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

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

L^2(0,\infty )

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, 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 and

L^2

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-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.

Keywords

Generalized Jacobi rational approximation / Fractional derivatives / Fractional Sturm-Liouville problems / Unbounded domains / Approximation results / Spectral accuracy / Mathematical Sciences / Pure Mathematics

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