Abstract
In this paper, we propose a numerical method for solving the time fractional Richards’ equation. We first approximate the time fractional derivative of the mentioned equations by a scheme of order O(τ2−α), 0 < a<1; then, we use the finite point method to approximate the spatial derivatives. Before the discrete spatial derivatives, we introduced the basic principles of the finite point method. We solve the one- and two-dimensional versions of these equations using the proposed method. Moreover, the stability properties of the discretized scheme related to time are theoretically analyzed. Numerical results showed the efficiency of the method presented in this paper.
Highlights
The theory of fractional calculus and fractional differential equation has been widely used in many fields, such as mechanics, physics, biomathematics, engineering, automatic control, fractal, and so on
Richards’ equation is a basic model for describing soil water movement. e multiscale heterogeneity of soil makes the nature of water diffusion process not consistent with the preconditions of applying Fick’s law, which often reflects the abnormal diffusion phenomenon of soil water infiltration
We show the results for two examples using the method described in previous sections
Summary
The theory of fractional calculus and fractional differential equation has been widely used in many fields, such as mechanics, physics, biomathematics, engineering, automatic control, fractal, and so on. Richards’ equation is a basic model for describing soil water movement. In response to the above phenomenon, the fractional Richards’ equation is proposed to describe the process of water movement in unsaturated soil [1, 2]. Pachepsky et al [1] used the finite difference method to solve the time fractional Richard’s equation. Chen et al [6] used the finite difference method and Kansa method to discretize the time fractional derivatives and the space fractional derivatives, respectively, in solving the time fractional diffusion equation. Freitas et al [7] proposed the modified fractional integral Richard’s equation to predict the anomalous diffusion process of horizontal infiltration in unsaturated media
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