American Journal of Applied Mathematics 2020; 8(2): 64-73 http://www.sciencepublishinggroup.com/j/ajam doi: 10.11648/j.ajam.20200802.13 ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online)
Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process
Abdugany Dzhunusovich Satybaev 1, Yuliya Vladimirovna Anishchenko 1, *, Ainagul Zhylkychyevna Kokozova 1, Aliyma Torozhanovna Mamatkasymova 2, Guljamal Abdazovna Kaldybaeva 3
1Department of Information Technology and Management, Faculty of Cybernetics and Information Technology, Osh Technological University, Osh, Kyrgyzstan 2Department of Informatics, Naturally-technical Faculty, Osh Technological University, Osh, Kyrgyzstan 3Department of Physics, Mathematics and Information Technology, Osh State University, Osh, Kyrgyzstan
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*Corresponding author To cite this article: Abdugany Dzhunusovich Satybaev, Yuliya Vladimirovna Anishchenko, Ainagul Zhylkychyevna Kokozova, Aliyma Torozhanovna Mamatkasymova, Guljamal Abdazovna Kaldybaeva. Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process. American Journal of Applied Mathematics . Vol. 8, No. 2, 2020, pp. 64-73. doi: 10.11648/j.ajam.20200802.13
Received : March 3, 2020; Accepted : March 23, 2020; Published : April 13, 2020
Abstract: We consider a one-dimensional inverse problem for a partial differential equation of hyperbolic type with sources - the Dirac delta-function and the Heaviside theta-function. The generalized inverse problem is reduced to the inverse problem with data on the characteristics using the method of characteristics and the method of isolation of singularities. At the beginning, the inverse problem of the wave process with data on the characteristics with additional information for the inverse problem without small perturbations is solved by the finite-difference method. Then, for the inverse problem of the wave process with data on the characteristics with additional information with small perturbations, that is, with small changes is used by the finite-difference regularized method, which developed by one of the authors of this article. The convergence of the finite-difference regularized solution to the exact solution of the one-dimensional inverse problem of the wave process on the characteristics is shown, and the theorem on the convergence of the approximate solution to the exact solution is proved. An estimate is obtained for the convergence of the numerical regularized solution to the exact solution, which depends on the grid step, on the perturbations parameter, and on the norm of known functions. From the equivalence of the problems, the one-dimensional inverse problem of the wave process with sources - the Dirac delta-function and the Heaviside theta-function and the one-dimensional inverse problem of the wave process with data on the characteristics, it follows that the solution of the last problem will be the solution of the posed initial problem. An algorithm for solving a finite-difference regularized solution of a generalized one-dimensional inverse problem is constructed. Keywords: One-dimensional Inverse Problem, Wave Process, Dirac Delta-Function, Heaviside Theta-Function, Method of Characteristic, Method of Isolation of Singularities, Finite-Difference Regularized Solution
Russian Academy of Sciences Valentin K. Ivanov [3]. 1. Introduction The inverse problems of wave processes were considered in Inverse problems are the so-called ill-posed problems, the theoretical terms by the Corresponding members of the foundations of which were laid by Academicians of the Russian Academy of Sciences Vladimir G. Romanov [4], Russian Academy of Sciences Andrei N. Tikhonov [1], Sergey I. Kabanikhin [5], professor Valery G. Yakhno [6], and Mikhail M. Lavrentiev [2], Corresponding Member of the they constructed solutions to the posed inverse problems. 65 Abdugany Dzhunusovich Satybaev et al. : Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process
