# Error estimates of the numerical method for the filtration problem with Caputo-Fabrizio fractional derivatives

## Authors

• Dossan Baigereyev S. Amanzholov East Kazakhstan University
• Nurlana Alimbekova S. Amanzholov East Kazakhstan University
• Nikolay Oskorbin Altai State University

## Keywords:

ақырлы элементтер әдісі, Капуто-Фабрицио бөлшек ретті туындысы, жинақтылық, фильтрация есебі, жарықшалы кеуекті орта

## Abstract

This paper investigates a model of fluid flow in a fractured porous medium under the assumption of a uniform distribution of fractures throughout the volume. This model is based on the use of a fractional differential analogue of Darcy's law, as well as on the assumption that the properties of rock and fluid depend on pressure and its fractional derivative. Unlike previous studies, this study uses a fractional derivative in the Caputo-Fabrizio sense with a non-singular kernel. In this paper, we propose a numerical method for solving this initial boundary value problem and theoretically investigate the order of its convergence. The formulation of a fully discrete scheme is based on application of the finite difference approximation for integer and fractional time
derivatives, and the Galerkin method in the spatial variable. A second-order formula is used to approximate both integer derivative and the fractional derivative in the sense of Caputo-Fabrizio. A priori estimates are obtained for both semi-discrete and fully discrete schemes, which imply their second-order convergence in time and space variables. A number of computational experiments were carried out on the example of a model problem to validate the accuracy of the scheme. The results of the numerical tests fully confirm the outcome of the theoretical analysis.

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