A Predictor–Corrector Compact Difference Scheme for a Nonlinear Fractional Differential Equation

Abstract

In this work, a predictor–corrector compact difference scheme for a nonlinear fractional differential equation is presented. The MacCormack method is provided to deal with nonlinear terms, the Riemann–Liouville (R-L) fractional integral term is treated by means of the second-order convolution quadrature formula, and the Caputo derivative term is discretized by the L1 discrete formula. Through the first and second derivatives of the matrix under the compact difference, we improve the precision of this scheme. Then, the existence and uniqueness are proved, and the numerical experiments are presented.

1. Introduction

This article investigates a predictor–corrector compact difference scheme for a nonlinear fractional differential equation

$$\begin{array}{ccc}\hfill & & {\mathcal{D}}_t^{\alpha }u+u{u}_x-u_{xx}-{\mathcal{I}}^{\beta }{u}_{xx}=f(x,t),0<t\le T,0<x<1,\hfill \end{array}$$

(1)

with the initial and boundary conditions

$$\begin{array}{c}\hfill u(x,0)=u^0\left(x\right),0\le x\le 1,\end{array}$$

(2)

$$\begin{array}{c}\hfill u(0,t)=u(1,t)=0,0\le t\le T,\end{array}$$

(3)

where $f(x,t)$ and $u^0\left(x\right)$ are given smooth functions, the Riemann–Liouville fractional [1] ${\mathcal{I}}^{\beta }\phi \left(t\right)$ is defined as

$$\begin{array}{c}\hfill {\mathcal{I}}^{\beta }\phi \left(t\right):=\frac{1}{\mathsf{\Gamma }\left(\beta \right)}{\int }_0^t{(t-s)}^{\beta -1}\phi \left(s\right)ds,0<\beta <1,t>0,\end{array}$$

(4)

and the Caputo fractional derivative of order $\alpha$$(0<\alpha \le 1)$ is defined as

$$\begin{array}{c}\hfill {\mathcal{D}}_t^{\alpha }u(x,t):=\left\{\begin{array}{c}\frac{1}{\mathsf{\Gamma }(1-\alpha )}{\int }_0^t\frac{\partial u(x,s)}{\partial s}\frac{ds}{{(t-s)}^{\alpha }},0<\alpha <1,\hfill \\ \\ \frac{\partial u(x,t)}{\partial t},\alpha =1.\hfill \end{array}\right.\end{array}$$

(5)

Time-fractional nonlinear differential equations are widely applied and researched in hydrodynamics [2], nonlinear acoustic waves [3,4], shallow water waves [5], etc. In order to obtain a greater understanding of the behavior and solutions for this type of equation, there has been significant research in the field of numerical methods for time-fractional nonlinear differential equations in recent years, such as finite difference methods [6,7,8,9], Galerkin methods [10,11], two-grid method [12,13], and the collocation method [14,15] and iterative algorithm [16,17]. To the authors’ best knowledge, there is limited work in the literature on the study of the generalized fractional nonlinear differential equation.

The application of the predictor–corrector method offers numerical stability and high accuracy, making it suitable for complex fluid dynamics problems. In [18], Diethelm et al. previously discussed the numerical solution of a fractional differential equation with the predictor–corrector method. In [19], Deng analyzed the short memory principle of fractional calculus and the predictor–corrector method, then presented the error analysis and numerical examples. In [20], Li et al. utilized the finite difference method with non-uniform grid to solve a nonlinear fractional differential equation and to establish the predictor–corrector scheme.

The MacCormack is a type of predictor–corrector method that was introduced by MacCormack in 1969 [21]. This method is particularly well-suited for handling nonlinear terms. In [22], Ngondiep et al. employed the MacCormack predictor–corrector method to solve the two-dimensional nonlinear coupled Burgers’ equation. This method achieves second-order convergence in both temporal and spatial directions, effectively reducing computational costs. In [23], Payri et al. used the MacCormack method for nonlinear noise calculations and found that this method has less dispersion and is faster compared to the Lax–Wendroff method.

The focus of the main work is to propose a predictor–corrector fourth-order compact difference scheme for solving generalized fractional nonlinear differential Equations (1)–(3). The MacCormack method is introduced to deal with nonlinear terms; the Riemann–Liouville fractional integral term is treated by using the second-order convolution quadrature formula; and the Caputo derivative term is handled by the L1 discrete formula. By combining compact difference methods under a matrix formula for the first derivative and the second derivative of space, we obtain the fully compact predictor–corrector scheme and improve the precision of this scheme. The existence and uniqueness are proved, and the numerical experiments are presented.

The following is the composition of this article. In Section 2, we introduce the construction of the proposed scheme. In Section 3, we provide the matrix form of the predictor–corrector compact difference scheme. In Section 4, the existence and uniqueness of the matrix form of the scheme are proved. In Section 5, numerical results are given. In Section 6, the work in this paper is summarized.

