Method of moments with a choice of special basic functions for fourth-order partial differential equations

Abstract

This paper presents an approximation method to solve the boundary value problem (BVP) for partial differential equations (PDEs) of a kind of the fourth order with the idea of discretization of a spatial variable by using the method of moments with a choice of special basic functions. A system of ordinary differential equations (ODEs) is obtained by multiplying the original equation by some auxiliary functions, followed by interpolation and integration over the spatial variable. Newton-Stirling, Hermite-Birkhoff and Hermite interpolations are flexibly applied to internal and pre-boundary nodes. In addition, boundary conditions are automatically satisfied without being approximated separately as in classical numerical methods (for example, method of grids, method of lines). Thus, the proposed schemes have a higher order of approximation. The main goal of the work is the construction of basic functions for a fourth-order differential operator and a possible increase in the order of the remainder term when passing to difference equations.