viscous liquids (Bocher-Bingaman liquids) typical to “soft” snow avalanches with significant humidity. The equations of basic physical-mechanical model are made of: the equation of the boundary state of a loose environment with the angle of internal friction and adherence coefficient (Coulomb-Tresk- St. Venant Condition), which defines the mutual connection between the components of tangential and normal stresses, as well as the link

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ON THE IMMERSION OF THE SURFACE IN THE 3D In the first example we study a continuous stream of matter on a

RIEMANN DIVERSITY conveyor belt going through a crusher so that the total number of

Tengiz Meunargia particles will change. In context with this example it will also I.Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State become clear that the traditional Lagrangian way of describing the University, Tbilisi, Georgia, [email protected] motion of solids is no longer adequate and must be replaced by the Eulerian point of view known from fluid mechanics. The second The surfaces with non-zero Gaussian curvature are the Rie- example deals with hollow particles, which rotate because of the mann’s diversity of 2-dimension, which are investment in the 3-D presence of body couples. Now a transient temperature field is supe- Euclidean space. Therefore, for these varieties of properties it is pos- rimposed, such that the moment of inertia field will change due to sible to construct quite clear representations. Further, it is shown that thermal expansion of the particles. This in turn will result in rotatio- any regular surface can be put in the Riemann’s 3-dimensional diver- nal motion that is no longer constant but varies in space and time. sity. Acknowledgement. This work was supported by a DFG grant Acknowledgement. The designated project has been fulfilled by a MU1752/43-1 and by Russian Foundation for Basic Research (16-01- financial support of Shota Rustaveli National Science Foundation 00815). (Grant SRNSF/FR/358/5-109/14).

INVESTIGATION OF THE BENDING DEFORMATION OF A FEW REMARKS ON RECENT DEVELOPMENTS IN CUSPED PRISMATIC PLATES BY FINITE ELEMENT MICROPOLAR CONTINUUM THEORY METHOD AND THE DEVELOPMENT OF THE REDUCTION

Wolfgang H. Müller*, Elena N. Vilchevskaya** METHOD WITH THE USE OF APPROXIMATION BY ANALYTIC FUNCTIONS * Institute of Mechanics, Chair of Continuum Mechanics and Constitutive Theory, Berlin Institute of Technology, Einsteinufer 5, 10587 Berlin, Giorgi Nozadze Germany, [email protected] LEPL G. Tsulukidze Mining Institute ** Institute for Problems in Mechanical Engineering of the Russian [email protected] Academy of Sciences, Bol’shoy pr. 61, V.O., 199178 St. Petersburg, Russia and Peter the Great Saint-Petersburg Polytechnic University, The talk deals with the bending deformation of a cusped Politekhnicheskaja 29, 195251 St.-Petersburg, Russia, prismatic plate, which is bounded by surfaces of different curvature [email protected] and has a plane of symmetry (see, e.g. [1]). It is well-known, that the

three-dimensional elasticity problem in such a case can be reduced to In this presentation we consider micropolar media that can a two-dimensional problem of the elasticity of a plane deformed state undergo structural changes and do not a priori consist of indestruc- of a body. tible material particles. Initially the pertinent literature will be revie- The problem of pure bending of a uniformly loaded special wed. Then the necessary theoretical framework for a continuum of (cusped) segment of the body region is considered. A discrete that type is presented: The standard macroscopic equations for mass, solution of the problem of elasticity in displacements can be obtained linear and angular momentum are complemented by a recently with the help of the finite element method. proposed balance for the moment of inertia tensor, which contains a production term. Two examples illustrate the effect of the production. 45 46

According to the discrete solutions, the displacement function is founded by means of an algorithm, the constituent parts of which can be represented by means of approximation by analytic functions. are the Galerkin method, a symmetric difference scheme and Jacobi In particular, sign variable functional power series with a finite iterative method. number of terms give satisfactory results at discrete points of the For both of these problems the new algorithms of approximate solution of the presented problem. solutions are constructed and numerical experiments are carried out. On the basis of the work performed, it can be shown that in The results of calculations are presented by tables and diagrams. solving elasticity problems in specific areas of geometric uncertainty, it is possible to develop reduction analytical methods that could be Reference used to improve the study of the deformed state of the body in these 1. Peradze J. A numerical algorithm for a Kirchhoff – type nonlinear regions, within acceptable results in practice. static beam. J. Appl. Math. 2009, Art.ID 818269, 12 pp. 2. Ma T.F. Positive solutions for a nonlocal fourth order equation of Reference Kirchhoff type. Discrete Contin. Dyn. Syst. (2007), 694-703. 1. Georg Jaiani - Cusped Shell-like Structures, Springer Briefs in Applied 3. Ball J. M. Stability Theory for an Extensible Beam, J. Diff. Eq., 14 Science and Technology, Springer-Heidelberg-Dordrecht-London-New (1973), 399-418. York, 2011, 84 p. 4. Papukashvili G., Peradze J., Tsiklauri Z. On a stage of a numerical algorithm for Timoshenko type nonlinear equation, Proc. A.Razmadze Math.Inst., Tbilisi, 158 (2012), 67-77. ON APPROXIMATE SOLUTION OF THE ALGORITHMS AND NUMERICAL COMPUTATIONS FOR SOME KIRCHHOFF TYPE NONLINEAR INTEGRO- MODELING AND ANALYSIS OF COMPLEX CABLE-ROD DIFFERENTION EQUATIONS STRUCTURES AND OTHER SIMILAR BUILDING STRUCTURES BASED ON DISCRETE REPRESENTATION Archil Papukashvili*, Giorgi Papukashvili**, Jemal Peradze*** AND SPECIAL ALGORITHMS * I. Javakhishvili Tbilisi State University, I.Vekua Institute of Applied Mathematics, Faculty of Exact and Natural Sciences David Pataraia ** V. Komarovi N 199 Public School, Georgian Technical University G. Tsulukidze Mining Institute, Tbilisi, Georgia, *** I. Javakhishvili Tbilisi State University, Faculty of Exact and Natural [email protected] Sciences, Georgian Technical University Tbilisi, Georgia The subject od research in the first place are complex cable-rod [email protected], [email protected], structures such as cableways, cable truss bridges, spacecraft cable [email protected] systems and antennas, protective roofs of stadiums and other having large sizes objects, avalanche-protection structures and arresting In the present talk we consider approximate solution issues for netting. In general, the subject of research of presented work is the following two problems: 1.Nonlinear boundary value problem for modeling and analysis of having similar complex configuration solid the Kirchhoff type static beam (see, for example [1], [2]). The deformable bodies. problem is reduced by means of Green’s function to a nonlinear The modeling and analysis of such objects by standard methods integral equation. To solve this problem we use the the Picard type and software is often associated with significant problems due to the iterative method; 2. Nonlinear initial-boundary value problem for the cause of hysterisis, clearances, friction and other essential nonlinear J.Ball dynamic beam (see, for example [3], [4]). Solution of problem relations. In addition, the analysis of large size objects, especially

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when the accuracy of the results is needed, also rises the increased sukzessiven Approximationsalgorithmus WORLD CONGRESS of and often hard-to-conduct requirements for applied computing O.I.T.A.F., Bolzano, Italy, 6-9 Juni, 2017 PAPERS OF THE methods and appropriate computer engineering. CONGRESS; http://oitaf2017.com/wp-content/uploads/2017/06/oitaf- The relevance of the research also is providing the complexity of kongr-bolcano-040617- vort-germ- JN-050617- 2.pdf adaptability and application for solution of specific tasks of appropriate universal methods and computational software packages, JACOBI ITERATION FOR A BEAM DYNAMIC PROBLEM especially when the object ubder study has non-linear characteristics and even large size. Jemal Peradze The novelty of the study lies in the method of static and dynamic I. Javakhishvili Tbilisi State University, Georgian Technical University, Tbilisi, Georgia, [email protected] calculation of complex rope-rod structures and other similar building structures developed by us, which is based on the discrete An initial boundary value problem for the differential equation representation of solid deformable bodies and a special computing [1, 2] 2 algorithm [1], [2], [3]. ∂ 2u ∂ 4u ∂ 4u ⎛ L⎛ ∂u ⎞ ⎞∂ 2u tx + − htx tx ),(),(),( ⎜λ +− ⎜ ⎟ dt ξξ⎟ tx 0=),(),( On the basis of the proposed approach, we developed an ∂t 2 ∂x 4 ∂∂ tx 22 ⎜ ∫⎜ ∂ξ ⎟ ⎟∂x 2 ⎝ 0 ⎝ ⎠ ⎠ algorithm and a countable program for the static and dynamic describing the dynamic behavior of beam is considered. As a result of calculation of solid deformable bodies with a complex configuration approximation of the solution with respect to the spatial and time and with nonlinear characteristics. The effectiveness and suitability variables a nonlinear system of discrete equations is obtained, which of the approach was confirmed by practical testing on specific objects is solved by iteration. The convergence conditions and the error and comparison with the results obtained by the standard software. estimate of the iteration are obtained.

Acknowledgement. A significant part of this work was accomp- References lished thanks to the support of the Rustaveli Foundation and 1. Henriques de Brito E.: A nonlinear hyperbolic equation, International cooperation with the staff of the Institute of Applied Mathematics of Journal of Mathematics and Mathematical Sciences, 3 (3), (1980), 505- Tbilisi State University and with its director Professor G. Jaiani. 520. 2. Timoshenko S.P., Young D.H., Weaver W.: Vibration problems on Reference engineering. John Wiley & Sons, Inc., 1974. 1. David Pataraia. The calculation of rope-rod structures of ropeways on the basis of the new approach.WORLD CONGRESS of O.I.T.A.F., RIO DE JANEIRO, BRAZIL, October, 2011, 24 – 27, PAPERS OF THE CONGRESS; http://www.oitaf.org/Kongress%202011/Referate/Pataraia.pdf 2. D. Pataraia. Ein Beitrag zur diskreten Darstellung des Seiles. Wissenschaftliche Arbeiten des Instituts fuer Eisenbahnwesen, Verkehrswirtschaft und Seilbahnen, Technische Univesitaet Wien, 1981. 3. David Pataraia. Dynamische Analyse und Simulation von Seil-Stab- Strukturen von Seilbahnen auf Basis diskreter Darstellung und

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ON SOLUTION OF A SYSTEM OF NONLINEAR THE BOUNDARY VALUE CONTACT PROBLEM OF

ALGEBRAIC EQUATIONS FOR A TIMOSHENKO BEAM ELECTROELASTICITY FOR PIECEWISE-HOMOGENEOUS

Jemal Peradze*’**, Zviad Kalichava* PLATE WITH ELASTIC *Iv. Javakhishvili Tbilisi State University, Tbilisi, Georgia, INCLUSION AND CUT

