ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia)

Stability and Convergence of Monotone Difference Schemes Approximating Boundary Value Problems for an Integro-differential Equation with a Fractional Time Derivative and the Bessel Operator

Author(s):

Junior Researcher, Department of Computational Methods,
Institute of Applied Mathematics and Automation, KBSC RAS

zarabaeva@yandex.ru

Murat Khamidbievich Beshtokov

Leading Researcher, Department of Computational Methods,
Institute of Applied Mathematics and Automation, KBSC RAS

beshtokov-murat@yandex.ru

Abstract:

Boundary value problems are studied for an integro-differential equation with a time fractional derivative and a Bessel operator. To solve the problems under consideration, a priori estimates are obtained in a differential interpretation, which implies the uniqueness and stability of the solution with respect to the initial data and the right-hand side. For the numerical solution of boundary value problems, monotone difference schemes with directed differences are constructed, analogs of a priori estimates are proved for them, and error estimates are given under the assumption that the solutions of the equations are sufficiently smooth. The obtained a priori estimates in the difference form imply the uniqueness and stability of the solution with respect to the initial data and the right-hand side, and also, by virtue of the linearity of the difference problems, the convergence with the second order in the parameters of the grid. An algorithm for the approximate solution of a boundary value problem with a condition of the third kind is proposed, and numerical calculations are carried out for a test example illustrating the theoretical results obtained in this work concerning the convergence and the order of approximation of the difference scheme.

Keywords

• a priori estimate
• boundary value problems
• fractional-order differential equation
• Gerasimov-Caputo fractional derivative
• integro-differential equation
• monotone schemes

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