Study of a Numerical Method for Solving a Boundary Value Problem for a Differential Equation with a Fractional Time Derivative

N.B Alimbekova,D.R Baigereyev,M.N Madiyarov

doi:10.14258/izvasu(2020)4-10

Abstract

Recently, there has been an increased interest in the problem of numerical implementation of multiphase filtration models due to its enormous economic importance in the oil industry, hydrology, and nuclear waste management. In contrast to the classical models of filtration, filtration models in highly porous fractured formations with the fractal geometry of wells are not fully understood. The solution to this problem reduces to solving a system of differential equations with fractional derivatives. In the paper, a finite-difference scheme is constructed for solving the initial-boundary value problem for the convection-diffusion equation with a fractional time derivative in the sense of Caputo-Fabrizio. A priori estimates are obtained for solving a difference problem under the assumption that there is a solution to the problem in the class of sufficiently smooth functions that prove the uniqueness of the solution and the stability of the difference scheme. The convergence of the solution of the difference problem to the solution of the original differential problem with the second order in time and space variables is shown. The results of computational experiments confirming the reliability of theoretical analysis are presented.

Highlights

В настоящее время замечается повышенный интерес к проблеме численной реализации моделей многофазной фильтрации в связи с ее огромной экономической значимостью в нефтедобывающей промышленности, гидрологии и управлении ядерных отходов
In contrast to the classical models of filtration, filtration models in highly porous fractured formations with the fractal geometry of wells are not fully understood. The solution to this problem reduces to solving a system of differential equations with fractional derivatives
A finite-difference scheme is constructed for solving the initial-boundary value problem for the convection-diffusion equation with a fractional time derivative in the sense of CaputoFabrizio

Summary

МАТЕМАТИКА И МЕХАНИКА

Исследование численного метода решения краевой задачи.д..ля дифференциального уравнения с дробной производной по времени*. Построена конечно-разностная схема для решения начально-краевой задачи для уравнения конвекциидиффузии с производной дробного порядка по времени в смысле Капуто-Фабрицио. In contrast to the classical models of filtration, filtration models in highly porous fractured formations with the fractal geometry of wells are not fully understood The solution to this problem reduces to solving a system of differential equations with fractional derivatives. При выполнении условий (5) для решения разностной задачи (6)-(8) справедливо неравенство. При выполнении условий (5) существует τ0 такое, что при τ ≤ τ0 для решения разностной задачи (6)-(8) справедлива оценка yn+1 2 ≤ μ7 y0 2 + max φm 2 , 0≤m≤n из которой следует единственность и устойчивость разностной схемы (6)-(8) по начальным данным и правой части.

Величина погрешности определялась по формуле

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