NUMERICAL METHODS AND THE BASIS FOR THEIR APPLICATION

Numerical solution of integro-differential equations of fractional moisture transfer with the Bessel operator

M. KH. Beshtokov

Institute of applied mathematics and automation, Kabardino-Balkarian scientific center of RAS, 89a Shortanova st., Nalchik, 360000, Russia

Abstract: The paper considers integro-differential equations of fractional order moisture transfer with the Bessel operator. The studied equations contain the Bessel operator, two Gerasimov – Caputo fractional differentiation operators with different orders $\alpha$ and $\beta$. Two types of integro-differential equations are considered: in the first case, the equation contains a non-local source, i.e. the integral of the unknown function over the integration variable $x$, and in the second case, the integral over the time variable $\tau$, denoting the memory effect. Similar problems arise in the study of processes with prehistory. To solve differential problems for different ratios of $\alpha$ and $\beta$, a priori estimates in differential form are obtained, from which the uniqueness and stability of the solution with respect to the right-hand side and initial data follow. For the approximate solution of the problems posed, difference schemes are constructed with the order of approximation $O(h^2+\tau^2)$ for $\alpha=\beta$ and $O(h^2+\tau^{2-max\{\alpha,\beta\}})$ for $\alpha\neq\beta$. The study of the uniqueness, stability and convergence of the solution is carried out using the method of energy inequalities. A priori estimates for solutions of difference problems are obtained for different ratios of $\alpha$ and $\beta$, from which the uniqueness and stability follow, as well as the convergence of the solution of the difference scheme to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme.

Keywords: moisture transfer equation, integro-differential equation, difference schemes, Bessel operator, a priori estimate, stability, convergence

UDC: 519.642

Received: 21.08.2022
Revised: 26.06.2023
Accepted: 18.01.2024

DOI: 10.20537/2076-7633-2024-16-2-353-373