Numerical Method to Modify the Fractional-Order Diffusion Equation

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: (1 + (1/12)δx)pj − (1 + (1/12)δ 2 x)p k− 1 j � μα 􏽐 k l�0 λ (l) α δ 2 xp k− l j + μβ 􏽐 k l�0 λ (l) β δ 2 xp k− l j + (τ/2)(1+ (1/12)δx)(f k− 1 j + f k j ), k � 1, 2, . . . , K; j � 1, 2, . . . , j − 1, p 0 j � ωj, j � 0, 1, . . . , J, p k 0 � φ(tk), p k j � ψ(tk), k � 0, 1, . . . , K, f l j � f(xj, tl),ωj � ω(xj). Accordingly, numerical methods can be built to modify FODEwith second-order time accuracy and fourthorder spatial accuracy in (zp(x, t)/zt) � ((z1− α/zt1− α) + B(z1− β/zt1− β))(z2p(x, t)/zx2) + f(x, t), 0< t≤ 1, 0<x< 1, p(x, 0) � 0, 0≤x≤ 1. p(0, t) � t2, p(1, t) � et2, 0≤ t≤ 1. Our suggested method can improve the time precision with a certain value.


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 [1][2][3]. (1) 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 m th 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 m th 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 secondorder 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.

Basic Concepts and Properties
Definition 1 (Grünwald-Letnikov fractional stratification number). Let α be a positive real number, and n − 1 ≤ α < n, n is a positive integer, and let the function f (x) be defined on the interval [a, b] as It is the α-order Grünwald-Letnikov (G-L) fractionalorder derivative of function f (x), [z] is the largest integer that does not exceeds z, and Definition 2 (Riemann-Liouville fractional stratification number). Let α be a positive real number, and n − 1 ≤ α < n, n is a positive integer, and let the function f(x) 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 f(x) has up to n − 1 continuous functions, and f (n) (x) is integrable in [a, b]. en, the Riemann-Liouville fractional derivative is equivalent to the Grünwald-Letnikov (GL) fractional derivative.

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): Suppose p(x, t) ∈ P(Ω), p(x, t) is the exact solution of problems (1)- (6) and Suppose that x j � jh, j � 0, 1, . . . , J, where h � L/j and τ � T/K are the space step and time step, respectively. We define and integrating the two sides of equation (5) with respect to t on the interval [t k− 1 , t k ], we get Hence, we get 2 Advances in Mathematical Physics Since p(x, t) ∈ P(Ω), then hence, Similarly, then, we have where Because (z 2 f(x, t)/zt 2 ) ∈ C(Ω), the following trapezoidal formula holds: and according to the above analysis, we have where Please note that 4 Advances in Mathematical Physics with kτ ≤ T; then, Now, the numerical methods should be taken into account for solving problems (5)- (7): k � 1, 2, . . . , K; j � 1, 2, . . . , j − 1, where f l j � f(x j, t l ), ω j � ω(x j ), and p l j is the approximation of the exact solution p(x j, t l ).

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 k � 0, 1, . . . , K, the following grid functions are defined, respectively: then, E k (x) and R k (x) can be expanded with the following Fourier series, respectively: where Let e following Parseval equation can be derived: Similarly, the following Parseval equation can also be derived: Notice that So far, we obtain
From (23), therefore, constant C 1 exists and makes

Again, paying attention to the first equation in (37), we have
By the convergence of the series at the left end of (37), there is a positive constant C 2 , so that Theorem 1. Assuming that p(x, t) ∈ P(Ω) is the exact solution of problems (5)- (7), (z 2 f(x, t)/zt 2 ) ∈ C(Ω), and time and space steps satisfy
According to Lemmas 1-3, we have through (40), 6 Advances in Mathematical Physics Conclusion (48) is proven by induction. From conclusions (36), (37), (45), and (48), we obtain the following one: where T. We know that because there are many high-precision numerical integration formulas (45) and (46), it is not difficult to continue to build up a high time precision numerical method for solving the modified fractional diffusion equation (5). However, the problem is that the qualitative analysis of convergence and stability can be very difficult.

Stability of the Numerical Methods
e stability of numerical format was discussed in (24)-(26); consider the following difference equation: where ρ k j � p k j − p k j , p k j is the approximation of p k j . For k � 0, 1, . . ., K, respectively, define the following grid function: en, ρ k (x) can be expanded by Fourier series: where Let Similar to the derivation of (36), the following Parseval equation can also be obtained: , k � 0, 1, . . . , K. (58)

Conclusion
As given in Table 1, our theoretical analysis results are strongly supported by the numerical experiments. Figure 1 shows that the numerical methods (24)  As shown in Figures 1 and 2, the numerical solutions of problems (73)-(75) and the exact solutions of problems (73)-(75) obtained by numerical methods (24)-(26) have a good approximation effect. is finding revealed that our theoretical analysis results are reliable.
In conclusion, one-dimensional fractional diffusion equations were studied in the present study, with the purpose that the construction techniques of numerical methods for solving one-dimensional fractional diffusion equations and the corresponding numerical analysis can be extended to multidimensional fractional diffusion equations.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e author declares that there are no conflicts of interest.