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

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.