2. Construction of the Proposed Scheme

Let J and N be positive integers; $h:=\frac{L}{J}$ is defined as the space-step size, and $x_j:=jh(0\le j\le J)$. Additionally, we take $k:=\frac{T}{N}$ as the time-step size, and $t_n:=nk(0\le n\le N)$. The grid functions in our study are defined as follows

$$\begin{array}{c}\hfill U_j^n:=u(x_j,t_n),f_j^n:=f(x_j,t_n),0\le j\le J,0\le n\le N.\end{array}$$

Considering the grid function $U=\{U_j^n|0\le j\le J,0\le n\le N\}$, various notations are defined as follows

$$\begin{array}{cc}\hfill & {\delta }_x{U}_j^n=\frac{1}{h}(U_j^n-U_{j-1}^n),\hfill \\ \hfill & {\delta }_x^2{U}_j^n=\frac{1}{h^2}(U_{j+1}^n-2U_j^n+U_{j-1}^n),{\mathbf{U}}^n={(u_j^n)}_{(J-1)\times 1},\hfill \\ \hfill & {({\mathbf{U}}^n)}^{{}^{″}}={({(u_j^n)}^{{}^{″}})}_{(j-1)\times 1},{({\mathbf{U}}^n)}^{{}^{\prime }}={({(u_j^n)}^{{}^{\prime }})}_{(j-1)\times 1},\hfill \\ \hfill & {(u_j^n)}^{{}^{″}}=\frac{{\partial }^2{u}_j^n}{\partial x^2},{(u_j^n)}^{{}^{\prime }}=\frac{\partial u_j^n}{\partial x},\hfill \\ \hfill & F^n={(f(x_j,t_n))}_{(j-1)\times 1},G^n={(g_j^n)}_{(j-1)\times 1}.\hfill \end{array}$$

(6)

Further, we can obtain the formula for calculating the second derivative of the matrix under compact difference

$$\begin{array}{c}\hfill M_2{\mathbf{U}}^{{}^{″}}=(A_2\mathbf{U}+H_2),\end{array}$$

(7)

and the formula for calculating the first derivative of the matrix under compact difference

$$\begin{array}{c}\hfill A_1{\mathbf{U}}^{{}^{\prime }}=(M_1\mathbf{U}+H_1),\end{array}$$

(8)

where

$$A_1={\left(\begin{array}{ccccc}4& 1& 0& \cdots & 0\\ 1& 4& 1& \cdots & ⋮\\ 0& \ddots & \ddots & \ddots & 0\\ ⋮& & 1& 4& 1\\ 0& \cdots & 0& 1& 4\end{array}\right)}_{(J-1)\times (J-1)},H_1=\frac{11}{12h}\times {\left(\begin{array}{c}-u_0\\ 0\\ ⋮\\ 0\\ u_J\end{array}\right)}_{(J-1)\times 1},$$

$$M_1=\frac{3}{h}\times {\left(\begin{array}{ccccccc}-\frac{4}{3}& 2& -\frac{4}{9}& \frac{1}{12}& 0& \cdots & 0\\ -1& 0& 1& 0& 0& & ⋮\\ 0& -1& 0& 1& 0& \ddots & ⋮\\ 0& 0& -1& 0& 1& \ddots & 0\\ ⋮& ⋮& \ddots & \ddots & \ddots & \ddots & 0\\ ⋮& ⋮& 0& 0& -1& 0& 1\\ 0& 0& 0& -\frac{1}{12}& \frac{4}{9}& -2& \frac{4}{3}\end{array}\right)}_{(J-1)\times (J-1)}.$$

$$A_2=\frac{12}{h^2}\times {\left(\begin{array}{ccccc}-2& 1& 0& \cdots & 0\\ 1& -2& 1& \cdots & ⋮\\ 0& \ddots & \ddots & \ddots & 0\\ ⋮& & 1& -2& 1\\ 0& \cdots & 0& 1& -2\end{array}\right)}_{(J-1)\times (J-1)},H_2=\frac{12}{h^2}\times {\left(\begin{array}{c}{u}_0\\ 0\\ ⋮\\ 0\\ u_J\end{array}\right)}_{(J-1)\times 1},$$

$$M_2={\left(\begin{array}{ccccccc}14& -5& 4& -1& 0& \cdots & 0\\ 1& 10& 1& 0& 0& & ⋮\\ 0& 1& 10& 1& 0& \ddots & ⋮\\ 0& 0& 1& 10& 1& \ddots & 0\\ ⋮& ⋮& \ddots & \ddots & \ddots & \ddots & 0\\ ⋮& ⋮& 0& 0& 1& 10& 1\\ 0& 0& 0& -1& 4& -5& 14\end{array}\right)}_{(J-1)\times (J-1)}.$$

We will use the second-order backward differentiation formula (BDF) quadrature rule [24,25] to approximate the R-L fractional integral ${\mathcal{I}}^{\beta }\phi \left(t_n\right)$

$$\begin{array}{c}\hfill {\mathcal{I}}^{\beta }\phi \left(t_n\right)\approx q_n^{\beta }\left(\phi \right)=k^{\beta }\sum _{p=1}^n{\omega }_{n-p}^{\beta }{\phi }^p+k^{\beta }{\tilde{\omega }}_n^{\beta }{\phi }^0;\end{array}$$

(9)

by the generating power series, we can obtain the quadrature weights ${\omega }_p^{\beta }$

$$\begin{array}{ccc}\hfill & & a_0^{\beta }=1,a_p^{\beta }=\frac{\beta (\beta +1)\cdots (\beta +p-1)}{p!},\hfill \\ \hfill & & {\omega }_p^{\beta }={(\frac{2}{3})}^{\beta }\sum _{i=0}^p\frac{a_{p-i}^{\beta }{a}_i^{\beta }}{3^i},p\ge 0.\hfill \end{array}$$

