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.
Highlights
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
The inverse problems of wave processes were considered in theoretical terms by the Corresponding members of the Russian Academy of Sciences Vladimir G
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
Summary
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. 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
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