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An optimization method in the Dirichlet problem for the wave equation

2012; De Gruyter; Volume: 20; Issue: 2 Linguagem: Inglês

10.1515/jip-2012-0025

1569-3945

Sergey Kabanikhin, Maktagali A. Bektemesov, Daniyar Nurseitov, Olga Krivorotko, Anel N. Alimova,

Elasticity and Wave Propagation

Abstract. A numerical method for solving the Dirichlet problem for the wave equation in the two-dimensional space is proposed. The problem is analyzed for ill-posedness and a regularization algorithm is constructed. The first stage in the regularization process consists in the Fourier series expansion with respect to one of the variables and passing to a finite sequence of Dirichlet problems for the wave equation in the one-dimensional space. Each of the obtained Dirichlet problems for the wave equation in the one-dimensional space is reduced to the inverse problem with respect to a certain direct (well-posed) problem. The degree of ill-posedness of the inverse problem is analyzed based on the character of decreasing of the singular values of the operator A . The numerical solution of the inverse problem is reduced to minimizing the objective functional . The results of numerical calculations are presented.