**Georgian Technical University, Tbilisi, Georgia Nugzar Shavlakadze [email protected], [email protected] Iv. Javakhishvili Tbilisi State University, A. Razmadze Mathematical Institute, Tbilisi, Georgia, [email protected] We consider an initial boundary value problem for the nonlinear dynamic beam [1, 2]. The solution is approximated by a We will consider a piecewise-homogeneous plate of piezo- variational method and implicit symmetric different scheme. The electric material, weakened by infinite crack and reinforced by an system of equations obtained by discretization is solved by an infinite inclusion (beam) as an electrode by a normal force of iteration method using Sherman-Morrison formula. The accuracy of intensity p0 ()x The normal stresses qx0 ()and the electric potential the iteration method is studied. are given at the edges of the crack. The problem consists of determining the expansion of cut xf )( References and the jump xp )( of normal contact stresses along the contact line, 1. Peradze J.: On the accuracy of the Galerkin method for a nonlinear dynamic beam equation. Math. Meth. Appl. Sci., 34, issue 14, (2011), of establishing their behavior in the neighborhood of singular points. 1725-1732. According to the equilibrium equation of inclusions elements we 2. Timoshenko S.P., Young D.H., Weaver W.: Vibration problems on have engineering. John Wiley & Sons, Inc., (1974). ddvx22(1)() 22Dx() = p0 (x)−> px ( ), x 0 (1) dx dx ON THE STABILITY OF NONIZOTHERMAL FLOWS and the equilibrium equation of the inclusion has the form ∞ ∞ BETWEEN POROUS CYLINDERS [p(tptdt )− ( )]= 0, t[p(tptdt )−= ( )] 0, (2) ∫ 0 ∫ 0 Luiza Shapakidze 0 0 I. Javakhishvili Tbilisi State University, A.Razmadze Mathematical )1( Institute, Tbilisi, Georgia, [email protected] where xv )( is the vertical displacement of inclusion points; xp )( is the jumps of normal contact stresses, subjects to determination. In this report we present the investigations of instability and xD )( is bending rigidity of the inclusions material. complex chaotic regimes arising after the loss of stability of viscous On the boundary of a crack we have heat-conducting fluids between two porous heated rotating vertical (2)+−(2) σσ()xxqxx+ ()=< 2 (), 0 (3) and fixed horizontal cylinders by pumping a fluid around the yy 0 annulus. On the interface of two material the following conditions are valid

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= )2()1( , =ττσσ)2()1( , = )2()1( , = vvuu )2()1( The crack opening behavior has the form xx xyxy −+12 ω )2()1( )2()1( fx()=→ Ox ( ), x 0,−< 0ω < 12. = yy , = DDEE xx (4) Reference where j ,τσj)()( are stress components , ,vu jj )()( are displacement 1. Parton V. Z., Kudriavzev B. A. Electromagneto-elasticity of piezo- x xy electric and electro-conductive bodies. Moscow, Nauka, (1988), p. 470 . j j)()( 2. Muskhelishvili N. Singular integral equations. Мoscow, Nauka, (1966), components, y , DE x are components of vectors of electrical p. 591. stress and of electrical inductive (j=1,2). 3. Gakhov F.D., Cherskii Yu.I. Convolution Type Equation. Moscow, On the bases of the conditions (1-4) introducing the notation Nauka, (1978), p.296. ψ −= tft )()( , we have the system of singular integral equations 4. Bantsuri R. The boundary problems of the theory of analytic with fixed singularity [1,2] functions.// Soobsh. Acad. Nauk. GSSR. 73 (3), (1974), 549-552. d ∞∞λ 1 xt ( [1 ++−=−R ( tx , )] ptdt ( ) R ( tx , )ψτ ( tdt ) ) dt[ p ( ) p (τ )] dτ , x > 0 ∫∫12 ∫∫0 dx00 t− x D() x 00 (5) BUCKLING PARADOX AND ANISOTROPIC PLASTIC ∞∞ PLATE BIFURCATION λ 3 (6) [−+−−R430 (,)]() t xψ t dt + R (,)p()q(),0 t − x t dt =−> x x Suresh Shrivastava ∫∫tx− 00 Department of Civil Engineering and Applied Mechanics For solving the system (5)-(6), when )( = = constDxD , McGill University tx Montreal, Quebec, Canada H3A 2K6 x > 0 , making notation ϕ = − )]()([)( dppdtx τττ, using [email protected] ∫∫ 0 0 0 generalized Fourier transform [3] for the function The plastic plate buckling paradox originated from the work of ∞ Handelman and Prager [1]. They found that the bifurcation stresses 1 Fs()= ϕ (eedςς )is ς , we obtain the boundary condition of predicted by the isotropic Mises incremental theory, the “correct” J ∫ 2 2π −∞ theory of strain hardening plasticity, were absurdly higher than those Karleman type problem for a strip from the “incorrect” J 2 deformation theory of plasticity. Experi- + − = sPsFisFsG ),()()3()( − ∞ < s < ∞ (7) ments generally favour the predictions of the deformation theory. Therefore, we consider the problem: find the function zF )( , Hence, the plastic plate buckling paradox: a correct theory yields the which is holomorphic in the strip 0< Imz <3, vanishing at infinity, wrong results, while an incorrect one gives the right results. Onat and continuously extendable on the border of the strip and satisfying Drucker [2] explained the paradox by showing that by taking condition (7). “unavoidable” out-of-plane geometric imperfection into account in a Using the method of factorization the solution of this problem is nonlinear growth analysis of a plate, the incremental theory gives represented in an explicit form [4]. maximum loads matching the deformation theory bifurcation loads. Applying the inverse integral transformation for normal contact The growth analysis is quite complicated, and can only be done stresses we obtain the following estimate: numerically. Efforts have been made to lower the buckling loads of −12 px0 ()− px ()==ϕ′′ () x Ox ( ), x →+ 0 53 54

the J 2 incremental theory by various other means, but without much DIFFRACTION OF NONLINEAR WAVES BY A SEMI- success. SUBMERGED HORIZONTAL RECTANGULAR CYLINDER

Here, using an analytical variational approach, bifurcation Wojciech Sulisz analyses for plate buckling are performed by considering the Polish Academy of Sciences, Institute of Hydroengineering anisotropic behaviour of the plate material. For this purpose Hill’s 80328 Gdansk, Poland, [email protected] incremental theory [3] for anisotropic strain hardening of sheet metals is used. The principal axis of anisotropy x is taken as the The design of ships and offshore structures requires information rolling direction; y is the transverse principal axis. These anisotropy regarding wave loads on a structure. A boundary-value problem for axes are assumed to remain fixed, uninfluenced by the loading of the the interaction of nonlinear water waves with a semi-submerged plate for bifurcation. Plane stress conditions are assumed to prevail. horizontal rectangular cylinder is formulated. An analytical solution is achieved up to second-order in wave steepness. The main attention The loading is biaxialσ = ασ , −≤11α ≤ , with ξν axes parallel ννξξ is paid to the modelling of large nonlinear wave load components. to the sides of the rectangular plate, simply supported at ξ = 0, a and The solution reveals a significant second-order load on a at ν = 0,b . The ξν axes are at an angle β (which may or may not cylinder. The second-order load component may exceed many times be zero) with respect to the xy principal axes of anisotropy. the corresponding first-order quantities. This phenomenon occurs within the commonly accepted range of the applicability of a second- Three cases are considered: (1) equibiaxial compression,α =1, order wave theory. (2) equal compression tension, α =−1 , and (3) uniaxial compression, Theoretical results are in a fairly good agreement with experi- α = 0 . The plastic plate buckling paradox is examined for each of the mental data. A reasonable agreement between theoretical results and cases. It is shown that by a suitable choice of initial anisotropic yield experimental data is observed even for steep waves. stresses, the resulting bifurcation stresses can be rendered quite close to those predicted by the J deformation theory, which in turn are 2 considered close to the experimental results. BOUNDARY VALUE PROBLEMS OF STEADY VIBRATIONS IN THE THEORY OF THERMOVISCOELASTICITY OF References BINARY MIXTURES 1. Handleman G. H. and Prager. W. Plastic buckling of a rectangular plate under edge thrusts: NACA Report 946 (1949). Maia Svanadze 2. Onat E. T. and Drucker. D. C. Inelastic instabilities and incremental Faculty of Exact and Natural Sciences, Tbilisi State University theories of plasticity: Journal of the Aeronautical Sciences, 20 (1953), I. Chavchavadze Ave., 3, Tbilisi 0179, Tbilisi, Georgia 181-186. [email protected] 3. Hill R. The Mathematical Theory of Plasticity, Oxford University Press, London (1950). The present talk concerns the linear theory of thermoviscoelasticity of binary mixture which is modelled as a mixture of an isotropic elastic solid and a Kelvin-Voigt material. The basic boundary value problems (BVPs) of steady vibrations of the considered theory are investigated and some basic results of the classical theory of thermoviscoelasticity are generalized. The fundamental solution of the system of equations of steady vibrations is constructed by

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elementary functions. The Green formulae and the formulae of integ- 3. Tsutskiridze V., Jikidze L. The conducting liquid flow between porous ral representations of regular vector and regular (classical) solutions wells with heat transfer. Proceedings of A. Razmadze Mathematical are obtained. The uniqueness theorems for solutions of the internal Institute, 167 (2015), 73-89. and external basic BVPs of steady vibrations are proved. The basic 4. Sharikadze J., Tsutskiridze V., Jikidze L. The flow conducting liquid properties of potentials and singular integral operators are presented. in magnetohydrodynamics tubes with heat transfer//International Scientific Journal IFToMM “Problems of Mechanics”, Tbilisi, 2011, Finally, the existence theorems for the above mentioned BVPs are № 3 (44), pp. 63--70. proved by means of the potential method and the theory of singular 5. Tsutskiridze V., Jikidze L. The nonstationary flow of a conducting integral equations. fluid a plane pipe in the presence of a transverse magnetic field. Elsevier (www. Elsevier. Com/locate/trmi) Proceedings of A. Acknowledgments. This research has been fulfilled by financial Razmadze Mathematical Institute, 170, Issue 2, September (2016), support of Shota Rustaveli National Science Foundation (Grant # 280-286. YS15_2.1.1_100).

ON THE JUSTFICATION AND STABILITY OF THE THE STATIONARY FLOW OF LAMINAR LIQUID REFINED AND HIERACHICAL THEORIES FOR THINWALLED ELASTIC STRUCTURES IN AN CIRCULAR PIPE OF INFINITE LENGTH

Varden Tsutskiridze*, Levan Jikidze**, Eka Elerdashvili*** Tamaz Vashakmadze * Department of mathematics, Georgian Technical University, I. Javakhishvili Tbilisi State University, I.Vekua Institute of Applied 77 M. Kostava st., 0175 Tbilisi, Georgia Mathematics, Tbilisi, Georgia, [email protected] [email protected], [email protected] **Department of engineering mechanics and technical expertise in In this report we consider the problems connected with construction, Georgian Technical University, 77 M. Kostava st., 0175 justification and stability of the finite (T. von Kármán–E. Reissner– Tbilisi, Georgia, [email protected] R. Mindlin type) and hierarchical (I. Vekua, I. Babuška) mathemati- ***Department of mathematics, Georgian Technical University, cal models, corresponding to elastic shells, plates and pivotal sys- 77 M. Kostava st., 0175 Tbilisi, Georgia, [email protected] tems. Based on [1, 2] we demonstrate that for boundaring variational methods of construction for these models is essential to a priori In the article the stationary flow of viscous incompressible define the admissible class of functions and existence of proof of the electrically is considered conducting liquid in infinite length pipe at desired extremals. These conditions are not sufficient for the stability existence of transversal magnet field. The motion is originated due of corresponding processes and require (i.e. in case of the natural applied in the initial moment of time constant longitudinal drop of boundary conditions) specific (see, e.g. [3,4]) corrections. pressure. The exact solution of the problem in the general form is obtained [1-5]. References 1. Young, L.: Lectures on the Calculus of Variations and Optimal Control References Theory, W.R. Sanders Company, 340 pp., Philadelphia-London- 1. Loitsiansky L.G. Mechanic of a fluid and gas. Moscow: Sciences, Toronto (1969). (1987), 840 p. (in Russian). 2. Vashakmadze, T.: On the Origin of the Theory of Distributions or the 2. Vatazin A. B., Lubimov G. A., Regirer C. A. Magnetohydrodynamic Heritage of Andrea Razmadze, Bull. Georg. Natl. Acad.Sci., 11 N2, Flow in Channels. Moscow: Nauka, (1970), 672 p. (In Russian). (2017), 12-14.

