2D space-time fractional diffusion on bounded domain — Application of the fractional Sturm-Liouville theory

In the paper, we construct a weak solution to a 2D space-time fractional diffusion problem in a bounded domain, provided the fractional orders of Riesz derivatives are in the range (1, 2). The spatial differential operator includes a non-symmetric combination of Riesz derivatives and variable diffusivities. In the construction, we apply eigenfunctions of the fractional Sturm-Liouville problem subjected to the mixed boundary conditions. In the main theorem we describe explicitly the unique, real-valued continuous solution to the problem.