Solving Diffusion and Wave Equations Meshlessly via Helmholtz Equations

TL;DR

This paper introduces a meshless method combining fundamental solutions and particular solutions of Helmholtz equations to efficiently solve diffusion and wave equations by discretizing them into Helmholtz problems.

Contribution

It presents a novel meshless approach that transforms time-dependent diffusion and wave equations into Helmholtz problems for efficient numerical solutions.

Findings

Method effectively solves diffusion and wave equations.

Numerical examples demonstrate high efficiency.

Approach is meshless and iterative.

Abstract

In this paper, using the approximate particular solutions of Helmholtz equations, we solve the boundary value problems of Helmholtz equations by combining the methods of fundamental solutions (MFS) with the methods of particular solutions (MPS). Then the initial boundary value problems of the time dependent diffusion and wave equations are discretized numerically into a sequence of Helmholtz equations with the appropriate boundary value conditions, which is done by either using the Laplace transform or by using time difference methods. Then Helmholtz problems are solved consequently in an iterative manner, which leads to the solutions of diffusion or wave equations. Several numerical examples are presented to show the efficiency of the proposed methods.