Symmetric Systems of Coordinate Functions for Homogeneous Boundary Value Problems on a Rectangle With Partial Differential Equations

Abstract

This paper is the first to propose the idea of ​​solving homogeneous boundary value problems with homogeneous partial differential equations on a selected class of functions of two variables while preserving symmetry. To achieve the best accuracy, 7 even or 7 odd coordinate functions are used for each variable. It is shown that an arbitrary continuous function on the right side of the equation can be represented as the sum of four continuous functions, one from each symmetry class. Using the Bubnov−Galerkin method, a solution is found numerically in each of the 4 symmetry classes of functions; the solution to the problem is equal to the sum of four partial solutions. In the first part of the work, 3 lemmas and a theorem are proved − a criterion for a solution to belong to a given parity class. We numerically solve four boundary value problems with the Poisson equation, the right side of which belongs to different parity classes. The uniform norm of the discrepancy in solving the problems is of the order of 10^-8. Another advantage of the proposed algorithm is its high-speed performance; for example, it outperforms the vector sweep method for solving the Poisson equation by a factor of 5 to 10.