An efficient series polynomial collocation method for solving matrix differential equations

Lakhlifa Sadek; Ibtisam Aldawish; Lakhlifa Sadek; Ibtisam Aldawish

doi:10.3934/math.2026054

AIMS Mathematics

2026, Volume 11, Issue 1: 1266-1286. doi: 10.3934/math.2026054

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Research article

An efficient series polynomial collocation method for solving matrix differential equations

Lakhlifa Sadek ^{1,2
,
,},
Ibtisam Aldawish ³

1.
Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamilnadu, India
2.
Department of Mathematics, Faculty of Sciences and Technology, BP 34. Ajdir 32003 Al-Hoceima, Abdelmalek Essaadi University, Tetouan, Morocco
3.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia

Received: 21 September 2025 Revised: 15 December 2025 Accepted: 24 December 2025 Published: 16 January 2026
MSC : 34A30, 41A10, 65L05

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.
- matrix differential equations,
- Kronecker product,
- collocation method,
- error analysis,
- spectral methods,
- numerical stability
Citation: Lakhlifa Sadek, Ibtisam Aldawish. An efficient series polynomial collocation method for solving matrix differential equations[J]. AIMS Mathematics, 2026, 11(1): 1266-1286. doi: 10.3934/math.2026054

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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.

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