Projective solutions method for solving multi-dimensional anisotropic Laplace equation, modified Helmholtz equation, and diffusion-advection equation

Abstract

The main issue of present paper is a new projective solutions method (PSM) to set up particular solutions as the bases for solving multi-dimensional linear partial differential equations with constant coefficients. For definiteness we are concerned with projective-type particular solutions (PTPSs) of the anisotropic Laplace equations, modified Helmholtz equations and two parabolic type equations. The PTPS is obtained via a projective function in terms of a projective variable; the governing equation of the projective function is a second-order ordinary differential equation (ODE) with constant coefficients. For the multi-dimensional anisotropic Laplace equations the PSM and the Trefftz projective solutions method (TPSM) are developed. The TPSM is extremely accurate. For the multi-dimensional modified Helmholtz equations the PSM is simple with rudimentary functions as the bases. Even for large wave number the numerical solution obtained by PSM is still very accurate. The exponential-cosine and exponential-sine functions are two linearly independent PTPSs for the heat equation, and for a linear diffusion-advection equation. Therefore, a powerful numerical method to solve these two parabolic type equations by means of meshless collocation technique is developed.