A new spectral framework for solving fractional-order differential equations: improved efficiency without residual functions

Volume 41, Issue 2, pp 207--221 https://dx.doi.org/10.22436/jmcs.041.02.05

Publication Date: October 03, 2025 Submission Date: May 23, 2025 Revision Date: June 28, 2025 Accteptance Date: July 28, 2025

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Authors

D. Zaidi - Department of Mathematics, University of Management and Technology, 54770 Lahore, Pakistan. I. Talib - Department of Mathematics, Nonlinear Analysis Group (NAG), Virtual University of Pakistan, 54-Lawrence Road, Lahore, Pakistan. M. B. Riaz - Department of Mathematics, University of Management and Technology, 54770 Lahore, Pakistan. - IT4Innovations, VSB-Technical University of Ostrava, Ostrava, Czech Republic. - Applied Science Research Center, Applied Science Private University, Amman, Jordan.

Abstract

The Spectral Tau method is a widely recognized numerical technique employed for solving fractional-order differential equations (FODEs). The central focus of the Spectral Tau method’s algorithm depends on the idea of operational matrices derived from the basis sets of orthogonal polynomials, which are utilized to approximate derivative terms in the problems. In this study, our main objective is to introduce a new numerical method within the class of spectral methods, but distinct in its formulation from the Spectral Tau method. While both approaches utilize operational matrices of orthogonal polynomials, the proposed method avoids the computation of residual functions, which is a key step in the Spectral Tau method. Another important feature of the proposed study is the construction of novel generalized integral operational matrices in the Riemann-Liouville sense, developed using a basis of orthogonal shifted Laguerre polynomials (OSLPs). This structure leads to simplified implementation, reduced computational cost, and enhanced spectral accuracy. To demonstrate the efficiency and practical applicability of the proposed method, we solve several test problems. Additionally, we compare the computational efficiency and the absolute errors obtained using our proposed method with those derived from the Spectral Tau method.

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Keywords

• Generalized fractional-order operators
• generalized operational matrices
• Sylvester equations
• Laguerre polynomials
• spectral methods

MSC

•  65L60
•  65N35
•  65M70
•  26A33

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