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3. Rectorys, K.: Variational Methods in Mathematics, Science and 2. Mao Bai, Derex Elsworth, Jean-Claude Roegiers: Multiporo- Engineering, Dr. Reidel Publ. Company, 592 pp., Prague (1980). sity\Multipermeability Approach to the Simulation of Naturally 4. Vashakmadze, T.: The Theory of Anisotropic Elastic Plates, Springer- Fractured Reservoirs. Water Resources Research, 29 (6) (1993), 1621- Sci.+Business Media,B.V.,256 pp, (2010). 1633.

STUDY OF STRESS STRAIN STA`TE OF SPONGY BONE AROUND IMPLANT UNDER OCCLUSAL LOAD

Natela Zirakashvili I. Javakhishvili Tbilisi State University, I.Vekua Institute of Applied Mathematics, Tbilisi, Georgia, [email protected]

The main purpose of this work is mathematical modeling and study of stress-strain state of spongy bone of the jaw with implant. Spongy bone may be considered as a multiporous medium where fractures and intervening porous blocks are the most obvious components of the dual-porosity system [1]. The spongy bone consists of solid and liquid phases. The paper, in the solid phase, presents the equations, which describe of the effect of fluid pressure on the solid deformation within each individual component and in the fluid phase, a separate equation written for each component of distinct porosity or permeability [2]. This paper studies the stress- strain state of the spongy bone of the jaw near the implant in the case of occlusal load. Mathematical model of this problem represents a contact problem of elasticity between the implant and the body of the jaw. Boundary element methods, which are based on the solutions to the problems of Flamant (BEMF) and Boussinesq (BEMB), are used to obtain numerical values of stresses in the bone tissue under the occlusal load on the implant. There are considered cases, when the diameter of implant is equal to 0.4 cm, 0.6 cm, 0.8 cm and 1 cm. The contours (isolines) of stresses in the bone tissue are constructed and the results, obtained through BEMF and BEMB for the implants with different diameters, are compared with each other.

References 1. Aifantis E.C. On the problem of diffusion in solids. Acta Mech., 37 (1980), 265-296.

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ilia vekuas cxovreba, moRvaweoba tis sisqis x cvladis mimarT, maTi integrebiT da da samecniero memkvidreoba 3 gadaadgilebis veqtor-funqciis, Zabvis da deforma- giorgi jaiani ciis tenzorebis komponentebis gaSliT furie-leJan- ivane javaxiSvilis saxelobis Tbilisis saxelmwifo dris orTogonalur mwkrivebSi x3 cvladis mimarT universitetis ilia vekuas saxelobis gamoyenebiTi da mwkrivebis kerZo jamebis ganxilviT agebuli iyo maTematikis instituti & firfitis organzomilebian modelTa ierarqia. Sev- zust da sabunebismetyvelo mecnierebaTa fakulteti niSnoT, rom klasikuri kirxhof-lavis da reisner- [email protected] mindlinis modelebi SeiZleba Caidgas i. vekuas mier

miRebul ierarqiaSi ise, rom isini SeiZleba ganvixi- moxseneba eZRvneba gamoCenili qarTveli mecni- loT rogorc firfitis gavrcelebuli sainJinro eris, msoflioSi aRiarebuli maTematikosisa da meqa- modelebis dazusteba. firfitebisa da garsebis i. ve- nikosis, akademikos ilia vekuas cxovrebis, samec- kuas mier agebuli modelebi moyvanilia mis mono- niero-organizaciuli da pedagogiuri moRvaweobisa grafiaSi [2]. da samecniero memkvidreobis mokle mimoxilvas. mox- i. vekuas mier agebuli maTematikuri modelebis seneba momzadda misi dabadebidan 110 wlisTavTan da- gamokvlevis pirveli Sedegebi miRebuli iyo d. gor- kavSirebiT. dezianis mier [3, 4], sadac damreci zogadi garsebis

statikuri organzomilebiani modelebi gamokvleuli

i. vekuas meTodis ganviTarebis Sesaxeb drekadi iyo sobolevis sivrceebSi da klasikur sakmarisad struqturebis ierarqiuli modelebis asagebad gluv funqciaTa sivrceebSi Seswavlili iyo urTi- erTkavSiri statikur samganzomilebian modelsa da gia avaliSvili*, mariam avaliSvili** organzomilebian modelebs Soris erTgvarovani *iv. javaxiSvilis saxelobis Tbilisis saxelmwifo izotropuli wrfivi drekadi mudmivi sisqis firfi- universiteti, zust da sabunebismetyvelo mecnierebaTa tebis SemTxvevaSi. SemdgomSi, variaciuli midgomis fakulteti, Tbilisi, saqarTvelo, [email protected] da i. vekuas ideis da misi ganzogadebis [5] gamoyene- **saqarTvelos universiteti, Tbilisi, saqarTvelo, biT firfitebis da garsebis organzomilebiani da [email protected] Zelebis erTgazomilebiani statikuri da dinamikuri modelebi agebuli da gamokvleuli iyo klasikuri warmodgenil moxsenebaSi moyvanilia Sedegebi wrfivi drekadobis Teoriis, Termodrekadobis, na- i. vekuas ganzomilebis reduqciis meTodis, misi gav- revTa Teoriis, mikrotemperaturebiani Termodreka- rcelebebis da ganzogadebebis gamoyenebis Sesaxeb dobis da Termodrekadobis araklasikuri Teoriebis drekadi struqturebis ierarqiuli modelebis asage- farglebSi. firfitebis, garsebisa da ZelebisaTvis bad. i. vekuas mier [1] naSromSi agebuli iyo organ- ganzomilebis reduqciis algoriTmebis gamokvlevi- zomilebian modelTa ierarqia wrfivi drekadi erT- sas miRebul Sedegebze dayrdnobiT agebuli da ga- gvarovani cvalebadi sisqis firfitebisaTvis. am naS- mokvleuli iyo sxvadasxva geometriuli formis mqo- romSi samganzomilebiani modelis Sesabamisi ganto- ne nawilebisagan Sedgenili drekadi multistruqtu- lebebis gamravlebiT leJandris polinomebze firfi- rebis pluri-ganzomilebiani ierarqiuli modelebi. 61 62

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[217596, Termodrekadi piezoeleqtruli struqtu- daduRebuli fenis daZabul-deformirebuli rebis ierarqiuli modelebis ageba da gamokvleva]. mdgomareobis gansazRvra mogorvisas cocvadobis TeoriiT literatura

1. Vekua, I.N.: On a way of calculating of prismatic shells, Proc. A. Razmadze Inst. Math. Georgian Acad. Sci., 21 (1955), 191-259 gulmira bulekbaeva*, omar kikviZe** (Russian). *S. esenovis sax. kaspiis inJineringisa da teqnologiebis 2. Vekua, I.N.: Shell theory: General methods of construction, Pitman saxelmwifo universiteti aqtau, yazaxeTis respublika, Advanced Publishing Program, Boston, 1985. [email protected] 3. Gordeziani, D.G.: On the solvability of some boundary value problems **akaki wereTlis saxelmwifo universiteti quTaisi, for a variant of the theory of thin shells, Dokl. Akad. Nauk SSSR, 215 saqarTvelo, [email protected] (1974), 6, 1289-1292 (Russian). 4. Gordeziani, D.G.: To the exactness of one variant of the theory of thin shells, Dokl. Akad. Nauk SSSR, 216 (1974), 4, 751-754 (Russian). daduRebis teqnologiuri procesiT xdeba detale- 5. G. Jaiani, On a mathematical model of bars with variable rectangular bis gacveTili zedapirebis aRdgena. daduRebuli fenis cross-sections, ZAMM, 81 (2001), 3, 147-173. meqanikuri maxasiaTeblebisa da zedapiris xarisxis ga- umjobesebis mizniT saWiroa fenis plastikuri defor-

zogierTi amonaxsnis ageba sicarielis mireba, rac xSirad xorcieldeba mogorviT. mogor- mqone sxeulebisaTvis visas plastikuri deformireba warmoebs xisti cilindruli gorgolaWiT, romelic asrulebs brtyel lamara biwaZe paralelur moZraobas mudmivi xazovani da kuTxuri i. javaxiSvilis saxelobis Tbilisis saxelmwifo siCqariT. universiteti, ganxilulia masis organzomilebiani, damyarebuli, i. vekuas saxelobis gamoyenebiTi maTematikis instituti, Tbilisi, saqarTvelo, [email protected] blant-plastikuri dineba ganmtkicebis Teoriis sa- fuZvelze [1]: moxseneba eZRvneba drekadobis Teoriis kvazista- = a κξσnm ee tikis gantolebebs sicarielis mqone sxeulebisaT- sadac = ξκdt - odkvistis parametria, σ - eqvivalen- vis. miRebulia amonaxsnis zogadi warmodgenis for- ∫ e e mulebi. agebulia fundamentur da singularul amo- turi Zabvaa, ξe - eqvivalenturi deformaciis siCqarea, naxsnTa matricebi elementaruli funqciebis saSua- ,, nma - masalis mudmivebia. lebiT. wonasworobis arawrfivi diferencialuri ganto- literatura lebebi Cawerilia dinebis siCqaris komponentebis 1. Ieshan D., A Theory of Thermoelastic Materials with voids, Acta Mechanica , 60 (1986),67-89. mimarT, romlebSic Sedis aseve saSualo Zabva [2]. 63 64

Cawerilia sasazRvro pirobebi gantolebebis ricxviTi nebismier procesSi, Tu viTarebaSi, konfliqtis integrirebisaTvis. dinebis siCqareebis povnis Semdeg warmoSobis sababad miCneulia raime saxis gansxva- sen-venan–levi–mizesis gantolebebiT gamoiTvleba vebuloba, romelsac kargad aCvenebs Sesabamisi maTe- Zabvis deviatoris komponentebi. amocanis amoxsniT matikuri matricis warmodgenis axali, veqtoruli ganisazRvreba fenis deformirebisaTvis saWiro Zalis forma. Camoyalibebulia pirobebi, romlebic uzrunvel- sidide da gorgolaWze moqmedi mgrexi momentis sidide, yofen ukonfliqto procesis garkveuli mimarTu- Sesabamisad teqnologiuri procesis energetikuli lebiT mdgrad ganviTarebas. maxasiaTebeli. miRebuli Sedegebi Sedarebulia adamianebis urTierTobis cnobil, magaliTad e.w. oqros wesTan literatura da TamaSebis Tanamedrove TeoriaSi ganxilul sxva Малинин Н.Н. Ползучесть в обработке металлов.-М.:Машинострое- 1. ние, 1986.- 216 с. modelebTan. 2. Киквидзе О.Г., Булекбаева Г.Ж., Кипиани П.Н. Пластическое деформирование наплавленного слоя на плоской поверхности. //Georgian Engineering News, №1 (Vol. 77).-2016.-pp.64-66. analizur funqciaTa i.vekuas Teoria: ganzogadebul analizur funqciaTa sivrce

konfliqtebi da katastrofebi grigor giorgaZe i. vekuas saxelobis gamoyenebiTi maTematikis instituti [email protected] guram gabriCiZe Tbilisi, saqarTvelo,