(10)

In order to move the integral to the second order in time, the correction quadrature weight ${\tilde{\omega }}_n^{\beta }$ is given to make the quadrature formula valid for the polynomial $\phi =1$; then, we have

$$\begin{array}{c}\hfill k^{\beta }\sum _{p=1}^n{\omega }_p^{\beta }+k^{\beta }{\tilde{\omega }}_n^{\beta }=\frac{1}{\mathsf{\Gamma }\left(\beta \right)}{\int }_0^{t_n}{(t_n-s)}^{\beta -1}ds=\frac{{\left(t_n\right)}^{\beta }}{\mathsf{\Gamma }(\beta +1)}.\end{array}$$

(11)

In more detail, the computational formula for finding the quadrature weights is given by

$$\begin{array}{c}\hfill {\tilde{\omega }}_n^{\beta }=\frac{n^{\beta }}{\mathsf{\Gamma }(\beta +1)}-\sum _{p=1}^n{\omega }_{n-p}^{\beta }.\end{array}$$

(12)

We obtain the quadrature error of $E\left(\phi \right)\left(t_n\right)={\mathcal{I}}^{\beta }\phi \left(t_n\right)-q_n^{\beta }\left(\phi \right)$ in the next lemma.

Lemma 1

([26]). Assuming u is a continuously differentiable function over the interval $t\in (0,T]$, and ${\partial }_t^{2}u$ is both continuous and integrable in the range of $t\in (0,T)$, the quadrature error under the second-order BDF can be expressed as

$$\begin{array}{ccc}\hfill |{\mathcal{I}}^{\beta }u\left(t_n\right)-q_n^{\beta }\left(u\right)|& \le & c_1{k}^2{t}_n^{\beta -1}|{\partial }_{t}u\left(0\right)|+c_2{k}^{\beta +1}{\int }_{t_{n-1}}^{t_n}|{\partial }_t^{2}u\left(s\right)|ds\hfill \\ & +& c_3{k}^2{\int }_0^{t_{n-1}}{(t_n-s)}^{\beta -1}|{\partial }_t^{2}u\left(s\right)|ds,\hfill \end{array}$$

where $q_n^{\beta }\left(u\right)$ is defined in (9), and the values of the positive constants $c_1,c_2,c_3$ are provided in [26].

We next use the L1 discrete formula [27,28] to discretize the Caputo fractional derivative ${\mathcal{D}}_t^{\alpha }u(·,t_n)$. We define the discrete operate $\mathfrak{D}$ for ${\mathcal{D}}_t^{\alpha }$

$$\begin{array}{ccc}\hfill \mathfrak{D}u(·,t_n):& =& \frac{k^{-1}}{\mathsf{\Gamma }(1-\alpha )}[b_{0}u(·,t_n)+\sum _{i=1}^{n-1}(b_{n-i}-b_{n-i-1})u(·,t_i)\hfill \\ \hfill & & -b_{n-1}u(·,t_0)],\hfill \end{array}$$

(13)

where $0<\alpha <1$, $b_i={\int }_{t_i}^{t_{i+1}}{z}^{-\alpha }dz=\frac{k^{1-\alpha }}{1-\alpha }[{(i+1)}^{1-\alpha }-i^{1-\alpha }]$, $i\ge 0$.

Lemma 2

([28]). Suppose that $u(·,t)\in C_t^2[0,1]$. We can obtain

$$\begin{array}{c}\hfill |{\mathcal{D}}_t^{\alpha }u(·,t_n)-\mathfrak{D}u(·,t_n)|=|\frac{1}{\mathsf{\Gamma }(1-\alpha )}{\int }_0^{t_n}\frac{{\partial }_{t}u(·,t_n)}{{(t_n-s)}^{\alpha }}ds-\mathfrak{D}u(·,t_n)|\le c_6{k}^{2-\alpha },\end{array}$$

where $c_6=\frac{1}{\mathsf{\Gamma }(2-\alpha )}[\frac{1-\alpha }{12}+\frac{2^{2-\alpha }}{2-\alpha }-(1+2^{-\alpha })]{max}_{0\le t\le t_n}\left|u_{tt}(·,t)\right|$, $0<\alpha <1$.

Consider the problem (1) on the node $(x_j,t_n)$; we have

$$\begin{array}{ccc}& & {\mathcal{D}}_t^{\alpha }u(x_j,t_n)+u(x_j,t_n){\partial }_{x}u(x_j,t_n)-{\partial }_x^{2}u(x_j,t_n)\hfill \\ & & -{\mathcal{I}}^{\beta }{\partial }_x^{2}u(x_j,t_n)=f(x_j,t_n),1\le j\le J-1,1\le n\le N.\hfill \end{array}$$

For nonlinear terms, it can be written in the following form

$$\begin{array}{ccc}\hfill & & {\mathcal{D}}_t^{\alpha }u(x_j,t_n)+{(\frac{\partial g}{\partial x})}_j^n-{\partial }_x^{2}u(x_j,t_n)-{\mathcal{I}}^{\beta }{\partial }_x^{2}u(x_j,t_n)\hfill \\ \hfill & & =f(x_j,t_n),1\le j\le J-1,1\le n\le N,\hfill \end{array}$$

(14)

where $g\left(u\right)=\frac{u^2}{2}$.

