Numerical method for solving the subdiffusion differential equation with nonlocal boundary conditions

This work was devoted to the construction of a numerical algorithm for solving the initial boundary value problem for the subdiffusion equation with nonlocal boundary conditions. For the case of not strongly regular boundary conditions, the well-known methods cannot be used. We applied an algorithm that consists of reducing the nonlocal problem to a sequential solution of two subproblems with local boundary conditions. The solution to the original problem was summed up from the solutions of the subproblems. To solve the subproblems, we constructed implicit difference schemes on the basis of the L1 formula for approximating the Caputo fractional derivative and central difference for approximating the space derivatives. Stability and convergence of the schemes were established. The Thomas algorithm was used to solve systems of linear algebraic equations. Numerical experiments were conducted to study the constructed algorithm. In terms of accuracy and stability, the algorithm performs well. The results of experiments confirmed that the convergence order of the method coincides with the theoretical one, $O({\tau }^{2-\alpha }+h^2)$.