An optimization method in the Dirichlet problem for the wave equation

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.