The regularization method for inverse problems was characteristics of real inhomogeneous layered media, and developed by Andrei N. Tikhonov [7], the method of small according to experimental information, clarified the time parameters was constructed by Mikhail M. Lavrentiev [8] and parameters of the source of disturbances, and also presented the quasi-solution method — by Valentin K. Ivanov [9]. The practical solutions to a number of actual inverse problems in projection-difference method for multidimensional inverse borehole exploration geophysics. problems was developed by Sergey I. Kabanikhin [10], the Dynamic inverse problems, definitions of one or several finite-difference regularization for Volterra integral equation coefficients of hyperbolic equations or systems, methods for of the first kind was investigated by Abdugany Dzh. Satybaev solving one-dimensional inverse problems, scalar inverse [11] and a regularization estimate is obtained. problems of wave propagation in layered media, inverse Aliyma T. Mamatkasymova and Abdugany Dzh. Satybaev problems for the theory of elasticity and acoustic equations are [12] constructed a finite-difference regularized solution and given in the monograph of Alexander S. Blagoveshchenskii [19]. obtained a convergence estimate for the inverse problem Alexander V. Avdeev, Vyacheslav I. Priimenko, E. V. arising in electromagnetic processes. Gorbunov, D. V. Zvyagin [20] considered the inverse problem The purpose of this work is to numerically solve a of electromagneto elasticity with electrodynamics of vibrating one-dimensional inverse problem of the wave process elastic media. proposed by the authors by a finite-difference regularized In the monograph of Sergey I. Kabanikhin [21] outlines method, which allows us to construct a numerical algorithm methods for studying and solving inverse and ill-posed for solving the problem. problems of linear algebra, integral and operator equations, integral geometry, spectral inverse problems and inverse 2. Research Overview scattering problems; linear ill-posed problems and coefficient inverse problems for hyperbolic, parabolic and elliptic Most of the scientific and technical aspects of the inverse equations were considered; given extensive reference material. problems of wave propagation in the medium and the A new globally convergent numerical method is developed mathematical connections between waves and scatterers are for a multidimensional coefficient inverse problem for a determined and presented in work G. Ghavent, G. C. hyperbolic PDE with applications in acoustics and Papanicolaou, P. Sacks, W. Symes [13]. electromagnetics. On each iterative step the Dirichlet The book, edited by G. Chavent and P. C. Sabatier [14], boundary value problem for a second-order elliptic equation is describes the current state of modeling and the numerical solved. The global convergence is rigorously established, and solution of wave propagation and diffraction, their numerical experiments are presented in works Larisa Beilina, applications, and features of inverse scattering problems on Michael V. Klibanov [22]. classical and distributed media. In the book by V. A. Burov and O. D. Rumyantseva [23], The article set forth by F. Natterer [15] three major methods inverse wave problems and their applied aspects related to linear for solving inverse problems: the method of ray tomography, and nonlinear acoustic tomography, as well as acoustic the method of single-particle emission tomography and the thermotomography, are considered. Part I briefly discusses the method of positron-emission tomography and described inverse coherent radiation problems, which are characterized by transfer equations transitions near the infrared region to incorrectness and a strong degree of non-uniqueness. Various elliptic equations