One non-local problem for the laplace equation in a disc

Abstract

This article deals with a non-local problem for the harmonic function in a disc, when the value of the function on one part of its semi-circle and the relation of unknown function values on the semi-circles of concentrated circles on the other part of the circle are given. This problem is reduced to the integral equation with a periodical kernel. To solve it, the analogue of the Wiener–Hopf method for a discrete case is developed. Its use allows one to obtain the problem equivalent to the factorization problem which has a unique solution. An effective solution is constructed.

Keywords:

• Poisson formula
• Laplace equation
• Fourier transform