saqarTvelos mecnierebaTa erovnuli akademia, analizur funqciaTa meTodebis gavrcelebis [email protected] idea funqciaTa ufro farTo klasze saTaves me-19 saukunidan iRebs da faqtiurad emTxveva kompleq- moxsenebis mizania, SeviswavloT nebismieri viTa- suri analizis rogorc maTematikis damoukidebeli rebis, procesis, Tu mdgomareobis mdgradi da dargis Camoyalibebas. kompleqsuri analizis meTo- ukonfliqto ganviTarebis pirobebi da maTi reguli- debis gavrcelebis pirveli mcdelobebi dakavSire- rebis SesaZleblobebi. buli iyo ganzogadebuli koSi-rimanis gantolebaTa meqanikaSi miRebuli Sedegebis eqstrapolacia da sistemis amonaxsnTa sivrcis Tvisebebis Seswavlas- ganzogadoeba safuZvels iZleva ganvacxadoT, rom Tan. nebismieri viTareba, procesi Tu mdgomareoba SesaZ- i. vekuasa da l. bersis cnobili monografiebis lebelia davaxasiaToT ori globaluri parametriT– Semdeg ganzogadebuli analizuri funqciebi, simravlis gamaerTianebeli raime saerTo Tvisebis i. vekuas terminologiiT da fsevdoanalizuri funq- SecvliTa da am gaerTianebaSi monawile obieqtebis, ciebi, bersis azriT, msoflios mravali wamyvani Tu subieqtebis damokidebulebiT am cvlilebis maTematikuri centris kvlevis obieqti gaxda. mimarT. moxsenebaSi ganxiluli iqneba ganzogadebul analizur funqciaTa Teoriis gamoyeneba konformu- 65 66

li velis TeoriaSi, kompleqsuri struqturebis de- amwe-satransporto da sagzao manqanebis samuSao formaciis Teoriasa da kvanturi meqanikis zustad aRWurvilobebis struqturulma kvlevam aCvena, rom amoxsnadi organzomilebiani modelebis TeoriaSi rig SemTxvevaSi aRniSnul konstruqciebSi adgili (ix.[1]). aqvs zedmeti bmebis arsebobas, ris gamoc rgolebis moZraoba saxsrebSi xdeba arsebuli RreCoebis meSve- literatura obiT, an liTonis konstruqciebis winaswar daZabu- 1. Akhalaia G., Giorgadze G., Kaldani N., Jikia V., Manjavidze N., li mdgomareobiT, rac moiTxovs konstruqciis dam- Makatsaria G. Elliptic systems on Riemann surfaces. Lecture Notes of zadebis sizustes da amcirebs manqanis simtkicesa da TICMI, vol. 13, pp.3-154, 2012. saimedoobas. amitom mizanSewonilia Seiqmnas iseTi racio- naluri konstruqciebi, romlebsac ar gaaCniaT amwe-satransporto da sagzao manqanebis samuSao zedmeti bmebi, TviTregulirebadia da ar iwvevs sa- mowyobilobebis srulyofis gzebi muSao aRWurvilobis liTonis konstruqciebis

winaswar daZabul mdgomareobas. vaJa gogaZe akaki wereTlis saxelmwifo universiteti Tbilisi, saqarTvelo, literatura

[email protected], [email protected] 1. vaJa gogaZe, giorgi gogaZe, buldozeris samuSao aRWurviloba, patenti P563; moxsenebaSi aRwerilia amwe-satransporto da sa- 2. vaJa gogaZe, giorgi gogaZe, cicxviani satvirTveli, pa- P5696; gzao manqanebis samuSao mowyobilobebis kinematiku- tenti U1552; ri da struqturuli kvleva. 3. vaJa gogaZe, TviTsacvleli, patenti 4. vaJa gogaZe, giorgi gogaZe, skreperis samuSao aRWur- cnobilia, rom amwe-satransporto da sagzao viloba, patenti P5634; manqanebis samuSao mowyobilobebi warmoadgens 5. vaJa gogaZe, mamuka gogaZe, aleqsandre kvirikaSvili, rTul sivrciT Rerovan sistemebs, romelTa rgo- safxvierebeli, patenti P6335; lebi saxsruladaa erTmaneTTan dakavSirebuli. rgo- 6. vaJa gogaZe, giorgi gogaZe, mamuka gogaZe, erTcicxvia- lebis SeerTebisas ZiriTadad gamoiyeneba cilindru- ni eqskavatoris samuSao aRWurviloba, patenti P6007; li saxsrebi, romelTa Tavisuflebis xarisxi erTis 7. vaJaAgogaZe, mamuka gogaZe, buTxuz paWkoria, Cangle- tolia da sferuli saxsrebi Tavisuflebis xarisxiT biani satvirTvelis samuSao aRWurviloba, patenti sami. P6380; mizanSewonilia aRniSnul manqanebSi gamoyenebul 8. vaJa gogaZe, mamuka gogaZe, ledi iremaZe, avtogreide- iqnas ormagi cilindruli saxsrebi, romelTa Tavi- ris samuSao aRWurviloba, patenti P6400. suflebis xarisxi oris tolia. ormagi cilindruli saxsrebis gamoyeneba zrdis manqanis funqcionalur daniSnulebas, mis mwarmoeblobas, amartivebs konst- ruqcias, ris Sedegadac izrdeba manqanis simtkice da saimedooba.

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garsTa Teoriis erTi amocanis amoxsna gazis dinebis modelireba ganStoebebisa da i. vekuas meTodiT N=3 miaxloebaSi daxris mqone satranzito milsadenSi

bakur gulua Teimuraz daviTaSvili iv. javaxiSvilis saxelobis Tbilisis saxelmwifo iv. javaxiSvilis saxelobis Tbilisis saxelmwifo universiteti, i. vekuas saxelobis gamoyenebiTi universiteti, maTematikis instituti, Tbilisi, saqarTvelo i. vekuas saxelobis gamoyenebiTi maTematikis instituti, soxumis saxelmwifo universiteti, Tbilisi, saqarTvelo Tbilisi, saqarTvelo, [email protected] [email protected] bunebrivi gazis ganawilebis qseli warmoadgens ganxilulia geometriulad arawrfivi aradamre- rTul sistemas asobiT da aTasobiT kilometris ci sferuli garsebi i. vekuas N=3 miaxloebaSi. mci- mqone milsadenebiT, sakompresio sadgurebiT da sxva re parametris meTodisa da kompleqsuri cvladis mravali meqanizmebiTa da mowyobilobebiT bunebrivi funqciebis gamoyenebiT amoxsnilia zogierTi amo- gazis transportirebisa da ganawilebis servisis- cana. Tvis. bunebrivi gazis transportirebisa da ganawilebis sistemaSi momxmareblisTvis saTanado piro- madloba. winamdebare naSromi Sesrulebuli iyo bebis Seqmna erT-erT praqtikulad mniSvnelovan da SoTa rusTavelis erovnuli samecniero fondis gansaxorcieleblad sakmaod rTul problemas war- finansuri mxardaWeriT (granti SRNSF/FR/358/5-109/14). moadgens. am problemis gadawyvetis gzaze gazis wnevisa da xarjis ganawilebis gansazRvra mTel milsa- denSi erT erTi mniSvnelovani nabijia. gazis wnevisa da xarjis ganawilebis gansazRvra daxrisa da gan- Stoebebis mqone milsadenSi kidev ufro rTul, magram metad praqtikul amocanas warmoadgens. am statiis mizania ganStoebebisa da daxris mqone milsadenSi ganvsazRvroT gazis wnevisa da xarjis ganawilebis sivrcul droiTi mdgomareoba gamartivebuli erT-ganzomilebiani kerZo warmoebuliani diferencialuri gantolebis safuZvelze, romelic aRwers gazis arastacionalur dinebas ganStoebebi- sa da daxris mqone milsadenSi. mocemulia analizuri amonaxsni gamartivebuli erT-ganzomilebiani ker- Zo warmoebuliani diferencialuri gantolebisa, romelic aRwers gazis arastacionalur dinebas gan- Stoebebisa da daxris mqone milsadenSi. aseve warmodgenilia ricxviTi Tvlis zogierTi Sedegebi.

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Txelkedlovani drekadi struqturebisaTvis implantis irgvliv Rrublisebri Zvlis daZabul- dazustebul da ierarqiul TeoriaTa dafuZnebisa deformirebuli mdgomareobis Seswavla da mdgradobis Sesaxeb okluziuri datvirTvisas

Tamaz vaSaymaZe, naTela ziraqaSvili iv. javaxiSvilis saxelobis Tbilisis saxelmwifo i. javaxiSvilis saxelobis Tbilisis saxelmwifo universiteti, universiteti, i. vekuas saxelobis gamoyenebiTi maTematikis instituti, i. vekuas saxelobis gamoyenebiTi maTematikis instituti, Tbilisi, saqarTvelo, [email protected] Tbilisi, saqarTvelo, [email protected]

drekad garsTa, firfitaTa da Zelovan sistemaTa moxsenebis mTavari mizania implantiani ybis Rru- Sesabamisi sasruli (T. fon karman–e. reisner r. min- blisebri Zvlis daZabul-deformirebuli mdgoma- dlinis tipis) da ierarqiuli (i. vekua, i. babuSka) ma- reobis maTematikuri modelireba da Seswavla. Rrub- Tematikuri modelebis magaliTze ganixileba maTi lisebri Zvali SeiZleba ganxilul iqnas, rogorc dafuZnebasa da mdgradobasTan dakavSirebuli sakiT- mravalforovani sxeuli, sadac Rarebi da forovani xebi. [1, 2] Sromebze dayrdnobiT, naCvenebi iqneba, blokebi arian orgvari forovani sistemis yvelaze rom aRniSnuli modelebis agebis variaciuli meTo- gamokveTili komponentebi [1]. Rrublisebri Zvali debis dafuZnebisaTvis arsebiTia apriori dasaSveb Sedgeba myari da Txevadi fazisagan. Zvlis myari funqciaTa klasis gansazRvra da saZiebeli eqstrema- fazisaTvis warmodgenilia gantolebebi, romlebic lis arseboba, rac Sesabamisi procesis mdgradobi- aRweren wnevis gavlenas myar deformaciaze yovel saTvis, mag., bunebrivi sasazRvro pirobebis SemTxve- komponentSi da Txevadi fazisTvis Cawerilia cal- vaSi, sakmarisi ar aris da moiTxovs garkveuli ti- keuli gantolebebi yoveli komponentisTvis, romle- pis (ix. mag., [3, 4] ) koreqcias. bic gansxvavdebian forovnebiTa da gamtarobiT [2]. Seswavlilia implantis maxloblad ybis Rru- literatura blisebri Zvlis daZabul-deformirebuli mdgomareo- 1. Young, L.: Lectures on the Calculus of Variations and Optimal Control ba okluziuri datvirTvis SemTxvevaSi. am amocanis Theory, W.R. Sanders Company, 340 pp., Philadelphia-London- maTematikur models warmoadgens drekadobis Teo- Toronto (1969). riis sakontaqto amocana implantsa da ybis Zvals 2. Vashakmadze, T.: On the Origin of the Theory of Distributions or the Soris. amocanis amosaxsnelad gamoyenebulia sasaz- Heritage of Andrea Razmadze, Bull. Georg. Natl. Acad. Sci., vol.11, N2, pp. 12-14(2017). Rvro elementTa meTodebi, romlebic dafuZnebulia 3. Rectorys, K.: Variational Methods in Mathematics, Science and flamanis (semf) da busineskis (semb) amocanebis amo- Engineering, Dr. Reidel Publ. Company, 592 pp., Prague (1980). naxsnebze. გanxilulia SemTxvevebi, rodesac implan- 4. Vashakmadze, T.: The Theory of Anisotropic Elastic Plates, Springer- tis diametri tolia 0.4sm, 0.6sm, 0.8sm da 1sm. agebu- Sci. + Business Media, B.V., 256 pp (2010). lia ZvalSi Zabvebis konturebi (izowirebi) da Seda- rebulia semf da semb-iT miRebuli Sedegebi sxvadasxva diametris mqone implantebisaTvis.

