Meshfree Method for Solving Partial Differential Equations Using Radial Basis Functions in MATLAB

Solving partial differential equations (PDEs) represents a fundamental requirement in engineering and scientific computing. Traditional mesh-based methods like finite element analysis require complex meshing procedures, while meshfree methods directly solve equations using discrete points, bypassing this bottleneck. Radial basis functions (RBF) serve as core tools in meshfree methodologies due to their excellent interpolation properties. The operational principle of MATLAB-based RBF meshfree solvers involves several key implementation steps: First, discrete nodes are distributed within the solution domain, requiring no regular arrangement. Next, appropriate radial basis functions (such as Gaussian functions, polyharmonic splines, etc.) are selected as interpolation basis functions. The PDE is then transformed into a system of linear equations using either collocation methods or Galerkin methods. MATLAB's powerful matrix computation capabilities enable efficient solution of these systems, where backslash operators (\) or dedicated linear algebra functions handle the numerical resolution. Key advantages of this approach include: 1) Adaptation to complex geometrical shapes through flexible node placement, 2) Native support for high-dimensional problems without mesh generation constraints, 3) Simplified implementation of h-p adaptivity through dynamic node refinement. However, attention must be paid to conditioning issues and the impact of parameter selection on solution accuracy. MATLAB's matrix manipulation capabilities and visualization functions (like meshgrid, surf, and contour plots) are particularly suitable for rapid algorithm validation and result analysis. For practical implementation, critical technical aspects involve optimizing basis function shape parameters using cross-validation techniques and properly handling boundary conditions through specialized collocation strategies. This methodology proves especially effective for irregular domains or moving boundary problems, where MATLAB's scripting environment facilitates quick prototyping and parameter tuning through iterative testing loops.