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

A new set of generalized Jacobi rational functions of the first and second kinds, GJRFs-1 and GJRFs-2, which are mutually orthogonal in \(L^{2}(0,\infty)\) $L^2(0,\infty )$ , 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 \(L^{2}\) $L^2$ -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.