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literatura madloba. winamdebare moxseneba Sesrulebuli iyo 1. Aifantis E.C. On the problem of diffusion in solids. Acta Mech., 37, SoTa rusTavelis erovnuli samecniero fondis 265-296, 1980 mxardaWeriT (granti SRNSF / FR / 358 / 5-109 / 14). 2. Mao Bai, Derex Elsworth, Jean-Claude Roegiers: Multiporosity

\Multipermeability Approach to the Simulation of Naturally Fractured Reservoirs. Water Resources Research, 29 (6), 1621-1633, 1993 zvavisebri nakadebis dinamikuri procesebis maTematikuri modelireba

drekadobis brtyeli Teoriis nawilobriv ucnobsazRvriani erTi amocanis Sesaxeb tariel kviciani saqarTvelos teqnikuri universiteti, giorgi kapanaZe Tbilisi, saqarTvelo, [email protected] i. javaxiSvilis saxelobis Tbilisis saxelmwifo universitetis i. vekuas saxelobis gamoyenebiTi zvavisebri nakadebis dinamikuri procesebis Ses- maTematikis instituti, Tbilisi, saqarTvelo wavlis mizniT moyvanilia cnobebi zvavsaSiSroebis i. javaxiSvilis saxelobis Tbilisis saxelmwifo pirobebTan dakavSirebiT. kerZod, zvavwarmoqmnis universitetis a. razmaZis maTematikis instituti Tbilisi, damokiduleba zedapiris qanobTan, romlebic zogad saqarTvelo, [email protected] warmodgenas iZlevian zvavebis warmoSobisa da gavrcelebis procesebze da sakmaod mniSvnelovanni ganxilulia drekadobis brtyeli Teoriis nawi- arian Teoriul-raodenobrivi Sefasebis SeqmnisaTvis. lobriv ucnobsazRvriani amocana (anu, rac igivea Tovlis zvavebis ZiriTad fizikur-meqanikur Tanabradmtkice konturis moZebnis amocana) modelad aRebulia uwyvet deformadi tanis modeli, marTkuTxa firfitisaTvis, romlidanac Tanabrad- sadac srulad vlindeba plastikurobis da sifxvie- mtkice konturiT (sazRvris ucnobi nawili) amoWri- ris xasiaTi. Tovlis zvavebis dinamikuri procesebis lia garkveuli nawili. vgulisxmobT, rom sazRvris aRsawerad gamoyenebulia kompoziciuri garemos (mew- sworxazovan monakveTebze moqmedeben mocemuli mTa- yer-Camonaqcevebis) gantolebaTa sistema, warmodge- vari veqtoris mqone normaluri mkumSavi Zalvebi, nili hidrodinamikis gantolebaTa formiT da misi xolo sazRvris ucnobi nawili Tavisufalia gare- erTganzomilebiani varianti [1]. es sistemebi iTva- gani datvirTvebisagan. ucnobi konturis Tanabrad- liswineben fxvier da plastikur Tvisebebs da amis simtkicis piroba gulisxmobs masze tangencialuri garda, plastikurobas, romelic maxasiaTebelia normaluri Zabvis mudmivobas. bingamisebri siTxeebisaTvis, dinamikur gantolebebSi amocanis amosaxsnelad gamoyenebulia kompleqsu- daZvris zRurbluri daZabulobis gaTvaliswinebis ri analizis meTodebi da saZiebeli kompleqsuri xarjze. marcvlovani fxvieri garemos parametrebis potencialebi da Tanabradmtkice konturis gantole- ganulebis dros es sistema avtomaturad gadadis ba agebulia efeqturad (analizuri formiT). genki-iliuSinis gantolebaSi, romelic aRwers e.w. plastikur-blant siTxeebs (boCer-bingamis siTxe), rac maxasiaTebelia „rbili“ Tovlis zvavebisaTvis, romlebsac gaaCniaT mniSvnelovani sinotive. 73 74

amosavali fizikur-meqanikuri modelis gantole- 3. Москалев Ю.Р. Возникновение и движение лавин Л.: bebi Sedgebian: fxvieri garemos zRvruli mdgomare- Гидрометеоиздат, 1976 272 с. obis gantolebisagan Sinagani xaxunis kuTxiT da SeWidulobis koeficientiT (kulon-treski-sen-venanis piroba), romliTac ganisazRvreba urTierTkavSiri zedapiris Cadgmis Sesaxeb rimanis mxebi da normaluri Zabvebis komponentebs Soris, 3-ganzomilebian mravalsaxeobaSi agreTve kavSiri Zabvebis tenzoris komponentebs So- ris, Siga xaxunis kuTxiT da siCqaruli deformaciis Tengiz meunargia tenzoris komponentebs Soris, romlebic gviCveneben i. javaxiSvilis saxelobis Tbilisis saxelmwifo daZvris deformaciis udidesi siCqaris mimarTule- universitetis i. vekuas saxelobis gamoyenebiTi maTematikis instituti, Tbilisi, saqarTvelo, bis Tanxvedras dacurebis wirebis ojaxis erTerT [email protected] mimarTulebasTan (iSlinski-genievis piroba), plius plastikurobis gaTvaliswineba, romelic ganpiro- gausis aranulovani simrudis zedapirebi warmo- bebulia Tovlis sinotiviT [1]. adgenen rimanis 2-ganzomilebian mravalsaxeobebs, zvavis sxeulis modelad aRebulia sasruli zo- romlebic Cadgmulia evklidis 3-ganzomilebian mebis mqone sxeuli. gaTvaliswinebulia misi geomet- sivrceSi. amis gamo, am mravalsaxeobebis Tvisebebze riuli zomebi, masaTa centrisa da frontaluri are- SesaZlebelia Seiqmnas sakmarisad naTeli warmodge- ebis gadaadgilebis siCqareebs Soris gansxvaveba. mi- nebi. Semdeg, naCvenebia, rom nebismieri regularuli Rebuli Sedegebi asruleben iseTi damxmare faqto- zedapiri SeiZleba Caidgas rimanis 3-ganzomilebian rebis rols, romlis gareSe SeuZlebelia amonaxsne- mravalsaxeobaSi. bis swori analizi da interpretacia. es exeba iseT parametrebs, rogorebicaa: zvavis sxeulis simaRle madloba. winamdebare moxseneba Sesrulebuli iyo SoTa da sigrZe; misi grZivi profilis koordinata; fron- rusTavelis erovnuli samecniero fondis finansuri tis siCqare, rogorc manZilisa da drois funqcia; mxardaWeriT (granti SRNSF/FR/358/5-109/14). winaaRmdegobebze zemoqmedebis Zaluri parametrebi; civi qarebis moqmedeba Tovlis safarze da a.S. rac arsebiT gavlenas axdens miRebuli Sedegebis utyua- robis dadgenaSi [2,3].

literatura 1. kviciani t. ferdos mdgradoba da zvavisebri nakadebi. gamomcemloba „teqnikuri universiteti“. Tbilisi, 2010. – 140 gv. 2. Лосов К.С.Некоторые воросы движения лавин и лругих аналогичных явлений. Труды всесоюзного совещания по лавинам.Л.: Гидрометеоиздат, 1978, с 64-69.

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wamaxvilebuli filebis Runvis deformaciebis sxeulebis deformaciis kvlevas praqtikaSi misaRebi kvelva sasrul elementTa meTodiT da sizustis farglebSi. reduqciuli meTodis damuSaveba analizuri funqciebiT aproqsimaciis gamoyenebiT Lliteratura 1. Georg Jaiani – “Cusped Shell-like Structures”, Springer Briefs in giorgi nozaZe Applied Science and Technology, Springer-Heidelberg-Dordrecht- ssip g. wulukiZis samTo instituti London-New York, 2011, 84 p. [email protected]

ganxilulia sxvadasxva simrudis zedapirebiT kirxofis tipis zogierTi arawrfivi integro- SemosazRvruli prizmuli wamaxvilebuli filebis diferencialuri gantolebis miaxloebiTi Runvis deformaciis amocana, romelTac gaaCniaT amoxsnis algoriTmebisa da ricxviTi simetriis sibrtye (ix., mag. [1]). rogorc cnobilia, gaTvlebis Sesaxeb aseT pirobebSi prizmuli filebis samganzomilebiani drekadobis amocana SesaZlebelia daviyvanoT or arCil papukaSvili*, giorgi papukaSvili**, ganzomilebian brtyeli deformirebuli mdgomareo- jemal feraZe*** bis drekadobis amocanaze. *iv. javaxiSvilis saxelobis Tbilisis saxelmwifo dasmulia wamaxvilebuli filis wveros, rog- universiteti, i. vekuas saxelobis gamoyenebiTi maTematikis instituti, orc gansakuTrebuli aris datvirTvis amocana. zust da sabunebismetyvelo mecnierebaTa fakulteti. simartivisaTvis ganxilulia sufTa Runvis amocana, **v. komarovis Tbilisis № 199 sajaro skola, rodesac gansakuTrebul areSi (wveros monakveTze) saqarTvelos teqnikuri universiteti modebulia ganawilebuli datvirTva simetriis sibr- ***iv. javaxiSvilis saxelobis Tbilisis saxelmwifo tyis marTobulad. miRebulia amocanis diskretuli universiteti, amonaxsni sasrul elementTa meTodis gamoyenebiT. zust da sabunebismetyvelo mecnierebaTa fakulteti, miRebuli diskretuli Sedegebis mixedviT gan- saqarTvelos teqnikuri universiteti, xorcielda sxeulis formis funqciis saTanado wer- Tbilisi, saqarTvelo, tilebis mixedviT gadaadgilebis uwyveti funqciis [email protected], [email protected], [email protected] ageba niSancvladi xarisxovani mwkrivebiT aproqsimaciis gamoyenebiT. naCvenebia, rom sasruli raode- ganxilulia miaxloebiTi amoxsnis sakiTxebi nobis wevrTa mqone niSancvladi xarisxovani mwkri- vebiT aproqsimacia iZleva karg miaxlovebas, raTa Semdegi ori amocanisTvis: drekadobis amocana gadaiWras gadaadgilebebSi gan- 1. arawrfivi sasazRvro amocana kirxofis tipis sakuTrebul areTa datvirTvis dros. statikuri ZelisTvis (ix. magaliTad [1], [2]). grinis Sesrulebuli kvleviTi samuSaos safuZvelze funqciebis gamoyenebiT amocana daiyvaneba arawrfiv naCvenebia, rom SesaZlebelia damuSavdes reduqciu- integralur gantolebaze, romlis amosaxsneladac li analizuri meTodebi, romelTa gamoyenebac gaaum- viyenebT pikaris tipis iteraciul meTods. jobesebs rTuli geomertriuli formis mqone 77 78

