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.