in the diffusion approximation. approaches to solving inverse wave problems of radiation and In the article by F. Natterer, A. Wiibbeling [16], a numerical incoherent problems of active-passive acoustic thermotomography calculation of the potential in the Helmholtz equation was laid are described. It is shown that the active-passive mode allows you out and a method was developed which possesses the stability to determine the set of acoustic and thermal characteristics of the of the solution and the convergence of the solution was shown medium within the framework of the general tomographic scheme. in the order O (h 4). Modern mathematical modeling of various wave processes 3. Formulation of the Problem (computed tomography, ultrasonic flaw detection, etc.) in the theory of inverse problems and their examples, main features, perspectives Wave processes of natural phenomena (earthquakes and are presented in the work of Alexander O. Vatulyan [17]. natural explosions, landslides and lavas), electrodynamics and In the Andrey V. Bayev’s dissertation work [18] developed geophysics, etc., are characterized by fields described by stable methods for solving inverse problems, determined the second-order partial differential equations: ϑ+= ϑ ϑ + ϑ + ϑ ∈ 2 ax()tt (,) xt bx ()(,) t xt cx () xx (,) xt dx () x (,) xt ex ()(,),(,) xt xt R + (1) where a(x), (xb ), (xc ), d(x), xe )( are the coefficients of the equation (physical parameters of the equation), ϑ tx ),( is the function describing the process. For (1) we consider the initial and boundary conditions of the form:
ϑ(,)xt≡ 0, ϑ (,) xt =+∈ htrttR δθ () (), (2) t<0x x = 0 0 0 + American Journal of Applied Mathematics 2020; 8(2): 64-73 66
δ Λ = { ∈ 6 ′ + = here h0,r0 are positive constant numbers, t)( is the 0 a x a x)(:)( C (R+ ), a ( O) ,0 θ < ≤ ≤ ≤ } Dirac delta-function, t)( is the Heaviside theta-function. 0 M a x)( M , a x)( 2 M . 1 2 C R+ )( 3 Let a time-like form be given with respect to the solution of the direct problem (1)–(2) In condition (4), the smoothness of the function is given an increased one to apply the finite-difference method. Equation ϑ(xt , )= ftt ( ), ∈ [0, T ] (3) x=0 (1) is a hyperbolic equation, and from the principle of dependence on the specification of data, the solution to Condition (2) means that the process has been at rest up to problem (1)–(3) can be considered in the region ∆ T )( [10]: the time t = 0 and since the time t = 0, in connection with the assignment to the boundary x = 0 of sources with forces ∆=()T{ (,): xtx ∈ (0,), Txt <<− 2 Tx } (5) h and r , the physical wave process is starting. 0 0 If condition (4) is met, the additional information is Usually, when considering inverse problems, the 4 correctness (existence, uniqueness, stability) of a solution is tf )( ∈C ,0( T) , and the solution to the problem is established; we established it in the works [24], [25]. ϑ tx ),( ∈ ∆(( T )) . A. When coefficients are not equal to zero, the direct The inverse problem consists in determining one of the problem (1)–(2) is a problem of hyperbolic type (problems: coefficients of the equation (it would be good if all the wave equation, acoustics, seismic, electrodynamics, etc.). = coefficients of the equation are determined simultaneously, B. When the coefficient xa )( 0 , we have a problem of but for these additional conditions must be specified). parabolic type (the problems of: diffusion, thermometry, the distribution of quasi-stationary electromagnetic fields, etc.). 