2. arawrfivi sawyis-sasazRvro amocana j. bolis nagebobebi da Semakavebeli badeebi. zogadad ki mo- dinamiuri ZelisTvis (ix. magaliTad [3], [4]). amocanis cemuli proeqtis kvlevis sagania msgavsi rTuli amonaxsni SeiZleba miviRoT algoriTmiT, romlis konfiguraciis myari deformirebadi tanebis mode- Semadgeneli nawilebia galiorkinis meTodi, simet- lireba da gaangariSeba. riuli sxvaobiani sqema da iakobis iteraciuli me- standartuli meTodebiT da programuli uzrun- Todi. velyofiT aseTi obieqtebis modelireba da gaangari- orive amocanis SemTxvevaSi gamowerilia miaxlo- Seba xSirad mniSvnelovan problemebTanaa dakavSire- ebiTi amoxsnis axali saTvleli algoriTmebi da Ca- buli histerezisis, folxvis, xaxunis da sxva arse- tarebulia ricxviTi eqsperimentebi. Sedegebi warmo- biTi arawrfivi damokidebulebebis mizeziT. garda dgenilia cxrilebisa da grafikebis saxiT. amisa, didi ganzomilebis obieqtebis gaangariSeba gansakuTrebiT ki, roca amave dros Sedegebis didi literaturა sizustea saWiro, aseve gazrdil da xSirad Znelad 1. J.Peradze, A numerical algorithm for a Kirchhoff – type nonlinear Sesasrulebel moTxovnebs uyenebs gamoTvliT meTo- static beam. J. Appl. Math. 2009, Art.ID 818269, 12pp. debsa da კომპiუტერულ teqnikas. 2. T.F.Ma, Positive solutions for a nonlocal fourth order equation of kvlevis aqtualurobas agreTve ganapirobebs sa- Kirchhoff type. Discrete Contin. Dyn. Syst. 2007, 694-703. Tanado universaluri meTodebisa da saTvleli 3. J. M. Ball, Stability Theory for an Extensible Beam, J. Diff. Eq., 14 programuli paketebis konkretuli amocanis gadasaW- (1973), 399-418. relad misadagebisa da gamoyenebis sirTule, gansa- 4. G.Papukashvili, J.Peradze, Z.Tsiklauri, On a stage of a numerical kuTrebiT ki, roca gamosakvlevi obieqti arawrfivi algorithm for Timoshenko type nonlinear equation, Proc. maxasiaTeblebiTaa da Tanac didi zomis. A.Razmadze Math.Inst., Tbilisi, 158, (2012), 67-77. kvlevis siaxle mdgomareobs rTuli bagirRero- vani struqturebis statikuri da dinamikuri gaanga- riSebis Cvens mier SemuSavebul meTodSi, romelic rTuli bagirRerovani struqturebis da sxva efuZneba myari deformirebadi tanis diskretul war- msgavsi samSeneblo konstruqciebis modelireba modgenasa da specialur saTvlel algoriTms [1], [2], da gaangariSeba diskretuli warmodgenisa da [3]. specialuri algoriTmis safuZvelze SemoTavazebuli midgomis safuZvelze Cvens mier damuSavebulia rTuli konfiguraciisa da arawrfi- daviT pataraia vi maxasiaTeblebis mqone sxeulebisa da samSeneblo grigol wulukiZis saxelobis samTo instituti, konstruqciebis statikuri da dinamikuri gaangari- Tbilisi, saqarTvelo, [email protected] Sebis algoriTmi da saTanado saTvleli programa, romelTa efeqtianoba da saimedooba dadasturda kvlevis sagans warmoadgens rTuli bagirRerova- realuri obieqtebis modelirebiT da konkretuli ni struqturebi, rogoricaa, magaliTad, sabagiro gaangariSebebiT. gzebi, vanturi xidebi, kosmosuri sabagiro sistemebi da antenebi, stadionebis da sxva didi zomis mqone obieqtebis damcavi gadaxurvebi, zvavsawinaaRmdego 79 80

madloba. winamdebare moxsenebis mniSvnelovani nawi- da ganzogadebulia blanti Termodrekadobis kla- li Sesrulebuli iyo rusTavelis fondis dafinan- sikuri Teoriis zogierTi ZiriTadi Sedegi. elemen- sebiTa da Tbilisis saxelmwifo universitetis gamo- taruli funqciebis saSualebiT agebulia mdgradi yenebiTi maTematikis institutis koleqtivTan da rxevis gantolebaTa sistemis fundamenturi amonaxs- misi xelmZRvanelTan profesor g. jaianTan TanamS- ni. miRebulia grinis formulebi da regularuli romlobiT. veqtorisa da klasikuri amonaxsnebis integraluri warmodgenebi. damtkicebulia mdgradi rxevis Siga literaturა da gare sasazRvro amocanebis amonaxsnebis erTader- 1. David Pataraia. The calculation of rope-rod structures of ropeways on Toba. dadgenilia potencialebisa da singularuli the basis of the new approach.WORLD CONGRESS of O.I.T.A.F., integraluri operatorebis ZiriTadi Tvisebebi. RIO DE JANEIRO, BRAZIL, October, 2011, 24 – 27, PAPERS OF bolos, potencialTa meTodisa da singularul THE CONGRESS; http://www.oitaf.org/Kongress%202011/Referate/Pataraia.pdf integralur gantolebaTa Teoriis gamoyenebiT damt- 2. D. Pataraia. Ein Beitrag zur diskreten Darstellung des Seiles. kicebulia zemoT aRniSnuli amocanebis amonaxsnebis Wissenschaftliche Arbeiten des Instituts fuer Eisenbahnwesen, arsebobis Teoremebi. Verkehrswirtschaft und Seilbahnen, Technische Univesitaet Wien, 1981. madloba. es kvleva Sesrulebulia SoTa rusTave- 3. David Pataraia. Dynamische Analyse und Simulation von Seil-Stab- lis erovnuli samecniero fondis finansuri mxarda- Strukturen von Seilbahnen auf Basis diskreter Darstellung und WeriT (granti # YS15_2.1.1_100). sukzessiven Approximationsalgorithmus WORLD CONGRESS of O.I.T.A.F., Bolzano, Italy, 6-9 Juni, 2017 PAPERS OF THE CONGRESS; http://oitaf2017.com/wp-content/uploads/2017/06/oitaf- kongr-bolcano-040617- vort-germ- JN-050617- 2.pdf iakobis iteracia Zelis dinamiuri amocanisTvis

mdgradi rxevis sasazRvro amocanebi binarul jemal feraZe narevTa blanti Termodrekadobis TeoriaSi iv. javaxiSvilis saxelobis Tbilisis saxelmwifo universiteti, Tbilisi, saqarTvelo maia svanaZe saqarTvelos teqnikuri universiteti, e-mail: [email protected] zust da sabunebismetyvelo mecnierebaTa fakulteti, Tbilisi, saqarTvelo, iv. javaxiSvilis sax. Tbilisis saxelmwifo universiteti, i. WavWavaZis gamz. 3, Tbilisi 0179, saqarTvelo, ganxilulia sawyis sasazRvro amocana diferen- [email protected] cialuri gantolebisaTvis [1, 2] 2 ∂ 2u ∂ 4u ∂ 4u ⎛ L⎛ ∂u ⎞ ⎞∂ 2u tx + ),(),( − htx tx ),( ⎜λ +− ⎜ ),( ⎟ dt ξξ⎟ tx ,0=),( moxsenebaSi ganxilulia binarul narevTa blan- 2 4 22 ⎜ ∫⎜ ⎟ ⎟ 2 ∂t ∂x ∂∂ tx 0 ⎝ ∂ξ ⎠ ∂x ti Termodrekadobis Teoria, roca narevis erTi ⎝ ⎠ komponenti izotropuli drekadi sxeulia, xolo romelic aRwers Zelis dinamiur mdgomareobas. sivr- meore - kelvin-foigtis masala. gamokvleulia am Te- culi da drois cvladebis mimarT amonaxsnis oriis mdgradi rxevis ZiriTadi sasazRvro amocanebi miaxloebis Sedegad miRebulia diskretuli ganto- 81 82

lebebis arawrfivi sistema, romlis amosaxsnelad wiboebiani garsebis angariSi gamoyenebulia iteracia. dadgenilia iteraciis kre- sasrul elementTa meTodiT badobis pirobebi da Sefasebulia misi cdomileba. gela yifiani, seiT bliaZe literaturა saqarTvelos saaviacio universiteti, Tbilisi, qeTevan. 1. Henriques de Brito E.: A nonlinear hyperbolic equation, International wamebulis q. №,16, 0144, saqarTvelo. [email protected], Journal of Mathematics and Mathematical Sciences, 3(3), 505-520, [email protected] 1980. 2. Timoshenko S.P., Young D.H., Weaver W.: Vibration problems on ganxilulia sixistis wiboebis mqone firfitebi- engineering. John Wiley & Sons, Inc., 1974. sa da garsebis Teoriis ganviTarebis ZiriTadi eta- pebi. aRniSnulia, rom wiboebis mqone garsebis Teo- ria warmoadgens erT-erT yvelaze dausrulebel da timoSenkos ZelisaTvis arawrfiv algebrul sadavo Teorias garsTa zogad TeoriaSi. moyvanilia gantolebaTa sistemis amoxsnis Sesaxeb wiboebis mqone garsebis angariSi sasrul elementTa meTodis gamoyenebiT. pirvel SemTxvevaSi wiboebi 1,2 1 jemal feraZe , zviad yaliCava warmodgenilia rogorc Zelis sasruli elementi 1 iv. javaxiSvilis saxelobis Tbilisis saxelmwifo (se), xolo garsi modelirebulia rogorc se firfi- universiteti, Tbilisi, saqarTvelo ta. meore SemTxvevaSi garsic da wiboebic modelire- 2 saqarTvelos teqnikuri universiteti, Tbilisi, saqarTvelo bulia rogorc se firfita. mesame SemTxvevaSi gar- [email protected]; [email protected] sic da wiboebic modelirebulia rogorc samganzomilebiani se. meoTxe SemTxvevaSi garsis wiboebi ganxilulia sawyis-sasazRvro amocana arawrfivi modelirebulia rogorc se Zeli, xolo garsi mode- dinamiuri ZelisaTvis [1, 2]. miaxloebiTi amonaxsnis lirebulia rogorc samganzomilebiani se. eqsperi- misaRebad gamoyenebulia variaciuli meTodi da si- mentuli kvleva Catarda BS EN ISO 9001 da EN 9100 metriuli aracxadi sxvaobiani sqema. diskretizaciis saerTaSoriso sertifikatebis mqone sawarmoos ss Sedegad miRebul gantolebaTa sistemis amosaxsne- “TbilaviamSeni”-s, simtkicis laboratoriaSi. moyva- lad gamoyenebulia iteraciuli meTodi da Serman- nilia eqsperimentuli da Teoriuli kvlevebis Sede- morisonis formula. Seswavlilia iteraciuli meTo- gebis SedarebiTi analizi. (cTomileba ≈ 5 %). dis sizuste. literatura literatura 1. Strang G., Fix G. J. An Analysis of the Finite Element Method. 1. Peradze J., On the accuracy of the Galerkin method for a nonlinear Prentice-Hall, inc. 1973. dynamic beam equation. Math. Meth. Appl. Sci. 34 (2011), no. 14, 1725- 1732. 2. Timoshenko S.P., Young D.H., Weaver W., Vibration problems on engineering. John Wiley & Sons, Inc., 1974.

