CONSTRUCTION OF A SOLUTION TO A BOUNDARY VALUE PROBLEM WITH N-D CONDITIONS FOR THE FOUR-DIMENSIONAL GELLERSTEDT EQUATION
Abstract
This work is a continuation of the study of a number of problems on the question of unique solvability for a degenerate equation of elliptic type. The article considers the Gellerstedt equation generalized to four variables in an infinite domain. This equation has four hypersurfaces of degeneracy. Earlier, sixteen fundamental solutions were constructed for the equation under consideration. From the basic theory of differential equations, it is known that each fundamental solution can be used in solving its own boundary value problem. Thus, by means of the obtained fundamental solutions the N problem, the Dirichlet problem, and two boundary value problems with mixed conditions have already been solved. The goal of this work is to construct a unique solution to a boundary value problem with mixed conditions, where one condition is the Neumann condition and three Dirichlet conditions. This is the first time that a problem with this formulation has been solved. We obtain solution of the problem in explicit form, which contains of second-order Gaussian hypergeometric series. In solving the problem, methods of partial differential equations, the method of differentiation of hypergeometric functions, the Gauss-Ostrogradsky formula and the Boltz autotransformation formula are used. The obtained results have a theoretical nature and can be used for further development of the theory of partial differential equations and the theory of special functions