# A numerical method for solving the Dirichlet problem for the wave equation

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## Abstract

In this paper we present a numerical method for solving the Dirichlet problem for a two-dimensional wave equation. We analyze the ill-posedness of the problem and construct a regularization algorithm. Using the Fourier series expansion with respect to one variable, we reduce the problem to a sequence of Dirichlet problems for one-dimensional wave equations. The first stage of regularization consists in selecting a finite number of problems from this sequence. Each of the selected Dirichlet problems is formulated as an inverse problem *Aq* = *f* with respect to a direct (well-posed) problem. We derive formulas for singular values of the operator *A* in the case of constant coefficients and analyze their behavior to judge the degree of ill-posedness of the corresponding problem. The problem *Aq* = *f* on a uniform grid is reduced to a system of linear algebraic equations *A* _{ ll } *q* = *F*. Using the singular value decomposition, we find singular values of the matrix *A* _{ ll } and develop a numerical algorithm for constructing the *r*-solution of the original problem. This algorithm was tested on a discrete problem with relatively small number of grid nodes. To improve the calculated *r*-solution, we applied optimization but observed no noticeable changes. The results of computational experiments are illustrated.

### Keywords

Dirichlet problem wave equation degree of ill-posedness singular value decomposition## Preview

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