Boundary value problems for Sobolev type equations of fractional order with memory effect

Boundary value problems are studied for a one-dimensional Sobolev type integro-differential equation with boundary conditions of the first and third kind with two fractional differentiation operators $\alpha$ and $\beta$ of different orders. Difference schemes of the order of approximation $O(h^2+\tau^2)$ for $\alpha=\beta$ and $O(h^2+\tau^{2-\max\{\alpha,\beta\}})$ are constructed for $\alpha\neq\beta$. Using the method of energy inequalities, a priori estimates are obtained in the differential and difference interpretations, from which the existence, uniqueness, stability, and convergence of the solution of the difference problem to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme follow. Numerical experiments were carried out to illustrate the results obtained in the paper.