This article is cited in 2 papers

Differential Equations and Mathematical Physics

Boundary value problems for Sobolev type equations of fractional order with memory effect

M. Kh. Beshtokov

Institute of Applied Mathematics and Automation of Kabardin-Balkar Scientific Centre of RAS, Nal'chik, 360000, Russian Federation

Abstract: Boundary value problems are studied for a one-dimensional Sobolev type integro-differential equation with boundary conditions of the first and third kind with two fractional differentiation operators $\alpha$ and $\beta$ of different orders. Difference schemes of the order of approximation $O(h^2+\tau^2)$ for $\alpha=\beta$ and $O(h^2+\tau^{2-\max\{\alpha,\beta\}})$ are constructed for $\alpha\neq\beta$. Using the method of energy inequalities, a priori estimates are obtained in the differential and difference interpretations, from which the existence, uniqueness, stability, and convergence of the solution of the difference problem to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme follow. Numerical experiments were carried out to illustrate the results obtained in the paper.

Keywords: Sobolev type equation, fractional derivative, memory effect, difference schemes, a priori estimate, stability and convergence.

UDC: 519.642.2

MSC: 65L05, 65N12, 65R20

Received: July 15, 2022
Revised: November 19, 2022
Accepted: December 16, 2022
First online: December 29, 2022

DOI: 10.14498/vsgtu1942

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