The Method of Fundamental Solutions with Eigenfunctions Expansion Method for 3D Nonhomogeneous Diffusion Equations

Abstract

After the successful applications of the combination of the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunctions expansion method (EEM) to solve 2D homogeneous and nonhomogeneous diffusion equations by Young et al. (Young et al., Numer Meth Part Differ Equat 22 (2006), 1173), this article intends to extend the same fundamental concepts to calculate more challenging 3D nonhomogeneous diffusion equations. The nonhomogeneous diffusion equations with time-independent source terms and boundary conditions are analyzed by the proposed meshless MFS-MPS-EEM model. Nonhomogeneous diffusion equation in any complex domains can be decomposed into a Poisson equation and a homogeneous diffusion equation by the principle of linear superposition. This approach is proved to be far better off than solutions by using classic method of separation of variables with inefficient multisummation of very sophisticated series expansion from special functions, which can only limit to treat very simple 3D geometries such as cube, cylinder, or sphere. Poisson equation is solved by using the MPS-MFS model, in which the source term in the Poisson equation is first handled by the MPS based on the compactly-supported radial basis functions and the Laplace equation is solved by the MFS. On the other hand, by utilizing the EEM, the homogeneous diffusion equation is first transformed into a Helmholtz equation, which is then solved by the MFS together with the technique of singular value decomposition (SVD) to acquire the eigenvalues and eigenfunctions. After the eigenfunctions are obtained, we can synthesize the diffusion solutions like the orthogonal Fourier series expansions but with only one summation for the series even for multidimensional problems. Numerical results for four case studies of 3D homogeneous and nonhomogeneous diffusion problems show good agreement with the analytical and other numerical solutions, such as finite element method (FEM). Thus, the present numerical scheme has provided a promising meshfree numerical approach to solve 3D nonhomogeneous diffusion equations with time-independent source terms and boundary conditions for very irregular domains. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009