An Eighth-Order Numerical Method for Spatial Variable-Coefficient Time-Fractional Convection–Diffusion–Reaction Equations

Abstract

In this paper, we propose a high-order compact difference scheme for a class of time-fractional convection–diffusion–reaction equations (CDREs) with variable coefficients. Using the Lagrange polynomial interpolation formula for the time-fractional derivative and a compact finite difference approximation for the spatial derivative, we establish an unconditionally stable compact difference method. The stability and convergence properties of the method are rigorously analyzed using the Fourier method. The convergence order of our discrete scheme is $\mathcal{O}({\tau }^{4-\alpha }+h^8)$, where $\tau$ and h represent the time step size and space step size, respectively. This work contributes to providing a better understanding of the dependability of the method by thoroughly examining convergence and conducting an error analysis. Numerical examples demonstrate the applicability, accuracy, and efficiency of the suggested technique, supplemented by comparisons with previous research.

1. Introduction

In the realm of contemporary scientific research and engineering applications, mathematical models are indispensable, with partial differential equations serving as a cornerstone. These equations are capable of effectively capturing the essential characteristics of dynamic behaviors in various systems, ranging from natural phenomena to engineering optimization. However, as the complexity of research subjects continues to grow, traditional integer-order partial differential equations are increasingly insufficient. This limitation has driven researchers to explore new mathematical frameworks to meet these challenges.

Fractional calculus has emerged as a promising solution to address these limitations. Fractional differential equations, by incorporating fractional-order derivatives, can naturally describe systems with memory effects, non-local characteristics, and complex dynamics. This makes them a more flexible tool for modeling. Among the various fractional models, fractional CDREs have garnered significant attention due to their wide applicability across multiple fields. These equations are capable of describing the transport of substances in complex media and effectively characterizing reaction kinetics. This makes them highly valuable in areas such as environmental science [1,2], materials science [3,4], and biomedical engineering [5,6].

Nevertheless, solving fractional CDREs presents formidable challenges. The non-local nature of fractional derivatives, combined with potential nonlinear terms and variable coefficients in the equations, renders traditional numerical methods largely inadequate for efficiently solving these equations. Among the numerical methods for constant-coefficient cases, one can refer to the adaptive moving mesh method [7], finite element method [8,9,10], finite volume method [11], and finite difference method [12,13]. In contrast, the more practically significant variable-coefficient cases have received relatively little attention. The presence of variable coefficients complicates the physical properties of the equations and poses greater difficulties for the design and analysis of numerical methods. In recent years, scholars have made significant strides in addressing these challenges. In 2021, Li and Wang [14] developed numerical methods for time-fractional CDREs. They utilized finite differences for the Caputo fractional derivative and the local discontinuous Galerkin method for the spatial derivative. In 2022, Ngondiep [15] introduced an advanced two-level fourth-order numerical scheme designed for time-fractional CDREs with variable coefficients. This scheme demonstrated unconditional stability and achieved a convergence rate of $\mathcal{O}({\tau }^{2-{\displaystyle \frac{\alpha }{2}}}+h^4)$. In 2023, Roul and Rohil [16] introduced a numerical method for multiterm time-fractional CDREs. It employs a graded mesh for the Liouville–Caputo fractional derivative and a compact finite difference method for spatial derivatives. Additionally, Hosseini et al. [17] proposed a meshless Generalized Finite Difference Method for nonlinear fractional CDREs. This approach leverages Taylor series and Moving Least Squares for spatial derivatives, ensuring reliability across complex domains. In 2024, Priyadarshana and Mohapatra [18] devised a numerical algorithm for 2D time-delayed parabolic CDREs. It utilizes an operator-splitting method and an upwind scheme on a Shishkin mesh. The Thomas algorithm is applied for computation efficiency, achieving first-order accuracy. Furthermore, Kumar et al. [19] presented a weak Galerkin finite-element method for singularly perturbed CDREs. This approach ensures higher-order convergence rates and is applicable to various polygonal mesh types, achieving $\mathcal{O}\left(h^k\right)$ in the triple-bar norm and $\mathcal{O}\left(h^{k+1}\right)$ in the $L^2$ norm, where k represents the highest degree of the polynomials employed. In 2025, Liu et al. [20] proposed a high-order compact finite difference method for CDREs with variable exponents and coefficients. They used a space–time transformation and interpolation quadrature rule to handle the complexities. The aforementioned research results provide reliable references for numerical methods of fractional differential equations, but the convergence rates in both the time and space directions need further improvement. Inspired by the above research, this paper builds upon the studies of previous scholars and enhances the convergence rates of the algorithm in both the time and space directions, with convergence of the $4-\alpha$ order in time and the eighth order in space.