4. The Inverse Problem C. When coefficients a x)( = b x)( = 0 we have a problem of elliptic type (the problems of: gravimetry, The inverse problem is to determine the function xс )( geoelectrics of stationary fields, etc.). from problem (1) - (2) when specifying the remaining The problem (1)–(2) with the following coefficients is coefficients of the equation and when specifying additional transformed: information about solving the direct problem at the boundary = = = Problem I. When coefficients xb )( d x)( xe )( 0 x = 0, that is, when specifying the function tf )( . = —to the wave equation problem, when xb )( 0 — to the The direct problem is to determine the function ϑ tx ),( oscillations problem; a(x), (xb ), (xc ), d(x), xe )( = = = when defining the functions Problem II. When a x)( b x)(,1 xe )( xc )(,0 is the satisfying condition (3). wave propagation velocity, d x)( is the density of the medium—to the acoustics problem; 4.1. Methods for Solving the Differential Problem = = Problem III. When b x)( xe )( xc )(,0 is the wave 4.1.1. The Method of Characteristic propagation velocity, a x)( = ρ x)( is the density of the We will enter the new variable of medium, d x)( is the Lame coefficient — to the seismic x a( x ) problem; z( x ) = dx (6) = = = ∫ Problem IV. When a x)( xe )(,1 d x)( b x)(,0 is 0 b( x ) the electrical conductivity of the medium — to the geoelectric problem; and we will enter the new functions of Problem V. When a x)( = xe )(,1 = d x)( = b x)(,0 is ba (( xz )) = b x a(/)( x),da (( xz )) = d x a(/)( x), the electric permeability — to the telegraph equation problem. ca (( xz )) = xc a(/)( x),ea (( xz )) = xe a(/)( x), . Problems I, II were investigated by A. Dzh. Satybaev, = ϑ Problem III — by A. A. Alimkanov, IV — by Yu. V. V (( xz t)), tx ).,( Anishchenko, V — by A. Zh. Kokozova. We make some calculations: The main problem in solving the inverse problems (1)–(3) ϑ = ϑ = is, firstly, the presence of the Dirac delta-function in the t tx ),( Vt (( xz ),t), tt tx ),( Vtt (( xz ),t), boundary condition and secondly, the problem is reduced to a ' ϑ tx ),( = V (( xz ),t) ⋅ z (x), system of nonlinear integral equations [10]. x z x Let in relation to the coefficients of equation a condition is ϑ = ⋅ 2 + ⋅ xx tx ),( Vzz (( xz ),t) zx x)( Vz (( xz ),t) zxx (x). executed Substituting the last calculations into equation (1), we ∈Λ (ax ( ), bx ( ), cx ( ), dx ( ), ex ( )) 0 (4) obtain
67 Abdugany Dzhunusovich Satybaev et al. : Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process
+ = + + + Vtt tz ),( ba )( Vz t tz ),( Vzz tz ),( ca )( zz Vzxx tz ),( da )( Vzz zx tz ),( ea Vz tz ),()( ′ (7) = + − 1 ac z z)( + da z)( + Vzz tz ),( Vz tz ),( ea Vz tz ),()( 2 ca z)( ca z)(
4.1.2. The Method of Isolation of Singularities We single out the singular and regular parts of the solution of the problem by the method of V. G. Romanov [4], representing the solution of the problem in the form: = ~ + θ − + θ − V tz ),( V tz ),( S z ()( t | z |) R z)( 1(t | z |) (8)
~ θ = ⋅θ where V tz ),( is a smooth function, 1 t)( t t)( . We make some calculations: ′ =ɶ +δ −+ θ − Vztt(,) Vzt z (,) Sz ()( t ||) z Rz ()( t ||), z =ɶ +δ′ −+ δ − Vzttt(,) Vzt zz (,) Sz ()( t ||) z Rz ()( t ||), z =ɶ +′ θ −− δ −+ θ −− θ − Vztz(,) Vzt z (,) Sz z ()( tz ||) Sz ()( tz ||) Rzz ()(1 tz ||) Rz ()( tz ||), =ɶ +′′ θ −− δ−+ δ ′ −+ Vztzz(,) Vzt zz (,) Sz zz ()( t ||)2 z Szz ()( tz ||) Sz ()( tz ||) +′′ θ −− θ −+ δ − Rzzz()(1 tz ||) 2 Rztz z ()( ||) Rztz ()( ||).
Substituting the last calculations into the obtained equation, we obtain
ɶ +δδ′ −+ −+ɶ + δθ −+ − Vzttt (,) Sz ()( t ||) z Rz ()( t ||) z bazVzt ()t (,) Sz ()( t ||) z Rz ()( t ||) z =VztSztzɶ (,) + ()(θ −− ||)2()( Sztz δ −+ ||) Sz ()( δ′ tz −+ ||) Rz ()( θ tz −− ||) zz zz z zz 1 (9) −θ −+ δ −+ɶ + θ−− δ −+ 2Rzz ()( t ||) z Rz ()( t ||) z gazVzt ()(,)z gazS () z ()(z t ||) z gazSz ()()( t ||) z +θθ −− −+ɶ +−+− θθ gazRz()z ()(1 t ||) z gaRz ()( t ||) z eazVzt () (,) Sz ()( t ||) z Rz ()(1 t ||) z
Here ′ 1caz ( z ) da ( z ) ga( z ) = − + (10) 2ca ( z ) ca( z ) δ − θ − θ − We collect terms with the same factors (t | z |), (t | z |), 1 (t | z |) and equate them to zero: δ ′ +[ +] = :2()Sz z baz () gaz () Sz ()0, θ ′++[] −− ′′ ′ − = :2()Rzz baz () gaz ()() Rz Szz () z gazSz ()() z eazSz ()()0, θ ′′+ ′ + = 1 :Rzz () z gazR ()() z z eazRz ()()0.