Next, for (14), we can discretize ${\mathcal{D}}_t^{\alpha }u(x_j,t_n)$ by using (13), discretize ${\partial }_x^{2}u(x_j,t_n)$ by the standard finite difference method, handle the integral term by the second-order convolution quadrature Formula (9), and use the MacCormack method for nonlinear terms. Specifically, backward difference is used in the predictor step, and forward difference is used in the corrector step. Then, we can obtain the predictor–corrector finite difference scheme by combining the initial boundary value conditions. The predictor–corrector process is as follows.

When $n=1$, the predictor step is:

$$\begin{array}{ccc}\hfill & & \frac{k^{-\alpha }}{\mathsf{\Gamma }(2-\alpha )}({\overline{u}}_j^1-u_j^0)+\frac{1}{h}(g_j^0-g_{j-1}^0)=\frac{k^{\alpha }{\omega }_0^{\alpha }}{h^2}({\overline{u}}_{j-1}^1-2{\overline{u}}_j^1\hfill \\ \hfill & & +{\overline{u}}_{j+1}^1)+\frac{k^{\alpha }{\tilde{\omega }}_1^{\alpha }}{h^2}(u_{j-1}^0-2u_j^0+u_{j+1}^0)+\frac{1}{h^2}({\overline{u}}_{j-1}^1-2{\overline{u}}_j^1+{\overline{u}}_{j+1}^1)+f_j^1.\hfill \end{array}$$

(15)

The corrector step is:

$$\begin{array}{ccc}\hfill & & \frac{k^{-\alpha }}{\mathsf{\Gamma }(2-\alpha )}({\tilde{u}}_j^1-{\overline{u}}_j^1)+\frac{1}{h}({\overline{g}}_{j+1}^1-{\overline{g}}_j^1)=\frac{k^{\alpha }{\omega }_0^{\alpha }}{h^2}({\tilde{u}}_{j-1}^1-2{\tilde{u}}_j^1\hfill \\ \hfill & & +{\tilde{u}}_{j+1}^1)+\frac{k^{\alpha }{\tilde{\omega }}_1^{\alpha }}{h^2}({\overline{u}}_{j-1}^1-2{\overline{u}}_j^1+{\overline{u}}_{j+1}^1)+\frac{1}{h^2}({\tilde{u}}_{j-1}^1-2{\tilde{u}}_j^1+{\tilde{u}}_{j+1}^1)+f_j^1.\hfill \end{array}$$

(16)

The temporal average is:

$$\begin{array}{c}\hfill u_j^1=\frac{1}{2}(u_j^0+{\tilde{u}}_j^1),1\le j\le J-1.\end{array}$$

(17)

When $n\ge 2$, the predictor step is:

$$\begin{array}{ccc}\hfill & & \frac{k^{-1}}{\mathsf{\Gamma }(1-\alpha )}[b_0{\overline{u}}_j^n-\sum _{i=1}^{n-1}(b_{n-i-1}-b_{n-i})u_j^i-b_{n-1}{u}_j^0]\hfill \\ \hfill & & +\frac{g_j^{n-1}-g_{j-1}^{n-1}}{h}=\frac{k^{\alpha }{\omega }_0^{\alpha }}{h^2}({\overline{u}}_{j-1}^n-2{\overline{u}}_j^n+{\overline{u}}_{j+1}^n)+\frac{k^{\alpha }}{h^2}\sum _{p=1}^{n-1}{\omega }_{n-p}^{\alpha }(u_{j-1}^p-2u_j^p\hfill \\ \hfill & & +u_{j+1}^p)+k^{\alpha }{\tilde{\omega }}_n^{\alpha }\frac{(u_{j-1}^0-2u_j^0+u_{j+1}^0)}{h^2}+\frac{1}{h^2}({\overline{u}}_{j-1}^n-2{\overline{u}}_j^n+{\overline{u}}_{j+1}^n)+f_j^n.\hfill \end{array}$$

(18)

The corrector step is:

$$\begin{array}{ccc}\hfill & & \frac{k^{-1}}{\mathsf{\Gamma }(1-\alpha )}[b_0{\tilde{u}}_j^n-\sum _{i=1}^{n-1}(b_{n-i-1}-b_{n-i}){\overline{u}}_j^{i+1}-b_{n-1}{\overline{u}}_j^1]\hfill \\ \hfill & & +\frac{{\overline{g}}_{j+1}^n-{\overline{g}}_j^n}{h}=\frac{k^{\alpha }{\omega }_0^{\alpha }}{h^2}({\tilde{u}}_{j-1}^n-2{\tilde{u}}_j^n+{\tilde{u}}_{j+1}^n)+\frac{k^{\alpha }}{h^2}\sum _{p=1}^{n-1}{\omega }_{n-p}^{\alpha }({\overline{u}}_{j-1}^{1+p}-2{\overline{u}}_j^{1+p}\hfill \\ \hfill & & +{\overline{u}}_{j+1}^{1+p})+k^{\alpha }{\tilde{\omega }}_n^{\alpha }\frac{({\overline{u}}_{j-1}^1-2{\overline{u}}_j^1+{\overline{u}}_{j+1}^1)}{h^2}+\frac{1}{h^2}({\tilde{u}}_{j-1}^n-2{\tilde{u}}_j^n+{\tilde{u}}_{j+1}^n)+f_j^n.\hfill \end{array}$$

(19)

The temporal average is:

$$\begin{array}{c}\hfill u_j^n=\frac{1}{2}(u_j^{n-1}+{\tilde{u}}_j^n),1\le j\le J-1,2\le n\le N.\end{array}$$

(20)

The initial and boundary value conditions are discretized by

$$\begin{array}{ccc}\hfill & & u_0^n=u_J^n=0,1\le n\le N.\hfill \\ & & u_j^0=u^0\left(x_j\right),1\le j\le J.\hfill \end{array}$$

(21)

3. Matrix Form of Predictor–Corrector Compact Difference Scheme

In the section, we write the predictor–corrector difference scheme as the matrix form and obtain the predictor–corrector compact difference scheme.

