Advances in Mathematical Physics

Advances in Mathematical Physics / 2022 / Article

Research Article | Open Access

Volume 2022 |Article ID 4846747 | https://doi.org/10.1155/2022/4846747

Yunkun Chen, "Numerical Method to Modify the Fractional-Order Diffusion Equation", Advances in Mathematical Physics, vol. 2022, Article ID 4846747, 10 pages, 2022. https://doi.org/10.1155/2022/4846747

Numerical Method to Modify the Fractional-Order Diffusion Equation

Academic Editor: Soheil Salahshour
Received30 Sep 2021
Accepted28 Jan 2022
Published07 Mar 2022

Abstract

Time or space or time-space fractional-order diffusion equations (FODEs) are widely used to describe anomalous diffusion processes in many physical and biological systems. In recent years, many authors have proposed different numerical methods to solve the modified fractional-order diffusion equations, and some achievements have been obtained. However, to our knowledge of the literature, up to date, all the proposed numerical methods to modify FODE have achieved at most a second-order time accuracy. In this study, we focus mainly on the numerical methods based on numerical integration in order to modify the fractional-order diffusion equation: , , . Accordingly, numerical methods can be built to modify FODE with second-order time accuracy and fourth-order spatial accuracy in . . Our suggested method can improve the time precision with a certain value.

1. Introduction

In recent years, fractional-order diffusion equations (FODEs) have been widely employed to describe anomalous diffusion processes in many physical and biological systems. By acting upon the diffusion factor with two levels of fractional-order time derivatives, models of special diffusion phenomena have been proposed [13].

Some scholars have constructed different numerical methods to solve the modified FODE, and accordingly, some useful achievements have been obtained. For one-dimensional positive FODE, Langlands [4] proposed an interpretation with an infinite series form of the Fox function over an infinite region. Liu et al. [5, 6] discussed numerical methods and analytical techniques to develop a finite element approximation with first-order time accuracy and mth order spatial accuracy, where m is the number of segmented polynomials. Mohebbi et al. [7] applied a fourth-order compact formula for the second-order spatial partial derivatives and the discretization of Riemann-Liouville fractional-order time derivatives to provide a higher-order and absolutely stable format with first-order time accuracy and fourth-order spatial accuracy. Bhrawy [8] studied the compact subdiffusion scheme including second-order time accuracy and fourth-order spatial accuracy. For the multirepair positive FODE, Zhang et al. [9] developed a finite difference/finite element method with (1 + min{α, β}) order time accuracy and mth order spatial accuracy, where m represents the number of segmented polynomials. Mohebbi et al. [10] proposed a fourth-order compact solution method with first-order time accuracy and fourth-order spatial accuracy. Abbaszadeh and Mohebbi [11] discussed a solution obtained by a radial basis function (RBF) meshless method with min{α, β} order time accuracy. Wang and Wang [12] analyzed the tight LOD method and its extrapolation method including 2 min{α, β} order time accuracy and fourth-order spatial accuracy. For two-dimensional variable-order modified FODE, Chen and Liu [13] examined a numerical method with first-order time accuracy and fourth-order spatial accuracy and further developed the numerical method to improve the time accuracy.

According to the literature, the existing numerical methods for the modified FODE can achieve at most second-order time accuracy. However, this study develops a new numerical method with not only second-order time accuracy but also fourth-order spatial accuracy based on the numerical integration of the modified FODE.

1.1. Basic Concepts and Properties

Definition 1. (Grünwald–Letnikov fractional stratification number). Let be a positive real number, and , n is a positive integer, and let the function f (x) be defined on the interval [a, b] asIt is the α-order Grünwald–Letnikov (G-L) fractional-order derivative of function f (x), [z] is the largest integer that does not exceeds z, and

Definition 2 (Riemann-Liouville fractional stratification number). Let be a positive real number, and , n is a positive integer, and let the function be defined on the interval [a, b], respectively. Here,represent the left and right α-order Riemann-Liouville fractional derivatives.

Property 1. Let α be a positive real number, and n − 1 ≤ α < n, n is a positive integer defined in the interval of [a, b], the function has up to n − 1 continuous functions, and is integrable in [a, b]. Then, the Riemann-Liouville fractional derivative is equivalent to the Grünwald–Letnikov (GL) fractional derivative.

1.2. Construction of Numerical Methods

In this research, the following numerical method is developed to modify the initial and boundary values of the fractional diffusion equation (MFDE):where is the fractional large derivative of .

Suppose is the exact solution of problems (1)–(6) and , where

Suppose thatwhere and are the space step and time step, respectively.

We defineand integrating the two sides of equation (5) with respect to t on the interval , we get

Hence, we get

Since , thenhence,where

Similarly, then, we havewhere

Because , the following trapezoidal formula holds:and according to the above analysis, we havewhere

Please note thatwith ; then,

Now, the numerical methods should be taken into account for solving problems (5)–(7):where , and is the approximation of the exact solution .

1.3. Convergence of Numerical Methods

In this section, we discuss the convergence of numerical equations (24)–(26). Subtract (24) from (20) and then obtain the following error equation:where

For , the following grid functions are defined, respectively:then, and can be expanded with the following Fourier series, respectively:where

Let

The following Parseval equation can be derived:

Similarly, the following Parseval equation can also be derived: