Nonlocal boundary value problems for Sobolev-type fractional equations and grid methods for solving them

We consider nonlocal boundary value problems for a Sobolev-type equation with variable coefficients with fractional Gerasimov–Caputo derivative. The main result of the article consists in proving a priori estimates for solutions to nonlocal boundary value problems both in differential and difference form obtained under the assumption of the existence of a solution $u(x,t)$ in a class of sufficiently smooth functions. These inequalities imply the uniqueness and stability of a solution with respect to the initial data and right-hand side and also the convergence of the solution to the difference problem to the solution to the differential problem.