Letting $R=k^{2\alpha }{\omega }_0^{\alpha }$, the above two Equations (15) and (16) can be converted into the following form

$$\begin{array}{ccc}\hfill & & {\overline{\mathbf{U}}}^1-R\mathsf{\Gamma }(2-\alpha ){({\overline{\mathbf{U}}}^1)}^{{}^{″}}-k^{\alpha }\mathsf{\Gamma }(2-\alpha ){({\overline{\mathbf{U}}}^1)}^{{}^{″}}={\mathbf{U}}^0-k^{\alpha }\mathsf{\Gamma }(2-\alpha ){(G^0)}^{{}^{\prime }}\hfill \\ & & +k^{2\alpha }\mathsf{\Gamma }(2-\alpha ){\tilde{\omega }}_1^{\alpha }{({\mathbf{U}}^0)}^{{}^{″}}+k^{\alpha }\mathsf{\Gamma }(2-\alpha )F^1,\hfill \end{array}$$

(22)

and

$$\begin{array}{ccc}\hfill & & {\tilde{\mathbf{U}}}^1-R\mathsf{\Gamma }(2-\alpha ){({\tilde{\mathbf{U}}}^1)}^{{}^{″}}-k^{\alpha }\mathsf{\Gamma }(2-\alpha ){({\tilde{\mathbf{U}}}^1)}^{{}^{″}}={\overline{\mathbf{U}}}^1-k^{\alpha }\mathsf{\Gamma }(2-\alpha ){({\overline{G}}^1)}^{{}^{\prime }}\hfill \\ & & +k^{2\alpha }\mathsf{\Gamma }(2-\alpha ){\tilde{\omega }}_1^{\alpha }{({\overline{\mathbf{U}}}^1)}^{{}^{″}}+k^{\alpha }\mathsf{\Gamma }(2-\alpha )F^1.\hfill \end{array}$$

(23)

Instituting ${\mathbf{U}}^{{}^{″}}=M_2^{-1}(A_2\mathbf{U}+H_2)$ into the above equations, then, for the predictor step, we have

$$\begin{array}{ccc}\hfill & & (M_2-R\mathsf{\Gamma }(2-\alpha )A_2-k^{\alpha }\mathsf{\Gamma }(2-\alpha )A_2){\overline{\mathbf{U}}}^1=R\mathsf{\Gamma }(2-\alpha )H_2\hfill \\ & & +k^{\alpha }\mathsf{\Gamma }(2-\alpha )H_2+M_2[{\mathbf{U}}^0-k^{\alpha }\mathsf{\Gamma }(2-\alpha ){(G^0)}^{{}^{\prime }}+k^{2\alpha }\mathsf{\Gamma }(2-\alpha ){\tilde{\omega }}_1^{\alpha }{({\mathbf{U}}^0)}^{{}^{″}}\hfill \\ & & +k^{\alpha }\mathsf{\Gamma }(2-\alpha )F^1].\hfill \end{array}$$

(24)

For the corrector step, we have

$$\begin{array}{ccc}\hfill & & (M_2-R\mathsf{\Gamma }(2-\alpha )A_2-k^{\alpha }\mathsf{\Gamma }(2-\alpha )A_2){\tilde{\mathbf{U}}}^1=R\mathsf{\Gamma }(2-\alpha )H_2\hfill \\ & & +k^{\alpha }\mathsf{\Gamma }(2-\alpha )H_2+M_2[{\overline{\mathbf{U}}}^1-k^{\alpha }\mathsf{\Gamma }(2-\alpha ){({\overline{G}}^1)}^{{}^{\prime }}+k^{2\alpha }\mathsf{\Gamma }(2-\alpha ){\tilde{\omega }}_1^{\alpha }{({\overline{\mathbf{U}}}^1)}^{{}^{″}}\hfill \\ & & +k^{\alpha }\mathsf{\Gamma }(2-\alpha )F^1].\hfill \end{array}$$

(25)

The above two equations can be used to find the correction value ${\tilde{\mathbf{U}}}^1$ of the first layer. Then, the numerical solution of the first layer is

$$\begin{array}{c}\hfill {\mathbf{U}}^1=\frac{1}{2}({\mathbf{U}}^0+{\tilde{\mathbf{U}}}^1),1\le j\le J-1.\end{array}$$

(26)

The Equations (24)–(26) combined with the initial boundary value condition can be solved for the first layer.

