A Spectral Approach to Variable-Order Fractional Differential Equations: Improved Operational Matrices for Fractional Jacobi Functions

Hany M. Ahmed, Mohammad Izadi and Carlo Cattani ()
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Hany M. Ahmed: Department of Mathematics, Faculty of Technology and Education, Helwan University, Cairo 11281, Egypt
Mohammad Izadi: Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 76169-14111, Iran
Carlo Cattani: Engineering School (DEIM), University of Tuscia, 01100 Viterbo, Italy

Mathematics, 2025, vol. 13, issue 16, 1-18

Abstract: The current paper presents a novel numerical technique to handle variable-order multiterm fractional differential equations (VO-MTFDEs) supplemented with initial conditions (ICs) by introducing generalized fractional Jacobi functions (GFJFs). These GFJFs satisfy the associated ICs. A crucial part of this approach is using the spectral collocation method (SCM) and building operational matrices (OMs) for both integer-order and variable-order fractional derivatives in the context of GFJFs. These lead to efficient and accurate computations. The suggested algorithm’s convergence and error analysis are proved. The feasibility of the suggested procedure is confirmed via five numerical test examples.

Keywords: collocation method; Liouville–Caputo derivative; variable-order fractional differential equations; fractional Jacobi functions; Jacobi polynomials; initial value problems; generalized hypergeometric functions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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