On a difference scheme for solution of the Dirichlet problem for diffusion equation with a fractional Caputo derivative in the multidimensional case in a domain with an arbitrary boundary

Z. V. Beshtokova   ; M. Kh. Beshtokov   ; M. H. Shkhanukov-Lafishev

Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 3, pp. 37-54

Résumé

In this paper, we study the Dirichlet problem for the diffusion equation with a fractional Caputo derivative in the multidimensional case in a domain with an arbitrary boundary. Instead of the original equation, we consider the diffusion equation with a fractional Caputo derivative with a small parameter. A locally one-dimensional difference scheme of A. A. Samarsky, the main essence of which is to reduce the transition from layer to layer to the sequential solution of a number of one-dimensional problems in each of the coordinate directions. Moreover, each of the auxiliary problems may not approximate the original problem, but in the aggregate and in special norms such an approximation takes place. These methods have been called splitting methods. Using the maximum principle, we obtain an a priori estimate in the uniform metric norm. The stability of the locally one-dimensional difference scheme and the uniform convergence of the approximate solution of the proposed difference scheme to the solution of the original differential problem for any $0<\alpha<1$ are proved. An analysis is made of the choice of optimal values of $\varepsilon$, at which the rate of uniform convergence of the approximate solution of the considered difference scheme to the solution of the original differential problem will be determined in the best way.

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