Let $R_1=k^{\alpha }{\omega }_0^{\alpha }$, and insert ${\mathbf{U}}^{{}^{″}}=M_2^{-1}(A_2\mathbf{U}+H_2)$ into the left of (18) and (19). Similarly, we can obtain for the predictor step:

$$\begin{array}{ccc}\hfill & & [M_2\frac{k^{-1}}{\mathsf{\Gamma }(1-\alpha )}{b}_0-R_1{A}_2-A_2]{\overline{\mathbf{U}}}^n=R_1{H}_2+H_2\hfill \\ \hfill & & +M_2[\frac{k^{-1}}{\mathsf{\Gamma }(1-\alpha )}\sum _{i=1}^{n-1}(b_{n-i-1}-b_{n-i}){\mathbf{U}}^i+\frac{k^{-1}}{\mathsf{\Gamma }(1-\alpha )}{b}_{n-1}{\mathbf{U}}^0-{(G^{n-1})}^{{}^{\prime }}\hfill \\ \hfill & & +k^{\alpha }\sum _{p=1}^{n-1}{\omega }_{n-p}^{\alpha }{({\mathbf{U}}^p)}^{{}^{″}}+k^{\alpha }{\tilde{\omega }}_n^{\alpha }{({\mathbf{U}}^0)}^{{}^{″}}+F^n].\hfill \end{array}$$

(27)

The corrector step is:

$$\begin{array}{ccc}\hfill & & [M_2\frac{k^{-1}}{\mathsf{\Gamma }(1-\alpha )}{b}_0-R_1{A}_2-A_2]{\tilde{\mathbf{U}}}^n=R_1{H}_2+H_2\hfill \\ \hfill & & +M_2[\frac{k^{-1}}{\mathsf{\Gamma }(1-\alpha )}\sum _{i=1}^{n-1}(b_{n-i-1}-b_{n-i}){\overline{\mathbf{U}}}^{i+1}+\frac{k^{-1}}{\mathsf{\Gamma }(1-\alpha )}{b}_{n-1}{\overline{\mathbf{U}}}^1-{({\overline{G}}^n)}^{{}^{\prime }}\hfill \\ \hfill & & +k^{\alpha }\sum _{p=1}^{n-1}{\omega }_{n-p}^{\alpha }{({\overline{\mathbf{U}}}^{p+1})}^{{}^{″}}+k^{\alpha }{\tilde{\omega }}_n^{\alpha }{({\overline{\mathbf{U}}}^1)}^{{}^{″}}+F^n].\hfill \end{array}$$

(28)

Combining the above two equations, the correction value ${\tilde{\mathbf{U}}}^n$ of the layer n can be obtained, and then the numerical solution of the layer n is

$$\begin{array}{c}\hfill {\mathbf{U}}^n=\frac{1}{2}({\mathbf{U}}^{n-1}+{\tilde{\mathbf{U}}}^n),1\le j\le J-1,2\le n\le N.\end{array}$$

(29)

The Equations (27)–(29), combined with the initial boundary value condition, can be solved for layer 2 to layer n.

4. Uniqueness and Existence

Theorem 1.

The difference scheme (24)–(29) is uniquely solvable.

Proof. 

When ${\mathbf{U}}^0$ is given and $n=1$, the coefficient matrix of the predictor step (24) is

$$\mathfrak{M}={\left(\begin{array}{ccccccc}\epsilon & \varrho & 4& -1& 0& \cdots & 0\\ \varsigma & \vartheta & \varsigma & 0& 0& & ⋮\\ 0& \varsigma & \vartheta & \varsigma & 0& \ddots & ⋮\\ 0& 0& \varsigma & \vartheta & \varsigma & \ddots & 0\\ ⋮& ⋮& \ddots & \ddots & \ddots & \ddots & 0\\ ⋮& ⋮& 0& 0& \varsigma & \vartheta & \varsigma \\ 0& 0& 0& -1& 4& \varrho & \epsilon \end{array}\right)}_{(J-1)\times (J-1)},$$

where

$$\epsilon =14+\frac{24R\mathsf{\Gamma }(2-\alpha )}{h^2}+\frac{24k^{\alpha }\mathsf{\Gamma }(2-\alpha )}{h^2},$$

$$\varrho =-5-\frac{12R\mathsf{\Gamma }(2-\alpha )}{h^2}-\frac{12k^{\alpha }\mathsf{\Gamma }(2-\alpha )}{h^2},$$

$$\varsigma =1-\frac{12R\mathsf{\Gamma }(2-\alpha )}{h^2}-\frac{12k^{\alpha }\mathsf{\Gamma }(2-\alpha )}{h^2},$$

$$\vartheta =10+\frac{24R\mathsf{\Gamma }(2-\alpha )}{h^2}+\frac{24k^{\alpha }\mathsf{\Gamma }(2-\alpha )}{h^2}.$$

Obviously, it is a diagonally dominant matrix. Therefore, ${\overline{\mathbf{U}}}^1$ is uniquely solvable. Substituting ${\overline{\mathbf{U}}}^1$ into the corrector step (25), the coefficient matrix of (25) is the same as (24), so it is also diagonally dominant. Accordingly, the ${\tilde{\mathbf{U}}}^1$ can be uniquely solved by (25), and finally the ${\mathbf{U}}^1$ can be uniquely solved by (25).