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eleqtrodrekadobis sasazRvro-sakontaqto ori masalis gamyof sazRvarze ZalaSia Semdegi piro- amocana drekadi CarTvisa da bzaris mqone uban- bebi uban erTgvarovani sibrtyisaTvis )2()1( )2()1( )2()1( )2()1( = xx , =ττσσxyxy , = , = vvuu (4)

= )2()1( , = DDEE )2()1( nugzar SavlayaZe yy xx j )( j )( j )( iv. javaxiSvilis saxelobis Tbilisis saxelmwifo sadac σ x x τ xy x)(),( - Zabvis komponentebia, xu ),( universiteti, a. razmaZis maTematikis instituti, Tbilisi, j )( xv ),( -gadaadgilebis komponentebi, j )( j )( xDxE )(),( saqarTvelo, [email protected] y x j = )2,1( eleqtruli Zabvis da induqciis veqtoris ganixileba piezo-eleqtruli masalis uban-uban komponentebia. (1-4) pirobebis ZaliT, SemoviRebT ra aRniSvnas erTgvarovani firfita, romelic Sesustebulia usas- ψ = −tft )()( , miviRebT singularul integralur ganto- rulo bzariT da gamagrebulia usasrulo CarTviT lebaTa sistemas uZravi singularobiT [1,2] (ZeliT), rogorc normaluri xp )( intensivobis mqone 0 d ∞∞λ 1 xt ( [1 + Rtx ( , )] ptdt ( )+− R ( tx , )ψτ ( tdt ) ) = dtp[ ( )− p (τ )] dτ , x > 0 ∫∫12 ∫∫ 0 ZaliT datvirTuli eleqtrodi. bzaris sazRvarze dx00 t− x D() x 00 mocemulia normaluri 0 xq )( Zalebi da eleqtruli (5) ∞∞λ potenciali. [(,)]()(,)p()q(),0− 3 +−−Rtxψ tdtRtxtdt + − =− x x > (6) ∫∫430 amocana mdgomareobs bzaris gaxsnis xf )( funqci- 00tx− isa da normaluri sakontaqto Zabvis xp )( naxtomis (5-6) sistemis amosaxsnelad, rodesac )( = = constDxD , x > 0 , gansazRvraSi, agreTve maTi yofaqcevis dadgenaSi ganzogadoebuli furies gardaqmnis gamoyenebiT [3] ∞ singularuli wertilebis midamoSi. 1 ςςis F ()se= ∫ ϕ ( )edς , CarTvis elementis wonasworobis gantolebis 2π −∞ ZaliT gvaqvs funqciisaTvis, sadac 22(1) tx ddvx() (1) , 22Dx() =− p0 (x) px ( ), x > 0 ϕ = − )]()([)( dppdtx τττ dx dx ∫∫ 0 0 0 da mTeli CarTvis wonasworobis gantolebebs ki aqvT miiReba karlemanis tipis sasazRvro amocana zoli- saxe ∞ ∞ saTvis [p(tptdt )− ( )]= 0, t[p(tptdt )−= ( )] 0, (2) ∫ 0 ∫ 0 + − = sPsFisFsG ),()()3()( − ∞ < s < ∞ (7) 0 0 maSasadame, ganixileba Semdegi amocana: vipovoT funq- sadac )1( xv )( aris CarTvis wertilebis vertikaluri cia zF )( , romelic holomorfulia < z < 3Im0 zolSi, gadaadgileba, xD )( aris CarTvis masalis sixiste Run- qrobadia usasrulobaSi, uwyvetad gagrZelebadia vaze. zolis sazRvarze da akmayofilebs (7) pirobas. bzaris sazRvarze gvaqvs faqtorizaciis meTodis gamoyenebiT am amocanis σσ(2)+−()xxqxx+ (2) ()=< 2 (), 0 (3) yy 0 amonaxsni warmoidgineba cxadi saxiT [4].

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Sebrunebuli gardaqnmiT normaluri sakontaqto laminaruli siTxis stacionaruli dineba ZabvisaTvis miiReba Semdegi Sefaseba usasrulo sigrZis wriuli formis milSi −12 px0 ()− px ()==ϕ′′ () x Ox ( ), x →+ 0 xolo bzaris gaxsnis funqcias aqvs Semdegi yofaqceva varden cucqiriZe*, levan jiqiZe,**, fx()=→ Ox (−+12 ω ), x 0,−< 0ω < 12. eka elerdaSvili* *maTematikis departamenti, saqarTvelos teqnikuri

literatura universiteti, saqarTvelo, 0175, Tbilisi, m. kostava 77, ** 1. Parton V. Z., Kudriavzev B. A. Electromagneto-elasticity of piezo- sainJinro meqanikisa da mSeneblobis teqnikuri electric and electro-conductive bodies. Moscow, Nauka, 1988. p. 470 . eqspertizis departamenti, saqarTvelos teqnikuri 2. Muskhelishvili N. Singular integral equations. Мoscow, Nauka, 1966. universiteti, saqarTvelo, 0175, Tbilisi, m. kostava 77, p. 591. [email protected], [email protected], 3. Gakhov F.D., Cherskii Yu.I. Convolution Type Equation. Moscow, [email protected] Nauka, 1978. p.296. 4. Bantsuri R. The boundary problems of the theory of analytic Seswavlilia eleqtrogamtari blanti arakumSva- functions.// Soobsh. Acad. Nauk. GSSR. 1974. 73(3), p. 549-552. di siTxis stacionaruli dineba usasrulo sigrZis mqone milSi, rodesac moqmedebs garegani magnituri

or forovan cilindrs Soris araizoTermuli veli. moZraoba gamowveulia wnevis mudmivi dacemiT. dinebebis mdgradobis Sesaxeb miRebulia dasmuli amocanis zusti amonaxsnebi zogadi saxiT [1-5]. Lluiza SafaqiZe i. javaxiSvilis saxelobis Tbilisis saxelmwifo literatura universiteti, a. razmaZis saxelobis maTematikis 1. Loitsiansky L.G. Mechanic of a fluid and gas. Moscow: Sciences, instituti, Tbilisi, saqarTvelo, [email protected] 1987. -840 p. (in Russian). 2. Vatazin A. B., Lubimov G. A., Regirer C. A. Magnetohydrodynamic

Flow in Channels. Moscow: Nauka, 1970, -672 p. (In Russian). moxsenebaSi warmodgenilia or forovan uZrav 3. V. Tsutskiridze, L. Jikidze. The conducting liquid flow between porous horizontalur cilindrs Soris am cilindrebis wells with heat transfer. Proceedings of A. Razmadze Mathematical gaswvriv datumbviT SeRweuli siTxis dinebisa da Institute Vol. 167, 2015, pp. 73-89, (Engl.). brunvad forovan vertikalur cilindrebs Soris 4. J. Sharikadze, V. Tsutskiridze, L. Jikidze. The flow conducting liquid dinebebis aramdgradobisa da qaosur moZraobaSi in magnetohydrodynamics tubes with heat transfer//International gadasvlis amocanebi. igulisxmeba rom dinebebze Scientific Journal IFToMM “Problems of Mechanics”, Tbilisi, 2011, moqmedeben radialuri dinebebi da radianuli tempe- № 3(44), pp. 63--70. raturuli gradienti. 5. V. Tsutskiridze and L. Jikidze, The nonstationary flow of a conducting fluid a plane pipe in the presence of a transverse magnetic field. Proceedings of A. Razmadze Mathematical Institute Vol. 170, Issue 2, September 2016, pp.280-286, (Engl).

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optikuri solitonebis aradrekadi vrcelebis Sesaswavlad gamoyenebuli sasrul sxvao- urTierTqmedebis modelireba biani meTodi aracxadi sqemiT. ricxviTi eqsperimentebiT naCvenebia, rom garemos oleg xarSilaZe*,****, vasil belaSovi**, araerTgvarovneba cvlis solitonebis da sxva sinaT- jemal rogava***, xaTuna Cargazia***,* lis impulsebis amplitudebs, maTi gavrcelebis siC- *fizikis departamenti, iv. javaxiSvilis saxelobis qareebs, maT raodenobas, rac ganpirobebulia araer- Tbilisis saxelmwifo universiteti, Tbilisi, saqarTvelo, Tgvarovan boWkoSi maTi aradrekadi urTierTqme- **yazanis federaluri universiteti, yazani ruseTi, debiT. garemos arastacionaloba cvlis impulsis ***i. vekuas saxelobis gamoyenebiTi maTematikis formas da mis speqtralur Tvisebebze axden gavle- instituti, iv. javaxiSvilis saxelobis Tbilisis nas. garemos parametrebis modulaciis cvlilebiT saxelmwifo universiteti, Tbilisi, saqarTvelo, SesaZlebelia solitonebis mizidva-ganzidvis arad- ****m. nodias saxelobis geofizikis instituti, iv. javaxiSvilis saxelobis Tbilisis saxelmwifo rekadi urTierTqmedebis xasiaTis cvlileba. mode- universiteti, Tbilisi, saqarTvelo, lirebiT naCvenebia optikuri solitonebis gaxleCis [email protected] da SeerTebis efeqtebi.

informaciis optikuri gadacemisTvis aqtualuria literatura optikur boWkoSi lazeruli solitonebis gavrce- 1. Agrawal, G.P. Nonlinear Fiber Optics. – Academic Press, – 2013. lebis da urTierTqmedebis Seswavla. es urTierTqme- 2. Belashov V.Yu., Belashova E.S. Solitons: theory, simulation, applications. Kazan, 2016. deba mniSvnelovnad cvlis sinaTlis velis maxasi- 3. Belashov V.Yu., Vladimirov S.V. Solitary Waves in Dispersiv aTeblebs da amaxinjebs gadacemul informacias. Complex Media. Theory, Simulation, Applications. Springer-Verlag solitonebis marTvis TvalsazrisiT aucilebelia GmbH&Co, 2005. boWkos defeqtebis, dispersiuli da arawrfivobis araerTgvarovnebebis, garemos parametrebis arastaci- onalurobis gavlenis Seswavla solitonebis gavrcelebaze. amocana daiyvaneba sinaTlis velis amplitudisTvis Sredingeris arawrfiv gantolebaze sivrcul koordinatze da droze damokidebuli koe- ficientebiT. Catarebuli ricxviTi modelirebisaTvis gamoyenebuli iyo fizikuri wevrebis mixedviT gantolebis gaxleCis furies meTodi, romelSic gaTvaliswinebu- lia koeficientebis araerTgvarovneba. gantoleba ga- yofilia or wrfiv da arawrfiv nawilad da cal- calkea ganxiluli dispersiuli da arawrfivi efeqtebi da Sesabamisi operatorebi CaTvlilia komuta- turebad. solitonebis arastacionalur garemoSi ga-

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friqciuli avtorxevebis dinamikuri analizi literatura oleg xarSilaZe*,***, xaTuna Cargazia**,***, nodar 1. N. Varamashvili, T. Chelidze, M. Devidze, Z. Chelidze, V. Chikhladze, varamaSvili***, dimitri amilaxvari*, levan dvali* A. Surmava, Kh. Chargazia, D. Tefnadze. Mass-movement and seismic *fizikis departamenti, iv. javaxiSvilis saxelobis processes study using Burridge-Knopoff laboratory and mathematical Tbilisis saxelmwifo universiteti, Tbilisi, saqarTvelo, models. Journal of Georgian Geophysical Society, Issue (A), Physics of **i. vekuas saxelobis gamoyenebiTi maTematikis instituti, Solid Earth, 2015, pp.19-25. Dieterich, J.H. (1979), Modeling of rock friction 1. Experimental results iv. javaxiSvilis saxelobis Tbilisis saxelmwifo 2. and constitutive equations. Journal of Geophysical Research, 84, B5, universiteti, Tbilisi, saqarTvelo, 2161-2168. *** m. nodias saxelobis geofizikis instituti, 3. Burridge R. and Knopoff L. 1967. Model and theoretical seismicity. iv. javaxiSvilis saxelobis Tbilisis saxelmwifo Bulletin of the Seismological Society of America, 57(3): 341-371.

