This article is cited in 2 papers

Difference schemes for partial differential equations of fractional order

A. K. Bazzaev^ab, I. D. Tsopanov<sup>b

a</sup> Khetagurov North-Ossetia State University, Vatutina str., 44-46, 362025, Vladikavkaz, Russia
^b Vladikavkaz Administration Institute, Borodinskaya str., 14, 362025, Vladikavkaz, Russia

Abstract: Nowadays, fractional differential equations arise while describing physical systems with such properties as power nonlocality, long-term memory and fractal property. The order of the fractional derivative is determined by the dimension of the fractal. Fractional mathematical calculus in the theory of fractals and physical systems with memory and non-locality becomes as important as classical analysis in continuum mechanics.
In this paper we consider higher order difference schemes of approximation for differential equations with fractional-order derivatives with respect to both spatial and time variables. Using the maximum principle, we obtain apriori estimates and prove the stability and the uniform convergence of difference schemes.

Keywords: initial-boundary value problem, fractional differential equations, Caputo fractional derivative, stability, slow diffusion equation, difference scheme, maximum principle, stability, uniform convergence, apriori estimate, heat capacity concentrated at the boundary.

UDC: 519.633

MSC: 65M12

Received: 31.05.2018

 English version: Ufa Mathematical Journal, 2019, 11:2, 19–33

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