Abstract
At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.
Highlights
The theory of fractional calculus is widely used in many fields of science for modeling various processes
Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source
There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater
Summary
Особенный интерес к дробным производным проявляют гидрогеологи в связи с проблемой обоснования безопасности хранения опасных химических и радиоактивных отходов в геологических формациях [1, 2]. В данной работе представлены результаты исследований по разработке численного метода и разностной схемы для решения одномерного дифференциального уравнения с дробной производной по времени. В данной статье рассматривается уравнение параболического типа, описывающее процесс диффузии, и один из подходов к построению разностных схем для такого типа уравнений – схема Кранка – Николсона, называемая также схемой с весом 1/2. Наиболее целесообразной является разработка неявной разностной схемы для решения дробного дифференциального уравнения на основе известной схемы Кранка – Николсона. Разработаем неявную разностную схему, являющуюся аналогом схемы Кранка – Николсона, для численного решения дробного дифференциального уравнения. Аналог неявной разностной схемы Кранка – Николсона для численного решения дробного дифференциального уравнения (1) с учетом выражения для дробной производной Капуто по времени (2) можно сформулировать следующим образом:. После преобразований (8)–(10) система разностных уравнений для первоначальной краевой задачи с дробной производной (1) имеет вид blk +1[(clk++11.
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