Grid Method for Solving Local and Nonlocal Boundary Value Problems for a Loaded Moisture Transfer Equation with Two Fractional Differentiation Operators

Abstract

This paper is devoted to the study of local and nonlocal boundary value problems for a loaded moisture transfer equation with two fractional Gerasimov-Caputo derivatives of different orders $\alpha$, $\beta$. Using the method of energy inequalities for various relations between $\alpha$ and $\beta$, a priori estimates in differential and difference interpretations are obtained for solving the problems under consideration, which implies the uniqueness and stability of the solution with respect to the initial data and the right-hand side, as well as the convergence of the solution of the difference problem to the solution of the differential problem with the rate $O(h^2+\tau ^2)$ for $\alpha =\beta$ and $O(h^2+\tau ^{2-\max \{\alpha ,\beta \}})$ for $\alpha \ne \beta$.