In the present work, we consider more general time-fractional CDREs with variable coefficients in order to better model complex physical phenomena [21].

$$\left\{\begin{array}{c}{}_0{}^{C}D_t^{\alpha }v(x,t)=d{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}(x,t)-p_1\left(x\right){\displaystyle \frac{\partial v}{\partial x}}(x,t)+p_2\left(x\right)v(x,t)+f(x,t),\hfill \\ v(0,t)={\phi }_1\left(t\right),v(L,t)={\phi }_2\left(t\right),\hfill \\ v(x,0)={\phi }_3\left(x\right),\hfill \end{array}\right.$$

(1)

where $(x,t)\in (0,L)\times (0,T]$, d is the constant diffusion coefficient, $p_1$ and $p_2$ are variable coefficients dependent on x, and $f(x,t)$ represents external influences. The boundary conditions are ${\phi }_1\left(t\right)$ at $x=0$ and ${\phi }_2\left(t\right)$ at $x=L$, and the initial condition is ${\phi }_3\left(x\right)$. All functions are assumed smooth enough to ensure the validity of the analysis. The Caputo fractional derivative ${}_0{}^{C}D_t^{\alpha }$ is defined as

$$\begin{array}{c}\hfill {}_0{}^{C}D_t^{\alpha }u\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{u}^{\prime }\left(s\right){(t-s)}^{-\alpha }ds,0<\alpha <1.\end{array}$$

To address the complexities of solving variable-coefficient time-fractional CDREs, this paper presents a novel numerical approach. This approach leverages the Lagrange polynomial interpolation formula for the time-fractional derivative and a compact finite difference approximation for the spatial derivative. This method not only achieves unconditional stability but also attains convergence rates of the $4-\alpha$ order in time and the eighth order in space, as rigorously analyzed through the Fourier method. By effectively addressing the challenges posed by variable coefficients, the proposed scheme significantly enhances numerical accuracy while maintaining computational efficiency. Three numerical experiments further validate the theoretical analysis, demonstrating superior performance compared to existing methods. This work thus offers a robust and efficient solution strategy for time-fractional CDREs, providing valuable support for modeling and analyzing complex systems in various fields.

This paper is organized as follows: Section 2 introduces the necessary mathematical notations and preliminary lemmas that form the foundation for developing our high-order compact finite difference scheme; Section 3 is dedicated to analyzing the stability and convergence properties of the proposed methods; Section 4 presents three numerical experiments that illustrate the practical application and effectiveness of our approach; and we conclude this paper with a summary of our findings and potential directions for future research in Section 5.

2. Construction of Full Discrete Scheme

To facilitate the analysis of Equation (1), we apply a series of transformations. These transformations are designed to address the challenges associated with variable coefficients and to simplify the equation by converting it into a more tractable form. Initially, we assume that the coefficient $p_1\left(x\right)$ is differentiable on the interval $[0,L]$ and define the following notations:

$$\begin{array}{cc}\hfill u(x,t)=k\left(x\right)v(x,t),& g(x,t)=k\left(x\right)f(x,t),\hfill \\ \hfill k\left(x\right)=exp\left(-{\displaystyle \frac{1}{2d}}{\int }_0^x{p}_1\left(s\right)ds\right),& q\left(x\right)=p_2\left(x\right)+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d{p}_1}{dx}}-{\displaystyle \frac{p_1^2}{2d}}\right).\hfill \end{array}$$

By addressing these complexities, Equation (1) is transformed into a more tractable form:

$$\left\{\begin{array}{cc}{}_0{}^{C}D_t^{\alpha }u(x,t)=d{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}(x,t)+q\left(x\right)u(x,t)+g(x,t),\hfill & x\in (0,L),t\in (0,T],\hfill \\ u(0,t)={\varphi }_1\left(t\right),u(L,t)={\varphi }_2\left(t\right),\hfill & t\in [0,T],\hfill \\ u(x,0)={\varphi }_3\left(x\right),\hfill & x\in [0,L],\hfill \end{array}\right.$$

(2)

where ${\varphi }_1\left(t\right)$, ${\varphi }_2\left(t\right)$, and ${\varphi }_3\left(x\right)$ are assumed to be sufficiently smooth. Clearly, $v(x,t)$ is a solution of Equation (1) if and only if $u(x,t)$ satisfies Equation (2). Consequently, the remainder of our study is dedicated to Equation (2).

