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

doi:10.1007/s42967-023-00337-y

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (4): 1419-1443.doi: 10.1007/s42967-023-00337-y

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

Tarek Aboelenen^1,2

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

收稿日期:2023-02-14 修回日期:2023-10-02 接受日期:2023-10-04 出版日期:2024-02-02 发布日期:2024-02-02
通讯作者: Tarek Aboelenen,E-mail:t.aboelenen@qu.edu.sa,tarek.aboelenen@aun.edu.eg E-mail:t.aboelenen@qu.edu.sa,tarek.aboelenen@aun.edu.eg
基金资助:
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.

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

Tarek Aboelenen^1,2

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:2023-02-14 Revised:2023-10-02 Accepted:2023-10-04 Online:2024-02-02 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.

摘要/Abstract

摘要： 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.

关键词: Generalized Jacobi rational approximation, Fractional derivatives, Fractional Sturm-Liouville problems, Unbounded domains, Approximation results, Spectral accuracy

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

中图分类号:

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.

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

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

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