Young等人成功应用基本解法（MFS）、特解法（MPS）和本征函数展开法（EEM）相结合求解二维齐次和非齐次扩散方程。（Young 等人，Numer Meth Part Differ Equat 22 (2006), 1173），本文旨在扩展相同的基本概念来计算更具挑战性的 3D 非均匀扩散方程。通过提出的无网格 MFS-MPS-EEM 模型分析具有时间无关源项和边界条件的非齐次扩散方程。根据线性叠加原理，任何复杂域中的非齐次扩散方程都可以分解为泊松方程和齐次扩散方程。事实证明，这种方法比使用经典方法分离变量的解决方案要好得多，该方法对特殊函数的非常复杂的级数展开进行低效的多重求和，这只能限制处理非常简单的 3D 几何图形，例如立方体、圆柱体或球体。泊松方程采用 MPSMFS 模型求解，其中泊松方程中的源项首先由基于紧支撑径向基函数的 MPS 处理，拉普拉斯方程由 MFS 求解。另一方面，利用EEM，首先将齐次扩散方程转化为亥姆霍兹方程，然后由MFS结合奇异值分解（SVD）技术求解，得到特征值和特征函数。得到特征函数后，我们可以像正交傅立叶级数展开式那样综合扩散解，但即使对于多维问题，该系列也只有一个求和。3D 均匀和非均匀扩散问题的四个案例研究的数值结果与解析解和其他数值解（如有限元法 (FEM)）非常吻合。因此，当前的数值方案提供了一种有前途的无网格数值方法，用于求解具有时间无关源项和非常不规则域的边界条件的 3D 非齐次扩散方程。© 2008 Wiley Periodicals, Inc. 数值方法偏微分方程：000–000，2008 3D 均匀和非均匀扩散问题的四个案例研究的数值结果与解析解和其他数值解（如有限元法 (FEM)）非常吻合。因此，当前的数值方案提供了一种有前途的无网格数值方法，用于求解具有时间无关源项和非常不规则域的边界条件的 3D 非齐次扩散方程。© 2008 Wiley Periodicals, Inc. 数值方法偏微分方程：000–000，2008 3D 均匀和非均匀扩散问题的四个案例研究的数值结果与解析解和其他数值解（如有限元法 (FEM)）非常吻合。因此，当前的数值方案提供了一种有前途的无网格数值方法，用于求解具有时间无关源项和非常不规则域的边界条件的 3D 非齐次扩散方程。© 2008 Wiley Periodicals, Inc. 数值方法偏微分方程：000–000，2008 目前的数值方案提供了一种很有前途的无网格数值方法来求解具有时间无关源项和非常不规则域的边界条件的 3D 非齐次扩散方程。© 2008 Wiley Periodicals, Inc. 数值方法偏微分方程：000–000，2008 目前的数值方案提供了一种很有前途的无网格数值方法来求解具有时间无关源项和非常不规则域的边界条件的 3D 非齐次扩散方程。© 2008 Wiley Periodicals, Inc. 数值方法偏微分方程：000–000，2008 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 timeindependent 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 MPSMFS 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 : 000–000, 2008