From the boundary condition (2), we obtain
ϑ=+= δ θ ' ⋅=+ ' ɶ ' δ + θ ⋅ a(0) x (,)xt= htrtVzt0 () 0 ()z (,) z x VtS z (0,) (0)() tR (0)() t . x 0 z=0 x = 0 c(0) ~ Here ' = is even function. Equating terms at δ t),( θ t)( , we obtain Vz t),0( 0
c )0( S )0( = h = ca h ,)0( a )0( 0 0
c )0( R )0( = r = ca r .)0( a )0( 0 0 American Journal of Applied Mathematics 2020; 8(2): 64-73 68
From the last obtained expressions, we obtain integral equations of the second kind with respect to S(z), R z)( :
z =−1 []λ + λ λλ ∈ Sz() cah (0)0 ∫ ba () ga ()(), S dz (0,) T (11) 2 0
1z 1 z =−+[]λ λλλ +++'' λ λ ' λλλ ∈ Rzcar() (0)0 ∫ ba () ga ()() Rd ∫ Sλλ () gaSeaS () λ ()() dz , (0,) T (12) 20 2 0
Taking into account that V tz ),( ≡ 0 , and also from the higher got calculations, we will get a next inverse problem with t<0 data on the characteristics:
S' ( z ) =−+⋅−z + ∈∆ Vzttt(,) Vzt zz (,) 2 baz () Vztz (,) bazVzt ()(,) z eazVzt ()(,), (,) zt () T (13) S( z )
Vzt(,)= Sz (), z ∈ [0,] T (14) t=| z |
Vzt(,)= ft (), t ∈ [0,2] T (15) z=0
Here the inverse problem is to determine the functions V tz ),,( S z)( when known coefficients ba(z),ea z)( (they depend on known functions (zb ), (za ), ze )( ), and with additional information about solving a direct problem (15). If we define the function zS )( from the problem (13)–(15), then by the formula
S′( z ) ga() z=− 2 − ba (), z z ∈ [0,] T (16) S( z ) we define the unknown function ga z)( . Using the d’Alembert formula for the direct problem (13) – (14), we obtain a solution of this problem
t t+ z − ξ ' 1 Sξ (ξ ) Vztftzftz(,)=++−+[] ( ) ( )/2 − 2 +⋅ ba ()ξ V (,) ξτ − baV ()(,) ξ ξτ + eaV ()(,) ξξττξ} dd (1 ∫ ∫ ξ ξ τ 20 t− z + ξ S ( ) 7) When t = z , we obtain:
t 2z−ξ S ' (ξ ) ==+ +−1 ξ +⋅−ξ ξτ ξ ξτ + ξξττξ Vzz(,) Szfzf ()[] (2) (0)/2 2 baV () ξ (,) baV ()(,) τ eaV ()(,) } dd (18) t= z ∫ ∫ ξ 20 ξ S ( )
4.2. Methods for Solving the Difference Problem 4.2.1. Finite-difference Solution To solve problem (13) – (15), we enter the grid region T ∆===()T z ih,t kh,h = ,i = 0 ,N;ih ≤≤− kh T ih } . h i k 2N where h is the grid step in z,t . The difference analogue of the differential equation (13):
Vk+1−+2 VV kk − 1 VVV kkk −+ 2 VV kk − − 1 i ii= iii+1 − 1 −Sba ii + eaVk ,(, ih kh ) ∈∆ h () T (19) h2 h 2 i h i i 69 Abdugany Dzhunusovich Satybaev et al. : Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process where S− S k = =i i −1 += = Vi VihkhSba(,), i 2 baba ii , baihea (), i eaih () (20) hS i
From (19) we obtain kkkk=+−+⋅+1 − 1 kk −++ kk −− 1 − 2 k =−=− Viiii+1 V V V − 1 SbahV iii V − 1 bahV iii V eahVi ii , 1, N 1, kiNi , (21)
From the last formula (21) we obtain recurrence formulas: kkkk++12=+−+ + 1 kk ++ 21 −+ kk + 1 −− 22 k + Viiii V−−−112 V V hSbaV iii −−− 112( V )() hbaV iii −−− 111 V eahV ii −− 11 ,
i=2, N − 2; ki =− 1, Ni −− 1
kkkk−1=+−+ −− 21 kk −− 11 −+ kk −− 12 −− 21 k − Viiii V−−−111 V V hSbaV iii −−− 112( V )( hbaV iii −−− 111 V ) eahV ii −− 11 ,
i=2, N − 2; ki =− 2, Ni −− 2
Substituting the last recurrence formulas into the right-hand side of formula (21), and then again writing down the recurrence formulas for the next term on the right-hand side and supplying again to the right-hand side of (21), etc. continuing the process, we obtain the difference analogue of the d’Alembert integral formula (17) (that is, the solution of the problem (13) – (14) in the difference form):
