Solving Diffusion and Wave Equations Meshlessly via Helmholtz Equations

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.

Figures (12)

• Figure 1: Collocation, interior, and source points for Example \ref{['helm1ex']}.
• Figure 2: Example \ref{['helm1ex']} errors on $\partial \Omega$ and $3$ interior points
• Figure 3: Domains and points of $I_n(\Omega_{\delta})$ for Example \ref{['nonhomo helm ex1']}
• Figure 4: Example \ref{['nonhomo helm ex1']} errors for different RBFs, $n$, and $c$ values
• Figure 5: The boundary $\partial \Omega$ and points in $I_n(\Omega_{\delta})$ for Example \ref{['nonhomohelm R3 ex']}

Theorems & Definitions (7)

• Example 5.1
• Example 5.2
• Example 5.3
• Example 5.4
• Example 5.5
• Example 5.6
• Example 5.7