Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)
An inverse Sturm-Liouville problem with a fractional derivative
Journal of Computational Physics
Exponentially accurate spectral and spectral element methods for fractional ODEs
Journal of Computational Physics
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We first consider a regular fractional Sturm-Liouville problem of two kinds RFSLP-I and RFSLP-II of order @n@?(0,2). The corresponding fractional differential operators in these problems are both of Riemann-Liouville and Caputo type, of the same fractional order @m=@n/2@?(0,1). We obtain the analytical eigensolutions to RFSLP-I & -II as non-polynomial functions, which we define as Jacobi poly-fractonomials. These eigenfunctions are orthogonal with respect to the weight function associated with RFSLP-I & -II. Subsequently, we extend the fractional operators to a new family of singular fractional Sturm-Liouville problems of two kinds, SFSLP-I and SFSLP-II. We show that the primary regular boundary-value problems RFSLP-I & -II are indeed asymptotic cases for the singular counterparts SFSLP-I & -II. Furthermore, we prove that the eigenvalues of the singular problems are real-valued and the corresponding eigenfunctions are orthogonal. In addition, we obtain the eigen-solutions to SFSLP-I & -II analytically, also as non-polynomial functions, hence completing the whole family of the Jacobi poly-fractonomials. In numerical examples, we employ the new poly-fractonomial bases to demonstrate the exponential convergence of the approximation in agreement with the theoretical results.