universiteti, Tbilisi, saqarTvelo, 4. Ruina A. 1983. Slip instability and state variable friction laws. Journal [email protected] of Geophysical Research, 88, 10359-10370. 5. Chelidze T., N. Varamashvili, M. Devidze, Z. Chelidze, V. Chikhladze miwisZvra ZiriTadad xasiaTdeba friqciuli dina- and T. Matcharashvili (2002), Laboratory study of electromagnetic mikiT, romelic iwvevs stik–slip (SeWideba–gasria- initiation of slip. Annals of Geophysics, 45, 587-599. leba) procesebs dedamiwis qerqSi arsebuli rRveve- bis gaswvriv. miwisZvris rekurentuli seismuri cik- li xasiaTdeba grZelperiodiani kvazi statikuri zogierTi samganzomilebiani sasazRvro amocana evoluciiT (teqtonikuri procesebi), romel-sac drekadi binaruli narevisTvis orgvari erTvis uecari gasrialebis efeqti Tanmxlebi dreka- forovnebiT di talRis gamosxivebiT: miwisZvriT (seismuri procesebi). naCvenebia, rom friqciuli meqanizmi gana- roman janjRava pirobebs arawrfiv relaqsaciur rxevebs. iv. javaxiSvilis saxelobis Tbilisis saxelmwifo warmodgenil moxsenebaSi ganxilulia arawrfivi universiteti, i. vekuas saxelobis gamoyenebiTi burij-knopovis modeli ori da sami blokis SemTxve- maTematikis instituti, Tbilisi, saqarTvelo, [email protected] visaTvis. Catarebul eqsperimentuli kvlevebSi gamov- lenil iqna aramdgradobebis trigerirebis meqanizmi moxseneba eZRvneba orgvari forovnebis mqone gasrialebis movlenebis Tanmxlebi akustikuri emi- drekadi sxeulebis wonasworobis wrfivi gantoleba- siis CanawerebSi. Catarebulia eqsperimentuli kvle- Ta sistemis ganxilvas, roca sxeulis myari ConCxi vis Sedegebis analizi. Seswavlilia miRebuli signa- warmoadgens ori izotropuli masalis narevs. am lebis speqtraluri Tvisebebi, rekurentuli Tvisob- gantolebaTa sistemis zogadi amonaxsni warmodgeni- rivi da raodenobrivi maxasiaTeblebi, veivlet diag- lia harmoniuli funqciebisa da metaharmoniuli ramebi. naCvenebia, rom stik-slip movlenas sistemaSi funqciis saSualebiT. agebuli zogadi amonaxsnis sa- adgili aqvs mxolod kritikuli siCqareebis viwro fuZvelze, cvladTa gancalebis meTodis gamoyenebiT, areSi. gare zemoqmedebas SeuZlia Secvalos stik- analizurad amoxsnilia sasazRvro amocanaTa klasi slipis dinamika. marTkuTxa paralelepipedisaTvis. 91 92

C O N T E N T S madloba. winamdebare moxseneba Sesrulebuli iyo SoTa rusTavelis erovnuli samecniero fondis fi- G. Jaiani nansuri mxardaWeriT (granti SRNSF/FR/358/5-109/14). Life, activities, and scientific heritage of Ilia Vekua 3 G. Avalishvili, M. Avalishvili literatura On development of I. Vekua method for construction of 1. Natroshvili D. G., Jagmaidze A. Ya., Svanadze M. Zh.: Some hierarchical models of elastic structures 29 problems of the linear theory of elastic mixtures, TSU press, 1986 (in V. Belashov, E. Belashova, O. Kharshiladze Russian). Solitary waves in fluids with variable dispersion 30 2. Khomasuridze N., Janjgava R. Solution of some boundary value thermoelasticity problems for a rectangular parallelepiped taking into V. Belashov, O. Kharshiladze Numerical modeling of interaction of vortex structures in fluids account microthermal effects, Meccacica, vol. 51, 2016, 211-221. and plasmas 31

L. Bitsadze On some solutions of elastic materials with voids 32 G. Bulyekbaeva, O. Kikvidze Determination of stress – strain state of welded layer at rolling by creep theory 33 N. Chinchaladze, R. Gilbert Cusped prismatic shell-fluid interaction problems 34 T. Davitashvili Transient gas flow modeling in inclined and branched pipeline 35 G. Gabrichidze Conflicts and catastrophes 36 G. Giorgadze I. Vekua's theory of the analytic functions: the space of generalized analytic functions 37 V. Gogadze Improvement ways of crane-transport and road-transport machines’ working appliances 37 B. Gulua The solution of one problem of theory of shells by method of I. Vekua for approximation N=3 38

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R. Janjgava J. Peradze Some three-dimensional boundary value problems for an Jacobi iteration for a beam dynamic problem 50 elastic binary mixtures with a double porosity 38 J. Peradze, Z. Kalichava G. Kapanadze On solution of a system of nonlinear algebraic equations for a On one problem of the plane theory of elasticity for the region Timoshenko beam 51 with a partially unknown boundary 39 L. Shapakidze O. Kharshiladze, V. Belashov, J. Rogava, K. Chargazia On stability of nonizothermal flows between porous cylinders 51 Modeling of nonelastic interactions of optical solitons 40 N. Shavlakadze O. Kharshiladze, K. Chargazia, N. Varamashvili, The boundary value contact problem of electroelasticity D. Amilaxvari, L. Dvali for piecewise-homogeneous plate with elastic Dynamical analysis of frictional auto-oscillations 41 inclusion and cut 52 G. Kipiani, S. Bliadze S. Shrivastava Calculation of ribbed shell by finite element method 42 Buckling paradox and anisotropic plastic plate bifurcation 54 T. Kvitsiani W. Sulisz Matehmatical modeling of the dynamical processes of Diffraction of nonlinear waves by a semi-submerged horizontal avalanche-like currents 43 rectangular cylinder 56 T. Meunargia M. M. Svanadze About immersion of the surface in the 3D Riemann diversity 45 Boundary value problems of steady vibrations in the theory W. H. Müller, E. N. Vilchevskaya of thermoviscoelasticity of binary mixtures 56 A few remarks on recent developments in micropolar V. Tsutskiridze, L. Jikidze, E. Elerdashvili continuum theory 45 The stationary flow of laminar liquid in an circular pipe of G. Nozadze infinite length 57 Investigation of the bending deformation of cusped prismatic T. Vashakmadze plates by finite element method and the development of the On the justfication and stability of the refined and hierachical reduction method with the use of approximation by analytic theories for thinwalled elastic structures 58 functions 46 N. Zirakashvili A. Papukashvili, G. Papukashvili, J. Peradze Study of stress-strain state of spongy bone around implant On approximate solution of the algorithms and numerical under occlusal load 59 computations for some Kirchhoff type nonlinear integro- differention equations 47 D. Pataraia Modeling and analysis of complex cable-rod structures and other similar building structures based on discrete representation and special algorithms 48

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s a r C e v i g. kapanaZe drekadobis brtyeli Teoriis nawilobriv g. jaini ucnobsazRvriani erTi amocanis Sesaxeb 73 ilia vekuas cxovreba, moRvaweoba da samecniero t. kviciani memkvidreoba 61 zvavisebri nakadebis dinamikuri procesebis g. avaliSvili, m. avaliSvili maTematikuri modelireba 74 i. vekuas meTodis ganviTarebis Sesaxeb drekadi struqturebis ierarqiuli modelebis asagebad 61 T. meunargia zedapiris Cadgmis Sesaxeb rimanis 3-ganzomilebian l. biwaZe mravalsaxeobaSi 76 zogierTi amonaxsnis ageba sicarielis mqone g. nozaZe sxeulebisaTvis 63 wamaxvilebuli filebis Runvis deformaciebis kvelva g. bulekbaeva, o. kikviZe sasrul elementTa meTodiT da reduqciuli meTodis daduRebuli fenis daZabul-deformirebuli damuSaveba analizuri funqciebiT aproqsimaciis mdgomareobis gansazRvra mogorvisas cocvadobis gamoyenebiT 77 4 TeoriiT 6 a. papukaSvili, g. papukaSvili, j. feraZe g. gabriCiZe kirxofis tipis zogierTi arawrfivi konfliqtebi da katastrofebi 65 integro_diferencialuri gantolebis miaxloebiTi g. giorgaZe amoxsnis algoriTmebisa da ricxviTi gaTvlebis analizur funqciaTa i. vekuas Teoria: Sesaxeb 78 ganzogadebul analizur funqciaTa sivrce 66 d. pataraia v. gogaZe rTuli bagirRerovani struqturebis da sxva msgavsi amwe-satransporto da sagzao manqanebis samuSao samSeneblo konstruqciebis modelireba da mowyobilobebis srulyofis gzebi 67 gaangariSeba diskretuli warmodgenisa da specialuri algoriTmis safuZvelze 79 b. gulua garsTa Teoriis erTi amocanis amoxsna i. vekuas m. svanaZe M meTodiT N=3 miaxloebaSi 69 mdgradi rxevis sasazRvro amocanebi binarul narevTa blanti Termodrekadobis TeoriaSi 81 T. daviTaSvili gazis dinebis modelireba ganStoebebisa da daxris j. feraZe mqone satranzito milsadenSi 70 iakobis iteracia Zelis dinamiuri amocanisTvis 82 T. vaSaymaZe j. feraZe, z. yaliCava Txelkedlovani drekadi struqturebisaTvis dazuste- timoSenkos ZelisaTvis arawrfiv algebrul bul da ierarqiul TeoriaTa dafuZnebisa da gantolebaTa sistemis amoxsnis Sesaxeb 83 mdgradobis Sesaxeb 71 g. yifiani, s. bliaZe n. ziraqaSvili wiboebiani garsebis angariSi sasrul elementTa implantis irgvliv Rrublisebri Zvlis daZabul- meTodiT 84 deformirebuli mdgomareobis Seswavla okluziuri datvirTvisas 72 97 98

n. SavlayaZe The selection and presentation of material and opinion expressed in eleqtrodrekadobis sasazRvro-sakontaqto amocana this publication are the sole responsibility of the authors concerned. drekadi CarTvisa da bzaris mqone uban-uban erTgvarovani sibrtyisaTvis 85 l. SafaqiZe or forovan cilindrs Soris araizoTermuli dinebebis mdgradobis Sesaxeb 87 v. cucqiriZe, l. jiqiZe, e. elerdaSvili English Editor: Ts. Gabeskiria laminaruli siTxis stacionaruli dineba usasrulo sigrZis wriuli formis milSi 88 Technical editorial board: M. Tevdoradze M. Sharikadze o. xarSilaZe, v. belaSovi, j. rogava, x. Cargazia optikuri solitonebis aradrekadi urTierTqmedebis modelireba 89 o. xarSilaZe, x. Cargazia, n. varamaSvili, d. amilaxvari, l. dvali friqciuli avtorxevebis inglisuris redaqtori: c. gabeskiria dinamikuri analizi 91 r. janjRava teqnikuri saredaqcio kolegia: m. TevdoraZe zogierTi samganzomilebiani sasazRvro amocana m. SariqaZe 2 drekadi binaruli narevisTvis orgvari forovnebiT 9

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