i p S− S Vffk=+−( k+ i +1 k − i − 1 )/2 h 2 µ µ −1 +⋅− baV( k− i −µ + 2 p i+1 ∑∑ µ µ p=1µ = 1 hS µ i p i p (22) −+kip−−+µ2 kipkip −−+ µ 2212 −− −−+− µ kip −−+µ 2 Vµ−1 )() h∑∑ baV µµµ V h ∑∑ eaV µµ , p=1µ = 1 p =1µ = 1 i=1, N − 1; kiNi = , −
In the last formula (22), setting k = i +1, we obtain the difference analogue of formula (18)
i p S− S Sff=+−(2i+ 2 0 )/2 h 2 µ µ −1 +⋅ baV( −µ +2p + 1 − i+1 ∑∑ µ µ p=1µ = 1 hS µ (23) i p i p −+−++µ21p −++−+ µµ 2122 pp −− −++ µ 21 p =− Vµ−1 )() h∑∑ baV µµµ V h ∑∑ eaV µµ , iN 1, 1 p=1µ = 1 p =1µ = 1
Formulas (22) and (23) constitute a system of difference non-linear equations of the second kind. The difference equations (22) and (23) are written without small values of O h)( , then for an exact solution with a small value of O h)( it is possible to obtain the same equations (22), (23) by a small quantity O h)( . Let us denote the solution of ~ k ~ equations with small quantity O h)( by Vi+1 , Si+1 . ~ k = ~ k − k ~ = ~ − Then for the difference of the exact approximate solution of the inverse problem (Vi+1 Vi+1 Vi+1, Si+1 Si+1 Si+1 ), we obtain the following expressions:
i p S− S k=−{ ( kipkip−−+µ2 −+ −−+ µ 2 ) µ µ −1 ( kipkip −−+µ2 −+ −−+ µ 2 ) VhbaVi+1 ∑∑ µ µ V µ−1 2 Vµ V µ − 1 p=1µ = 1 hS µ
ɶ ɶ i p Sµ− S µ − +2 1 (Vkipkip−−+µ2 −+ V −−+ µ 2 ) hbaV( kipkip −−+µ2 −− V −−+− µ 2 1 ) (24) ɶ µµ−1 ∑∑ µµ µ hS µ p=1µ = 1 i p −2k− i −µ + 2 p + = − = − h∑∑ eaVµ µ Oh( ), i1, N 1; kiNi , p=1µ = 1 American Journal of Applied Mathematics 2020; 8(2): 64-73 70
i p S− S =−{ ( −++µ21p −+ −++ µ 21 p ) µ µ −1 ( −++µ 21p −+ −++ µ 21 p ) ShbaVi+1 ∑∑ µ µ V µ−1 2 Vµ V µ − 1 p=1µ = 1 hS µ
ɶ ɶ i p Sµ− S µ − +2 1 (V−++µ2121p −+ V −++ µ p ) hbaV( −++µ 21p −− V −+ µ 2 p ) (25) ɶ µµ−1 ∑∑ µµµ hS µ p=1µ = 1 i p −2−µ + 2p + 1 + = − h∑∑ eaVµ µ Ohi( ), 1, N 1 p=1µ = 1
To estimate (24), (25), we enter the notation = = BAmax | bai |, EA max | ea i |, iN=0, iN = 0, k Zi =max | Vi |, iN = 0, ; (26) ki=,2 Ni − = = Simax| S i |, S i min| S i | i=0, N i=0, N
We estimate the expressions (24), (25)
iiS i S i ≤⋅ +p + p +⋅ +×2 Zi+1 2 hNBAZ∑∑ p hN 8 VhN p 8 ∑ V p 2 hNBAZ ∑ p hN pp==11hSp p = 1 hS p p = 1 (27) i ii i i × +=⋅ + +⋅ +⋅⋅ + =− EAZ∑p Oh()2 TBAZ ∑∑ ppp 16 N SZp 2 TBAZTEAh ∑ ∑ Z p Oh (), i 1,1 N p=1 pp == 11 p = 1 p = 1
iiS i S i ≤⋅+p + p +⋅+×2 Si+1 2 hNBAZ∑∑ p hN 8 VhN p 8 ∑ V p 2 hNBAZ ∑ p hN pp==11hSp p = 1 hS p p = 1 (28) i ii i i × +=⋅ + +⋅ +⋅⋅ + =− EA∑ Zp Oh()2 TBA ∑∑ Z ppp 16 N SZp 2 TBA ∑ Z TEAh ∑ Z p Ohi (),1,1 N p=1 pp == 11 p = 1 p = 1
S p = = + Here S p , if we denote by Zi+1 max (Zi+1,S i 1 ), then i= N−1,0 S p
i i i ≤⋅ + +⋅⋅ + Zi+1 4 TBA∑ Z pp 16 NS ∑ Z TEAh ∑ Z p Oh ( ) (29) p=1 p = 1 p = 1
Using the discrete analogue of the Gronwall – Bellman lemma from the last estimate, we obtain O( h ) = ⋅( ⋅+ +⋅ ) Zi+1 Oh() exp4 TBA 16 NShTEA (30)
Theorem 4.1. The conditions (4) and (6), (solution V tz ),( ∈C4 (∆(T)) ) are fulfilled, then the finite-difference solution k ~ k ~ (Vi ,Si ) of the inverse problem (13)–(15) constructed converges to the exact solution ( Vi ,Si ) of the inverse problem with a speed of order O h)( and has the estimate (30).
