Finite Element Method for Solving ODEs with MATLAB Toolbox Implementation

This article introduces a MATLAB-based finite element toolbox designed for numerically solving ODEs (Ordinary Differential Equations), PDEs (Partial Differential Equations), and BVPs (Boundary Value Problems). The toolbox supports problem domains in one-dimensional, two-dimensional, and multi-dimensional spaces, with current version designation 2.01. Key implementation features include: - Galerkin method implementation for weak formulation of differential equations - Isoparametric element support for geometric flexibility - Gaussian quadrature integration for stiffness matrix assembly - Sparse matrix solvers for efficient large-scale computations The toolbox streamlines solution processes for diverse mathematical problems, ranging from simple linear equations to complex nonlinear systems. Users can implement custom governing equations through modular function interfaces while leveraging built-in routines for mesh generation, boundary condition application, and post-processing visualization. Adaptive mesh refinement algorithms ensure precision optimization for problems with sharp gradients or boundary layers.