The method of approximate particular solutions for solving certain partial differential equations

Abstract

A standard approach for solving linear partial differential equations is to split the solution into a homogeneous solution and a particular solution. Motivated by the method of fundamental solutions for solving homogeneous equations, we propose a similar approach using the method of approximate particular solutions for solving linear inhomogeneous differential equations without the need of finding the homogeneous solution. This leads to a much simpler numerical scheme with similar accuracy to the traditional approach. To demonstrate the simplicity of the new approach, three numerical examples are given with excellent results.

Compactly supported kernels method of approximate particular solutions for solving elliptic problems

“…If we write a radial kernel K in f -form with s = r 2 /2 = ∥ξ − η∥ 2 2 /2, its d-variate Laplacian follows via …”

“…In this approach the approximate particular solution is obtained in such a way to satisfy the differential equation as well as the boundary conditions [2]. In this numerical approach very simple and accurate kernel based numerical scheme is obtained.…”

The method of particular solutions for solving scalar wave equations

“…In this paper we will deal with (2) by using several versions of the method of particular solutions [12,14]. As we shall see in the following sections, more details about these approaches will be provided.…”

“…Various kinds of numerical schemes can be used to approximate the particular solutions as mentioned earlier. In [11,12], various numerical schemes using the approximate particular solution as a basis function have been implemented to solve the inhomogeneous problems without the need of obtaining a homogeneous solution. In this way, the MFS or other boundary methods are not required since the homogeneous solution is not needed in the solution process.…”

Two-dimensional convection—diffusion problem solved using method of localized particular solutions

“…The local method of approximated particular solution (LMAPS) was proposed by Cheng et al and was applied to elliptic problems [5] and non-linear problems [6]. In LMAPS the domain is covered by cloud of scattered nodes.…”