On an ill-posed boundary value problem for the Laplace equation in a circular cylinder

Abstract

In this paper, we consider a mixed problem for the Laplace equation in a region in a circular cylinder. On the lateral surface of a cylidrical region, the homogeneous boundary conditions of the first kind are given. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i.e. a function and its normal derivative are given. The other border of the cylindrical area is free. This problem is ill-posed, and to construct its approximate solution in the case of Cauchy data known with some error it is necessary to use regularizing algorithms. In this paper, the problem is reduced to a Fredholm integral equation of the first kind. Based on the solution of the integral equation, an explicit representation of the exact solution of the problem is obtained in the form of a Fourier series with the eigenfunctions of the first boundary value problem for the Laplace equation in a circle. A stable solution of the integral equation is obtained by the Tikhonov regularization method. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem in the whole is constructed. The theorem on convergence of the approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is matched with the error in the data is given. The results can be used for mathematical processing of thermal imaging data in medical diagnostics.