An efficient series polynomial collocation method for solving matrix differential equations

Abstract

This paper introduces a numerical method based on series polynomials and collocation techniques for the solution of first-order linear matrix differential equations. The proposed framework reformulates the original problem into a system of algebraic equations through structured matrix operations, including the use of Kronecker products. A rigorous error analysis is conducted to establish the accuracy and stability of the methods. Comprehensive numerical experiments are presented, comparing the performance of the series-based collocation approach with the Bernstein polynomial method. The results demonstrate notable improvements in accuracy, particularly for higher approximation orders, thereby validating the theoretical findings and confirming the superior precision of the proposed series-based techniques.