Development of a Finite-difference Regularized Solution of the One-Dimensional Inverse Problem of the Wave Process

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.


Introduction
Inverse problems are the so-called ill-posed problems, the foundations of which were laid by Academicians of the Russian Academy of Sciences Andrei N. Tikhonov [1], Mikhail M. Lavrentiev [2], Corresponding Member of the Russian Academy of Sciences Valentin K. Ivanov [3].
The inverse problems of wave processes were considered in theoretical terms by the Corresponding members of the Russian Academy of Sciences Vladimir G. Romanov [4], Sergey I. Kabanikhin [5], professor Valery G. Yakhno [6], and they constructed solutions to the posed inverse problems. The regularization method for inverse problems was developed by Andrei N. Tikhonov [7], the method of small parameters was constructed by Mikhail M. Lavrentiev [8] and the quasi-solution method -by Valentin K. Ivanov [9]. The projection-difference method for multidimensional inverse problems was developed by Sergey I. Kabanikhin [10], the finite-difference regularization for Volterra integral equation of the first kind was investigated by Abdugany Dzh. Satybaev [11] and a regularization estimate is obtained.
Aliyma T. Mamatkasymova and Abdugany Dzh. Satybaev [12] constructed a finite-difference regularized solution and obtained a convergence estimate for the inverse problem arising in electromagnetic processes.
The purpose of this work is to numerically solve a one-dimensional inverse problem of the wave process proposed by the authors by a finite-difference regularized method, which allows us to construct a numerical algorithm for solving the problem.

Research Overview
Most of the scientific and technical aspects of the inverse problems of wave propagation in the medium and the mathematical connections between waves and scatterers are determined and presented in work G. Ghavent, G. C. Papanicolaou, P. Sacks, W. Symes [13].
The book, edited by G. Chavent and P. C. Sabatier [14], describes the current state of modeling and the numerical solution of wave propagation and diffraction, their applications, and features of inverse scattering problems on classical and distributed media.
The article set forth by F. Natterer [15] three major methods for solving inverse problems: the method of ray tomography, the method of single-particle emission tomography and the method of positron-emission tomography and described transfer equations transitions near the infrared region to elliptic equations in the diffusion approximation.
In the article by F. Natterer, A. Wiibbeling [16], a numerical calculation of the potential in the Helmholtz equation was laid out and a method was developed which possesses the stability of the solution and the convergence of the solution was shown in the order O (h 4 ).
Modern mathematical modeling of various wave processes (computed tomography, ultrasonic flaw detection, etc.) in the theory of inverse problems and their examples, main features, perspectives are presented in the work of Alexander O. Vatulyan [17].
In the Andrey V. Bayev's dissertation work [18] developed stable methods for solving inverse problems, determined the characteristics of real inhomogeneous layered media, and according to experimental information, clarified the time parameters of the source of disturbances, and also presented practical solutions to a number of actual inverse problems in borehole exploration geophysics.
Dynamic inverse problems, definitions of one or several coefficients of hyperbolic equations or systems, methods for solving one-dimensional inverse problems, scalar inverse problems of wave propagation in layered media, inverse problems for the theory of elasticity and acoustic equations are given in the monograph of Alexander S. Blagoveshchenskii [19].
In the monograph of Sergey I. Kabanikhin [21] outlines methods for studying and solving inverse and ill-posed problems of linear algebra, integral and operator equations, integral geometry, spectral inverse problems and inverse scattering problems; linear ill-posed problems and coefficient inverse problems for hyperbolic, parabolic and elliptic equations were considered; given extensive reference material.
A new globally convergent numerical method is developed for a multidimensional coefficient inverse problem for a hyperbolic PDE with applications in acoustics and electromagnetics. On each iterative step the Dirichlet boundary value problem for a second-order elliptic equation is solved. The global convergence is rigorously established, and numerical experiments are presented in works Larisa Beilina, Michael V. Klibanov [22].
In the book by V. A. Burov and O. D. Rumyantseva [23], inverse wave problems and their applied aspects related to linear and nonlinear acoustic tomography, as well as acoustic thermotomography, are considered. Part I briefly discusses the inverse coherent radiation problems, which are characterized by incorrectness and a strong degree of non-uniqueness. Various approaches to solving inverse wave problems of radiation and incoherent problems of active-passive acoustic thermotomography are described. It is shown that the active-passive mode allows you to determine the set of acoustic and thermal characteristics of the medium within the framework of the general tomographic scheme.

Formulation of the Problem
Wave processes of natural phenomena (earthquakes and natural explosions, landslides and lavas), electrodynamics and geophysics, etc., are characterized by fields described by second-order partial differential equations: For (1) we consider the initial and boundary conditions of the form: Usually, when considering inverse problems, the correctness (existence, uniqueness, stability) of a solution is established; we established it in the works [24], [25].
A. When coefficients are not equal to zero, the direct problem (1)-(2) is a problem of hyperbolic type (problems: wave equation, acoustics, seismic, electrodynamics, etc.).
B. When the coefficient we have a problem of parabolic type (the problems of: diffusion, thermometry, the distribution of quasi-stationary electromagnetic fields, etc.).
C. When coefficients we have a problem of elliptic type (the problems of: gravimetry, geoelectrics of stationary fields, etc.).
The problem (1)-(2) with the following coefficients is transformed: Problem I. When coefficients is the density of the medium-to the acoustics problem; Problem III. When is the electrical conductivity of the medium -to the geoelectric problem; Problem V. When The main problem in solving the inverse problems (1)-(3) is, firstly, the presence of the Dirac delta-function in the boundary condition and secondly, the problem is reduced to a system of nonlinear integral equations [10].
Let in relation to the coefficients of equation a condition is executed In condition (4), the smoothness of the function is given an increased one to apply the finite-difference method. Equation (1) is a hyperbolic equation, and from the principle of dependence on the specification of data, the solution to problem (1)-(3) can be considered in the region ) (T ∆ [10]: If condition (4) is met, the additional information is , and the solution to the problem is The inverse problem consists in determining one of the coefficients of the equation (it would be good if all the coefficients of the equation are determined simultaneously, but for these additional conditions must be specified).

The Inverse Problem
The inverse problem is to determine the function ) (x с from problem (1)

The Method of Characteristic
We will enter the new variable of and we will enter the new functions of ). , ( ) ), We make some calculations: Substituting the last calculations into equation (1), we obtain

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: We make some calculations: Substituting the last calculations into the obtained equation, we obtain  (2), we obtain is even function. Equating terms at Taking into account that , and also from the higher got calculations, we will get a next inverse problem with data on the characteristics: Here the inverse problem is to determine the functions When z t = , we obtain:

Finite-difference Solution
To solve problem (13) -(15), we enter the grid region The difference analogue of the differential equation (13): From (19) we obtain From the last formula (21) we obtain recurrence formulas: 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): In the last formula (22) Formulas (22) and (23) we obtain the following expressions: To estimate (24), (25), we enter the notation 0, 0, We estimate the expressions (24), (25)

Finite-difference Regularized Solution
Let the additional information for the inverse problem be given with an error of ε , that is satisfied: Comment. Other coefficients of equation (1) can also be recovered by the same method and obtain their finite-difference regularized solutions.

Conclusion
This article investigates the inverse problem of the wave process with boundary data of the Dirac delta-function and the Heaviside theta-function. The problem is reduced to a problem with data on characteristics using the method of characteristics and the method of isolation of singularities.
To the last problem with the data on the characteristics is applied the finite-difference method, and to relatively small changes with additional information is applied the finite-difference regularized method, which developed by A. Dzh. Satybaev. For numerical implementation constructed an algorithm for solving the problem.