On an approximate solution to an ill-posed mixed boundary value problem for the Laplace equation in a cylindrical domain with homogeneous conditions of the second kind on the lateral surface of the cylinder

Abstract

We consider a problem mixed in boundary conditions for the Laplace equation in a domain that is a part of a cylinder of a rectangular cross-section with homogeneous boundary conditions of the second kind on the side surface of the cylinder. The cylindrical region is limited on one side by surface of a general kind on which the Cauchy conditions are specified, i.e. a function and its normal derivative are given, and the other boundary of the cylindrical region is free. In this case, the problem has the property of instability of the Cauchy problem for the Laplace equation with respect to the error in the Cauchy data, i.e. is ill-posed, and its approximate solution, robust to errors in Cauchy data, requires the use of regularization methods. The problem under consideration is reduced to the Fredholm integral equation of the first kind. Based on the solution of the integral equation obtained in the form of a Fourier series on the eigenfunctions of the second boundary value problem for the Laplace equation in a rectangle, an explicit representation of the exact solution of the problem was constructed. A stable approximate solution to the integral equation was constructed using the Tikhonov regularization method. The extremal of the Tikhonov functional is considered as an approximate solution to the integral equation. Based on the approximate solution of the integral equation, an approximate solution of the boundary value problem as a whole is constructed. A theorem is proved for the convergence of an approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is consistent with the error in the data.