When the solution ${\mathbf{U}}^{n-1}$ of the $n-1$$(2\le n\le N)$ layer is known, for the solution ${\mathbf{U}}^n$ of the n layer, the coefficient matrix of the predictor step (27) is

$$\mathfrak{n}={\left(\begin{array}{ccccccc}\varpi & \aleph & 4\kappa & -\kappa & 0& \cdots & 0\\ \chi & \iota & \chi & 0& 0& & ⋮\\ 0& \chi & \iota & \chi & 0& \ddots & ⋮\\ 0& 0& \chi & \iota & \chi & \ddots & 0\\ ⋮& ⋮& \ddots & \ddots & \ddots & \ddots & 0\\ ⋮& ⋮& 0& 0& \chi & \iota & \chi \\ 0& 0& 0& -\kappa & 4\kappa & \aleph & \varpi \end{array}\right)}_{(J-1)\times (J-1)},$$

where

$$\kappa =\frac{k^{-1}}{\mathsf{\Gamma }(1-\alpha )}{b}_0,$$

$$\varpi =14\kappa +2R_1+2,$$

$$\aleph =-5\kappa -R_1-1,$$

$$\chi =\kappa -R_1-1,$$

$$\iota =10\kappa +2R_1+2.$$

Clearly, it is also a diagonally dominant matrix, so ${\overline{\mathbf{U}}}^n$ can be uniquely solved. Similarly, the coefficient matrix of (28) is the same as that of (27), so ${\tilde{\mathbf{U}}}^n$ is uniquely solvable. Combined with (29), ${\mathbf{U}}^n$ is uniquely solvable.

Therefore, from what has been discussed above, the existence and uniqueness are proven. □

5. Numerical Results

Now we will use the scheme (24)–(29) to solve problems (1)–(3). Denote

$$\begin{array}{ccc}& & E(h,k)=\underset{1\le j\le M-1,1\le n\le N}{max}|U_j^n-u_j^n|,\hfill \\ & & rat{e}^x=lo{g}_{\frac{b}{a}}(\frac{E(\frac{b}{a}h,k)}{E(h,k)}),rat{e}^t=lo{g}_2(\frac{E(h,2k)}{E(h,k)}),\hfill \end{array}$$

where $\frac{b}{a}$ is the space-step ratio.

Example 1.

For problems (1)–(3), we consider the initial condition $u^0\left(x\right)=2sin\pi x$, and the source term is

$$\begin{array}{ccc}& & f(x,t)=\frac{-tsin2\pi x}{\mathsf{\Gamma }(2)}+(2sin\pi x-\frac{t^{\alpha +1}sin2\pi x}{\mathsf{\Gamma }(\alpha +2)})(2\pi cos\pi x\hfill \\ & & -\frac{2\pi t^{\alpha +1}cos2\pi x}{\mathsf{\Gamma }(\alpha +2)})+2{\pi }^{2}sin\pi x-\frac{4{\pi }^2{t}^{\alpha +1}sin2\pi x}{\mathsf{\Gamma }(\alpha +2)}+\frac{2{\pi }^2{t}^{\beta }sin\pi x}{\mathsf{\Gamma }(\beta +1)}\hfill \\ & & -\frac{4{\pi }^{2}sin2\pi x{t}^{\beta +\alpha +1}}{\mathsf{\Gamma }(\beta +\alpha +2)}.\hfill \end{array}$$

The reference solution to this problem is

$$\begin{array}{c}\hfill u(x,t)=2sin\pi x-\frac{t^{\alpha +1}}{\mathsf{\Gamma }(\alpha +2)}sin2\pi x,0<\alpha <1.\end{array}$$

The number of steps in fixed time is $N=200,000$, and the maximum error and corresponding spatial convergence order of the predictor–corrector compact difference scheme at $\alpha =0.2,0.5,0.75$ and $\beta =0.8,0.75,0.5$ are given in

Table 1. The maximum error and convergence order in space for N = 200,000 for Example 1.

$\mathsf{\alpha }$	$\mathsf{\beta }$	M	E	${\mathit{rate}}^{\mathit{x}}$
0.5	0.75	8	$0.0118$	−
		16	1.5163× 10${}^{-4}$	6.287
		24	1.9402 × 10${}^{-5}$	5.071
		32	4.6915 × 10${}^{-6}$	4.935
0.75	0.5	8	$0.0094$	−
		16	1.0868 × 10${}^{-4}$	6.440
		24	1.4319 × 10${}^{-5}$	4.999
		32	3.6362 × 10${}^{-6}$	4.764
0.2	0.8	8	$0.0139$	−
		16	1.7837 × 10${}^{-4}$	6.284
		24	1.8787 × 10${}^{-5}$	5.551
		32	4.4745 × 10${}^{-5}$	4.987

. The numerical results in the following table show that the spatial convergence order fluctuates around order 4. The number of fixed space steps $M=32$.

Table 2. The maximum error and convergence order in time for $M=32$ for Example 1.

$\mathsf{\alpha }$	$\mathsf{\beta }$	N	E	${\mathit{rate}}^{\mathit{t}}$
0.5	0.75	$2^{11}$	4.8620 × 10${}^{-4}$	−
		$2^{12}$	2.1650 × 10${}^{-4}$	1.167
		$2^{13}$	9.6739 × 10${}^{-5}$	1.162
		$2^{14}$	4.2779 × 10${}^{-5}$	1.177
0.75	0.5	$2^{11}$	3.2513 × 10${}^{-4}$	−
		$2^{12}$	1.5519 × 10${}^{-4}$	1.067
		$2^{13}$	7.4252 × 10${}^{-5}$	1.064
		$2^{14}$	3.4999 × 10${}^{-5}$	1.085
0.2	0.8	$2^{11}$	$0.0010$	−
		$2^{12}$	5.0799 × 10${}^{-4}$	1.031
		$2^{13}$	2.4559 × 10${}^{-4}$	1.049
		$2^{14}$	1.1657 × 10${}^{-4}$	1.075

shows the maximum error and corresponding time convergence order of the predictor–corrector compact difference scheme when $\alpha =0.2,0.5,0.75$ and $\beta =0.8,0.75,0.5$.

