In the paper, we show a connection between a regular fractional Sturm-Liouville problem with left and right Caputo derivatives of order in the range (1/2, 1) and a 1D space-time fractional diffusion problem in a bounded domain. Both problems include mixed boundary conditions in a finite space interval. We prove that in the case of vanishing mixed boundary conditions, the Sturm-Liouville problem can be rewritten in terms of Riesz derivatives. Then, we apply earlier results on its eigenvalues and eigenfunctions to construct a weak solution of the 1D fractional diffusion equation with variable diffusivity. Adding an assumption on the summability of the eigenvalues’ inverses series, we formulate a theorem on a strong solution of the 1D fractional diffusion problem.
Optimal control of a fractional diffusion Sturm-Liouville problem on a star graph
