Fractional Sturm–Liouville eigen-problems: Theory and numerical approximation

We first consider a regular fractional Sturm–Liouville problem of two kinds RFSLP-I and RFSLP-II of order ν ∈ ( 0 , 2 ) . The corresponding fractional differential operators in these problems are both of Riemann–Liouville and Caputo type, of the same fractional order μ = ν / 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.