Higher-order approximation difference scheme for the generalized Aller equation of fractional order

In this paper, the first boundary value problem for the Aller equation of fractional time order with generalized memory functions was considered. For the numerical solution of the problem, two difference schemes of an increased order of approximation are constructed. In the case of variable coefficients, a second-order difference scheme of approximation is proposed, both in time and in space. A compact difference scheme of the fourth order of approximation in space and the second order in time for the generalized Aller equation with constant coefficients is proposed. A priori estimates for solutions of the mentioned difference schemes are obtained by the method of energy inequalities. Their unconditional stability and convergence are proved. It is shown that the convergence rate coincides with the order of approximation error in the case of a sufficiently smooth solution of the original problem. On the basis of the proposed algorithms, numerical calculations of test problems were carried out, confirming the obtained theoretical results. All calculations were performed using the Julia v1.5.1 programming language.