NUMERICAL SOLUTION OF THE INVERSE PROBLEM FOR THE ACOUSTIC EQUATION

Abstract

This article presents the continuation problem for one-dimensional equations of acoustics. Currently, one of the most difficult areas of research in applied mathematics is the reverse calculation of wave propagation, which has an important area of application in geophysics and medicine. The propagation of an acoustic wave in an inhomogeneous medium and the measurement of the return of some part of the sound by special devices leads to the calculation of the initial-boundary for the partial derivative equation. Most of these problems are ill-posed. One such ill-posed problem is the continuation problem. The continuation problem is based on finding the value of the desired function in the rest of the boundary using additional data in a certain part of the boundary. In this regard, we present the continuation problems to the inverse boundary value problem. There are many methods for solving general boundary value inverse problems, for example, the gradient method. In this paper, we construct a finite-difference scheme for this inverse problem and find an unknown function on the characteristic from this difference equation by inverting the difference scheme. The effectiveness of this method lies in a simple and accurate fast calculation algorithm. To test the correctness of any method or algorithm, we first solve a direct problem by specifying a exact function. Then we find additional information and solve the inverse problem. Let us compare the inverse problem with solving a problem with an exact function. However, in practice it often happens that additional information is provided with certain errors. Therefore, the correctness of the algorithm was verified by adding different error levels to the additional information. Quantitative data and graphs of these quantitative results are compared.