To construct a high-precision compact finite difference scheme for Equation (2), we define M and N as the numbers of spatial and temporal subdivisions, respectively, and $h={\displaystyle \frac{L}{M}}$ and $\tau ={\displaystyle \frac{T}{N}}$ as the spatial and temporal step sizes. Let $\left[M\right]=\{1,2,\dots ,M\}$. We define $u_j^n=u(x_j,t_n)$ and $g_j^n=g(x_j,t_n)$, where $x_j$ and $t_n$ are the spatial and temporal grid points. The difference operators are defined as follows:

$$\begin{array}{c}\hfill {\delta }_x^2{u}_j=u_{j+1}-2u_j+u_{j-1},\mathcal{L}={\displaystyle \frac{1}{h^2}}{\left(1+{\displaystyle \frac{1}{560}}{\delta }_x^6\right)}^{-1}{\delta }_x^2\left(1-{\displaystyle \frac{1}{12}}{\delta }_x^2+{\displaystyle \frac{1}{90}}{\delta }_x^4\right).\end{array}$$

Here, we introduce several lemmas that are employed in the subsequent construction of the finite difference scheme.

Lemma 1

([22]). For the difference operator $\mathcal{L}$, it holds that $u_{xx}(x_j,t_n)=\mathcal{L}{u}_j^n+\mathcal{O}\left(h^8\right)\mathit{.}$

Lemma 2

([23]). The discretized form of the Caputo derivative on a uniform mesh is given by

$$\begin{array}{cc}\hfill {}_0{}^{C}D_t^{\alpha }{u}_j^n& ={\displaystyle \frac{1}{\Gamma (1-\alpha )}}[{\int }_{t_0}^{t_1}{(t_n-s)}^{-\alpha }{u}^{\prime }\left(s\right)ds+{\int }_{t_1}^{t_2}{(t_n-s)}^{-\alpha }{u}^{\prime }\left(s\right)ds\hfill \\ \hfill & +\sum _{k=3}^n{\int }_{t_{k-1}}^{t_k}{(t_n-s)}^{-\alpha }{u}^{\prime }\left(s\right)ds]\hfill \\ \hfill & ={\displaystyle \frac{1}{\Gamma (2-\alpha )}}\sum _{k=0}^n{\lambda }_k{u}_j^{n-k}+\mathcal{O}\left({\tau }^{4-\alpha }\right)\mathit{.}\hfill \end{array}$$

Here, the coefficients ${\lambda }_k$ are consistent with those reported in Reference [23].

Lemma 3

([23]). Let ${\lambda }_k$ be defined as ${\tau }^{-\alpha }{\psi }_k$. The coefficients ${\psi }_k$ exhibit the following properties:

(i) 

For $n\ge 3$ and any α in the interval $(0,1)$:

(i-1) 

${\psi }_0$ is calculated as ${\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{2-\alpha }}+{\displaystyle \frac{1}{(2-\alpha )(3-\alpha )}}$, and it falls within the range $\left(1,{\displaystyle \frac{11}{6}}\right)$.

(i-2) 

${\psi }_2$ is positive.

(i-3) 

${\psi }_k$ is negative for $k\ge 1$ and $k\ne 2$.

(ii) 

For any order $\alpha \in (0,1)$ and for all n, the sum of coefficients equals zero, i.e., ${\sum }_{k=0}^n{\psi }_k=0$.

Now, using Lemmas 1 and 2, Equation (2) can be rewritten as

$${\displaystyle \frac{1}{\Gamma (2-\alpha )}}\sum _{k=0}^n{\lambda }_k{u}_j^{n-k}=d\mathcal{L}{u}_j^n+q_j{u}_j^n+g_j^n+\mathcal{O}({\tau }^{4-\alpha }+h^8).$$

(3)

By denoting $U_j^n$ as the numerical estimate of $u_j^n$ and omitting the truncation errors $\mathcal{O}({\tau }^{4-\alpha }+h^8)$ in Equation (3), we obtain

$${\displaystyle \frac{1}{\Gamma (2-\alpha )}}\sum _{k=0}^n{\lambda }_k{U}_j^{n-k}=d\mathcal{L}{U}_j^n+q_j{U}_j^n+g_j^n.$$

