Newly generated path graph matching-polynomial matrix of integration to solve initial-value, finite-value and infinite-boundary-value problems

Abstract

This work introduces a new class of matching polynomials from path graphs, embedded into a closed-form operational matrix of integration for solving differential equations. Unlike conventional spectral methods, the approach avoids orthogonalization, uses symbolic–numeric computation, and reuses a precomputed integration matrix to reduce computational cost. The method effectively handles both linear and nonlinear differential equations, offering flexibility in addressing diverse boundary conditions, including transformed infinite boundaries. The computational efficiency is further enhanced by the reuse of precomputed integration matrices, leading to reduced memory usage and faster runtimes. The method is applied to initial-value, finite-boundary, and transformed-infinite boundary problems, with examples from Williamson fluid flow and a nonlinear atmospheric model. Comparisons with exact solutions, numerical solvers, and previous wavelet-based approaches confirm spectral-like accuracy, stability, and efficiency, demonstrating its versatility for nonlinear dynamical systems across diverse scientific domains. Furthermore, the method is applied to climate modeling, illustrating its broader applicability to problems in environmental science and public health. The method robustness, fast convergence, and ease of implementation make it an effective tool for solving complex real-world differential equations across diverse scientific and engineering disciplines.