Regular Sturm-Liouville Problem with Riemann-Liouville Derivatives of Order in (1,2): Discrete Spectrum, Solutions and Applications

摘要

We study a regular fractional Sturm-Liouville problem formulated using left and right Riemann-Liouville derivatives of order in the range. We prove a theorem describing the eigenvalues and eigen-functions of such a problem considered on the space of functions continuously differentiable in a finite interval and obeying vanishing Dirichlet and fractional Neumann boundary conditions. It appears that the spectrum of eigenvalues is discrete and that the eigenfunctions form a basis in the space of square-integrable functions. We also show applications of the derived eigenfunctions in the theory of partial fractional differential equations.