Initial-boundary value problems to the time-space nonlocal diffusion equation

This article investigates a time-fractional space-nonlocal diffusion equation in a bounded domain.The fractional operators are defined rigorously, using the Caputo fractional derivative of order  and the Riemann-Liouville fractional integral of order  , where 0 <  <  1 .The solution is expressed as a series involving the two-parameter Mittag-Leffler function and orthonormal eigenfunctions of the Sturm-Liouville operator.The convergence of the series is investigated, and conditions for the solution to belong to a specific function space are established.The uniqueness of the solution is demonstrated and the continuity of the solution in the specified domain is confirmed through the uniform convergence of the series.