4.2.2. Finite-difference Regularized Solution Let the additional information for the inverse problem be given with an error of ε , that is satisfied:
ε |ft ( )− f ( t ) | < ε , ε is a small number (31) 71 Abdugany Dzhunusovich Satybaev et al. : Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process
ˆ k,ε ˆ ε Then for a finite-difference regularized solution of the inverse problem (we denote it by Vi+1 ,Si+1 , we can also obtain formulas of the form (22) and (23):
ε ε i p Sˆ− S ˆ Vffˆ k,ε=+( k+ i + 1, ε k − i − 1, ε )/2 − h 2 µ µ −1 +⋅ baV ( ˆ k− i −µ + 2 p , ε − i+1 ∑∑ µ µ p=1µ = 1 hS µ (32) i p i p −+ˆkip−−+µε2, ˆ kipkip −−+ µεµε 2, − ˆ −−+− 21,2 − ˆ kip −−+ µε 2, =−=− Vµ−1 )() h∑∑ baV µµµ V h ∑∑ eaV µµ , iNkiNi 1, 1; , p=1µ = 1 p =1µ = 1
ε ε i p Sˆ− S ˆ Sffˆε=+−(2i+ 2, ε 0, ε )/2 h 2 µ µ −1 +⋅ baV ( ˆ −µ +2p + 1, ε − i+1 ∑∑ µ µ p=1µ = 1 hS µ (33) i p i p −+ˆ−++µε2p 1, ˆ −++−+ µεµε 2p 1, −− ˆ 2 p , 2 ˆ −++ µε 2 p 1, =− Vµ−1 )() h∑∑ baV µµµ V h ∑∑ eaV µµ , iN 1, 1 p=1µ = 1 p =1µ = 1
Taking now from formulas (22), (23) of formulas (32), (33), we obtain
⌣ i p kkk=−=−+−ˆ ,ε ki++ 1 ki ++ 1, ε ki −− 1, ε ki −− 1 −ˆ kipkip −−+µ2 − ˆ −−+ µ 2 + VVVfi+1 i + 1 i + 1 ( f f f )/2 h∑∑ { baVµ( µ V µ − 1 ) p=1µ = 1 ˆ− ˆ − SSµ µ−1 −−+µεµε −−+ SSµ µ − 1 −−+ µ −−+ µ +2⌣ (VVkip2 , −+ kip 2 , ) 2 ( VVˆ kipkip2 −+ ˆ 2 ) (34) µ µ−1 µ µ − 1 hSµ hS µ i p i p + ˆkipkip−−+µ2− ˆ −−+− µ 212 − ˆ kip −−+µ 2 =−=− h∑∑ baVµµ( V µ) h ∑∑ eaV µµ , iNkiNi 1, 1; , p=1µ = 1 p =1µ = 1
⌣ i p =−=−+−−ˆε2i+ 2 2 i + 2, ε 0, ε 0 ˆ−++µ21p − ˆ −++ µ 21 p + SSSi+1 i + 1 i + 1 ( f f ff )/2 h∑∑ { baVµ( µ V µ − 1 ) p=1µ = 1 ˆ ˆ SSµ− µ− SSµ − µ − 1 −++µεµε21,p −++ 21, p 1 ˆ −++ µ 21p ˆ −++ µ 21 p +2 (VVµ −+ µ− ) 2 ( VVµ −+ µ − ) (35) ˆ 1 1 hS µ hS µ i p i p +ˆ−++µ2p 1 − ˆ − µ +22p−ˆ −++µ 21 p = − h∑∑ baµ( V µ V µ ) h ∑∑ eaVµ µ , i 1, N 1 p=1µ = 1 p =1µ = 1
Taking into account the introduced notation (26), we estimate the last equations: ⌣ii ⌣⌣ i i ⌣ ≤+⋅ε + +⋅ +⋅⋅ =− Zi+1 2 TBAZ∑∑ ppp 16 NSZp 2 TBAZTEAhZ ∑ ∑ p , i 1, N 1 (36) pp==11 p = 1 p = 1
⌣ ii⌣ ⌣ i i ⌣ ≤+⋅ε + +⋅ +⋅⋅ =− S hTBAZ∑∑p16 N SZp p 2 TBAZTEAhZ ∑ p ∑ p , i 1, N 1 (37) pp==11 p = 1 p = 1 ⌣ ⌣ δ = Denote by Zi+1 max{Zi+1, Si+1}, then from the last expressions we get: i= N −1,0
i i i ε≤+⋅ε εε + +⋅⋅ ε =− Zi+1 4 TBAZ∑ pp 16 NSZTEAhZ ∑ ∑ p , i 1,1 N (38) p=1 p = 1 p = 1
Using the Gronwall – Bellman inequality, we obtain American Journal of Applied Mathematics 2020; 8(2): 64-73 72
z =⋅δ [ ⋅+ +⋅ ] Zi+1 exp 4 T BA 16 NS hT BA (39)
Then the estimate of a finite-difference regularized solution of the inverse problem has the form: z, O ( h ) =+ε δ ⋅++⋅ Zi+1 ( Oh ())exp(4 TBA 16 NShTEA ) (40)
