Об одной некорректно поставленной краевой задаче для метагармонического уравнения в круговом цилиндре

Abstract

In this paper, we consider a mixed problem for a metaharmonic equation in a domain in a circular cylinder. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i. e. the function and its normal derivative are set. The other border of the cylindrical area is free. On the lateral surface of the cylindrical domain, homogeneous boundary conditions of the first kind are given. The problem is illposed and its approximate solution, stable to errors in the Cauchy data, is constructed using regularization methods. The problem is reduced to a first kind Fredholm integral equation. Based on the solution of the integral equation obtained in the form of a Fourier series by the eigenfunctions of the first boundary value problem for the Laplace equation in a circle, an explicit representation of the exact solution of the problem is constructed. A stable solution of the integral equation is obtained by the method of Tikhonov regularization. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem as a whole is constructed. A 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 early diagnostics in medicine.