SOLUTION OF A MIXED BOUNDARY VALUE PROBLEM FOR THE LAPLACE EQUATION BY THE METHOD OF DIVIDING AN OBJECT IN A RECTANGULAR PARALLELEPIPED

Abstract

The aim of the study is to solve the boundary value problem for the Laplace equation in a rectangular parallelepiped by the method of separation of variables and to estimate the obtained Fourier variable separation constants.
 
 Materials and methods. Methods of mathematical physics were used to solve the boundary value problem for the Laplace equation. The initial problem was divided into three standard ones, in which the inhomogeneous boundary conditions were given only on two parallel sides, for the rest of the problem they being assumed to be equal to 0.
 
 Results. The boundary value problem for the Laplace equation in a rectangular parallelepiped has been broken down into three problems. Partial solutions to these problems under given boundary conditions have been obtained. The Fourier variable separation constants are estimated.
 
 Findings. The solution to the Laplace’s equation for a parallelepiped is the sum of the solutions to three partial problems. The boundary functions of a parallelepiped are odd periodic over two variable functions whose periods are equal to the lengths of the corresponding sides of the parallelepiped. The Fourier constants of partial solutions to the problem are the coefficients of the expansion of the boundary periodic functions of two variables into a trigonometric Fourier series. In two-dimensional series of the solution to the Laplace’s equation for odd-numbered harmonics and for a set of simultaneously even and odd harmonics, the Fourier constants differ only in signs.