Communications on Applied Mathematics and Computation >

2025 , Vol. 7 >Issue 4: 1419 - 1443

DOI: https://doi.org/10.1007/s42967-023-00337-y

ORIGINAL PAPERS

Efficient and Accurate Spectral Method for Solving Fractional Differential Equations on the Half Line Using Orthogonal Generalized Rational Jacobi Functions

  • Tarek Aboelenen
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  • 1. Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Qassim, Saudi Arabia;
    2. Department of Mathematics, Assiut University, Assiut, 71516, Egypt

Received date: 2023-02-14

  Revised date: 2023-10-02

  Accepted date: 2023-10-04

  Online published: 2024-02-02

Supported by

The author would like to express special thanks to anonymous referees for their valuable comments and suggestions, which significantly improved the quality of this paper.

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Abstract

A new set of generalized Jacobi rational functions of the first and second kinds, GJRFs-1 and GJRFs-2, which are mutually orthogonal in L2(0, ∞), are introduced and they are analytical eigensolutions to a new family of singular fractional Sturm-Liouville problems (SFSLPs) of the first and second kinds as non-polynomial functions. We establish some properties and optimal approximation results for these GJRFs-1 and GJRFs-2 in non-uniformly weighted Sobolev spaces involving fractional derivatives, which play important roles in the related spectral methods for a class of fractional differential equations. We develop Jacobi rational-Gauss quadrature type formulae andL2-orthogonal projections based on GJRFs-1 and GJRFs-2. As examples of applications, the two quadrature rules are proposed for Fermi-Dirac and Bose-Einstein integrals. Using various orthogonal properties of GJRFs-1 and GJRFs-2, the Petrov-Galerkin methods are proposed for fractional initial value problems and fractional boundary value problems. Numerical results demonstrate its efficient algorithm, and spectral accuracy for treating the above-mentioned classes of problems. The suggested numerical scheme provides an applicable substitutional to other competitive methods in the recent method-related accuracy.

Key words： Generalized Jacobi rational approximation; Fractional derivatives; Fractional Sturm-Liouville problems; Unbounded domains; Approximation results; Spectral accuracy

Cite this article

Tarek Aboelenen . Efficient and Accurate Spectral Method for Solving Fractional Differential Equations on the Half Line Using Orthogonal Generalized Rational Jacobi Functions[J]. Communications on Applied Mathematics and Computation, 2025 , 7(4) : 1419 -1443 . DOI: 10.1007/s42967-023-00337-y

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