Figure 1 provides the comparison curve between the numerical solution and the exact solution when $x_j=7/8$ and $t_n=1$ for $\alpha =0.5,\beta =0.75$, $k=1/8000$, and $h=1/20$. Figure 2 shows the absolute error of the numerical solution and the exact solution with $\alpha =0.5,\beta =0.75$ for $t_n=1$, $x_j=\frac{19}{20}$. Figure 3 shows the three-dimensional surface plot of the numerical solution and the exact solution of the absolute error of the predictor–corrector compact difference method at $\alpha =0.5$, $\beta =0.75$ for $h=\frac{1}{20}$, $k=\frac{1}{8000}$. It can be seen that the absolute errors of the numerical solution and the exact solution are very small, and the numerical solution is consistent with the exact solution. From these figures, it is clear that there is a strong alignment between the numerical solution and the exact solution.

Example 2.

For problem (1)–(3), we consider the initial condition $u^0\left(x\right)=2sin\pi x$, and the source term is

$$\begin{array}{ccc}& & f(x,t)=\frac{-t^{\beta }\mathsf{\Gamma }(\alpha +\beta +1)sin\pi x}{\mathsf{\Gamma }(\beta +1)\mathsf{\Gamma }(\alpha +2)}+{(1-\frac{t^{\alpha +\beta }}{\mathsf{\Gamma }(\alpha +2)})}^2\pi sin\pi xcos\pi x\hfill \\ & & +{\pi }^2(1-\frac{t^{\alpha +\beta }}{\mathsf{\Gamma }(\alpha +2)})sin\pi x+\frac{t^{\beta }{\pi }^{2}sin\pi x}{\mathsf{\Gamma }(\beta +1)}-\frac{{\pi }^2{t}^{\alpha +2\beta }sin\pi x\mathsf{\Gamma }(\alpha +\beta +1)}{\mathsf{\Gamma }(\alpha +2\beta +1)\mathsf{\Gamma }(\alpha +2)}.\hfill \end{array}$$

The reference solution to this problem is

$$\begin{array}{c}\hfill u(x,t)=(1-\frac{t^{\alpha +\beta }}{\mathsf{\Gamma }(\alpha +2)})sin\pi x,0<\alpha ,\beta <1.\end{array}$$

In

Table 3. The maximum error and convergence order in space for N = 100,000 for Example 2.

$\mathsf{\alpha }$	$\mathsf{\beta }$	M	E	${\mathit{rate}}^{\mathit{x}}$
0.25	0.75	6	$0.0026$	−
		8	4.8407 × 10${}^{-4}$	5.869
		12	3.1571 × 10${}^{-5}$	6.733
		16	9.5981 × 10${}^{-6}$	4.139
0.5	0.75	6	$0.0024$	−
		8	4.4901 × 10${}^{-4}$	5.838
		12	3.0650 × 10${}^{-5}$	6.621
		16	7.4286 × 10${}^{-6}$	4.927
0.2	0.8	6	$0.0027$	−
		8	4 × 10${}^{-4}$	6.284
		12	1.8787 × 10${}^{-5}$	5.551
		16	4.4745 × 10${}^{-5}$	4.987

and

Table 4. The maximum error and convergence order in time for $M=32$ for Example 2.

$\mathsf{\alpha }$	$\mathsf{\beta }$	N	E	${\mathit{rate}}^{\mathit{t}}$
0.25	0.75	$2^8$	$0.0051$	−
		$2^9$	$0.0023$	1.124
		$2^{10}$	$0.0011$	1.128
		$2^{11}$	4.9100 × 10${}^{-4}$	1.128
0.5	0.75	$2^8$	$0.0028$	−
		$2^9$	$0.0013$	1.142
		$2^{10}$	5.8217 × 10${}^{-4}$	1.113
		$2^{11}$	2.7434 × 10${}^{-4}$	1.086
0.2	0.8	$2^8$	$0.0060$	−
		$2^9$	$0.0028$	1.091
		$2^{10}$	$0.0013$	1.099
		$2^{11}$	6.1016 × 10${}^{-4}$	1.102

, we present the error and convergence for different $\alpha$ and $\beta$ for Example 2. It can be seen from Table 3 and Table 4 that the numerical results are almost in agreement with the theoretical results. The comparison graphs between the numerical solution and the exact solution in time and space are drawn in Figure 4 for $k=1/8000$, $h=1/20$.

6. Concluding Remarks

In this article, we proposed a predictor–corrector compact finite difference scheme for a nonlinear fractional differential equation. The MacCormack method was employed to handle the nonlinear terms, while the Riemann–Liouville fractional integral term was handled using a second-order convolution quadrature formula, and the fractional Caputo derivative term was discretized using the L1 discrete formula. We also proved the existence and uniqueness of the provided scheme. Numerical examples further verified the effectiveness of the proposed method.