Abstract
Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.
Highlights
Ordinary and partial fractional differential equations (FDEs) have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics and engineering [1] [2]
Several numerical methods to solve FDEs have been given such as variational iteration method [5], homotopy perturbation method [3] [6], Adomian decomposition method [7] [8], homotopy analysis method [9], collocation method [10] [11] and finite difference method [12]-[17]
These random walks extended the predictive capability of models built on the stochastic process of Brownian motion, which is the basis for the classical advectiondispersion equation (ADE)
Summary
Ordinary and partial fractional differential equations (FDEs) have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, viscoelasticity, biology, physics and engineering [1] [2]. Similar to integer-order differentiation, Caputo fractional derivative operator is a linear operation. Non-Fickian, dispersion has been an active area of research in the physics community since the introduction of continuous time random walks (CTRW) by Montroll and Weiss [1965] These random walks extended the predictive capability of models built on the stochastic process of Brownian motion, which is the basis for the classical advectiondispersion equation (ADE). When a fractional Fick’s law replaces the classical Fick’s law in an Eulerian evaluation of solute transport in a porous medium, the result is a fractional ADE. It describes the spread of solute mass over large distances via a convolutional fractional derivative.
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