Open AccessNUMERICAL CALCULATION OF NONSTATIONARY FRACTIONAL DIFFERENTIAL EQUATION IN PROBLEMS OF MODELING TOXIC SUBSTANCES DISTRIBUTION IN GROUND WATERS
Author(s) -
Alexandra Alekseyevna Afanasyeva,
T. N. Shvetsova-Shilovskaya,
Dmitriy Ivanov,
D. I. Nazarenko,
Elena Viktorovna Kazarezova
Publication year - 2019
Publication title -
vestnik astrahanskogo gosudarstvennogo tehničeskogo universiteta. seriâ: upravlenie, vyčislitelʹnaâ tehnika i informatika
Language(s) - English
Resource type - Journals
eISSN - 2224-9761
pISSN - 2072-9502
DOI - 10.24143/2072-9502-2019-4-70-80
Subject(s) - mathematics , tridiagonal matrix , differential equation , partial differential equation , crank–nicolson method , matrix (chemical analysis) , numerical analysis , basis (linear algebra) , fractional calculus , finite difference method , mathematical analysis , geometry , physics , eigenvalues and eigenvectors , materials science , quantum mechanics , composite material
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.