Theorem 4.2. Suppose that conditions (4), (6) and (31) are satisfied, then the constructed finite-difference regularized solution converges to the exact solution of the inverse problem with a speed of order O h)( and has the estimate (40).
4.2.3. The Algorithm for Obtaining a Finite-Difference Regularized Solution of the Generalized One-Dimensional Inverse Problem (1)–(3) ε hO )(, = ε hOi )(,, Thus, the finite-difference regularized solution of the inverse problem (13) – (15) is Si Vi , then by formula (16) we have:
ε,()Oh ε ,() Oh ε S− S ,O ( h ) =−i i −1 − = gai 2ε ,O ( h ) baiNi , 0, (41) hS i
We note that it is possible to obtain from the formula
' = ⋅ − caz ()2() z da z ca ()2 z ga ()() z ca z (42) we integrate
z =+ λλ −⋅ λ λλ] caz() ca (0)2∫ ca ()() da ca () ga () d (43) 0
Using the quadrature formulas for the integral, we obtain the solution of the inverse problem (7), (14), (15):
i caε,()Oh=+ ca(0) 2 ca εεε ,() Oh da − ca ,(),() OhOh ga ⋅= h , i 1, N i∑ iiii (44) p=1
ε hO )(, Having determined cai we can get:
ε ε ,()Oh= ,() Oh ⋅ = (45) ci ca i ai i , 0, N
ε hO )(, algorithm for solving the problem. ci is a finite-difference regularized solution of the equivalent inverse problem (1) – (3). Comment. Other coefficients of equation (1) can also be References recovered by the same method and obtain their finite-difference regularized solutions. [1] A. N. Tikhonov and V. Ya. Arsenin, Methods for solving ill-posed problems, Moscow, The science, 1979 (in Russian). 5. Conclusion [2] M. M. Lavrentiev, V. G. Romanov and S. P. Shishatsky, Ill-posed problems of mathematical physics and analysis, This article investigates the inverse problem of the wave Moscow, The science, 1980 (in Russian). process with boundary data of the Dirac delta-function and [3] V. K. Ivanov, V. V. Vasin and V. P. Tanana, The theory of the Heaviside theta-function. The problem is reduced to a linear ill-posed problems and its applications, Moscow, The problem with data on characteristics using the method of science, 1978 (in Russian). characteristics and the method of isolation of singularities. To the last problem with the data on the characteristics is [4] V. G. Romanov, The stability in inverse problems, Moscow, The scientific world, 2005 (in Russian). applied the finite-difference method, and to relatively small changes with additional information is applied the [5] S. I. Kabanikhin, Inverse and ill-posed problems, Novosibirsk, finite-difference regularized method, which developed by A. Siberian Scientific Publishing House, 2009 (in Russian). Dzh. Satybaev. For numerical implementation constructed an 73 Abdugany Dzhunusovich Satybaev et al. : Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process
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