Fractional Laguerre spectral methods and their applications to fractional differential equations on unbounded domain

In this article, we first introduce a singular fractional Sturm-Liouville problem (SFSLP) on unbounded domain. The associated fractional differential operator is both Weyl and Caputo type. The properties of spectral data for fractional operator on unbounded domain have been investigated. Moreover, it has been shown that the eigenvalues of the singular problem are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions of SFSLP are obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived. We construct an efficient generalized Laguerre fractional-polynomials-Petrov–Galerkin methods for a class of fractional initial value problems and fractional boundary value problems. As a numerical example, we examine space fractional advection–diffusion equation. Our theoretical results are confirmed by associated numerical results

Spectral Theory and Numerical Approximation for Singular Fractional Sturm-Liouville eigen-Problems on Unbounded Domain

In this article, we first introduce a singular fractional Sturm-Liouville eigen-problems (SFSLP) on unbounded domain. The associated fractional differential operators in these problems are both Weyl and Caputo type . The properties of spectral data for fractional operators on unbounded domain has been investigated. Moreover, it has been shown that the eigenvalues of the singular problems are real-valued and the corresponding eigenfunctions are orthogonal. The analytical eigensolutions to SFSLP is obtained and defined as generalized Laguerre fractional-polynomials. The optimal approximation of such generalized Laguerre fractional-polynomials in suitably weighted Sobolev spaces involving fractional derivatives has been derived, which is also available for approximated fractional-polynomials growing fast at infinity. The obtained results demonstrate that the error analysis beneficial of fractional spectral methods for fractional differential equations on unbounded domains. As a numerical example, we employ the new fractional-polynomials bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results.Comment: This paper has been withdrawn by the author due to a crucial sign error in equation