(4)

By multiplying both sides of Equation (4) by $1+{\displaystyle \frac{1}{560}}{\delta }_x^6$ and introducing the parameters $\mu ={\displaystyle \frac{1}{\Gamma (2-\alpha )}}$ and $\theta ={\displaystyle \frac{d}{h^2}}$, we obtain the final discrete formulation

$$\begin{array}{cc}\hfill & \left({\displaystyle \frac{\mu {\lambda }_0}{560}}-{\displaystyle \frac{q_j}{560}}-{\displaystyle \frac{\theta }{90}}\right)U_{j+3}^n+\left(-{\displaystyle \frac{3\mu {\lambda }_0}{280}}+{\displaystyle \frac{3q_j}{280}}+{\displaystyle \frac{3\theta }{20}}\right)U_{j+2}^n+\left({\displaystyle \frac{3\mu {\lambda }_0}{112}}-{\displaystyle \frac{3q_j}{112}}-{\displaystyle \frac{3\theta }{2}}\right)U_{j+1}^n\hfill \\ \hfill & +\left({\displaystyle \frac{27\mu {\lambda }_0}{28}}-{\displaystyle \frac{27q_j}{28}}+{\displaystyle \frac{49\theta }{18}}\right)U_j^n+\left({\displaystyle \frac{3\mu {\lambda }_0}{112}}-{\displaystyle \frac{3q_j}{112}}-{\displaystyle \frac{3\theta }{2}}\right)U_{j-1}^n\hfill \\ \hfill & +\left(-{\displaystyle \frac{3\mu {\lambda }_0}{280}}+{\displaystyle \frac{3q_j}{280}}+{\displaystyle \frac{3\theta }{20}}\right)U_{j-2}^n+\left({\displaystyle \frac{\mu {\lambda }_0}{560}}-{\displaystyle \frac{q_j}{560}}-{\displaystyle \frac{\theta }{90}}\right)U_{j-3}^n\hfill \\ \hfill & =+\left({\displaystyle \frac{1}{560}}{g}_{j+3}^n-{\displaystyle \frac{3}{280}}{g}_{j+2}^n+{\displaystyle \frac{3}{112}}{g}_{j+1}^n+{\displaystyle \frac{27}{28}}{g}_j^n+{\displaystyle \frac{3}{112}}{g}_{j-1}^n-{\displaystyle \frac{3}{280}}{g}_{j-2}^n+{\displaystyle \frac{1}{560}}{g}_{j-3}^n\right)\hfill \\ \hfill & -\mu \sum _{k=1}^n{\lambda }_k({\displaystyle \frac{1}{560}}{U}_{j+3}^{n-k}-{\displaystyle \frac{3}{280}}{U}_{j+2}^{n-k}+{\displaystyle \frac{3}{112}}{U}_{j+1}^{n-k}+{\displaystyle \frac{27}{28}}{U}_j^{n-k}+{\displaystyle \frac{3}{112}}{U}_{j-1}^{n-k}\hfill \\ \hfill & -{\displaystyle \frac{3}{280}}{U}_{j-2}^{n-k}+{\displaystyle \frac{1}{560}}{U}_{j-3}^{n-k}),\hfill \end{array}$$

(5)

with the initial and boundary conditions

$$\begin{array}{c}\hfill U_j^0={\tilde{\varphi }}_3\left(x_j\right),\mathrm{and}{U}_0^n={\tilde{\varphi }}_1\left(t_n\right)={\tilde{\varphi }}_1^n\mathrm{and}{U}_M^n={\tilde{\varphi }}_2\left(t_n\right)={\tilde{\varphi }}_2^n.\end{array}$$

For the computation of ghost-point values, we utilize the extrapolation formulas presented below:

$$\begin{array}{cc}\hfill U_{-2}^k=& 36U_0^k-168U_1^k+378U_2^k-504U_3^k+420U_4^k-216U_5^k+63U_6^k-8U_7^k+\mathcal{O}\left(h^8\right),\hfill \\ \hfill U_{-1}^k=& 8U_0^k-28U_1^k+56U_2^k-70U_3^k+56U_4^k-28U_5^k+8U_6^k-U_7^k+\mathcal{O}\left(h^8\right),\hfill \\ \hfill U_{M+1}^k=& 8U_M^k-28U_{M-1}^k+56U_{M-2}^k-70U_{M-3}^k+56U_{M-4}^k-28U_{M-5}^k+8U_{M-6}^k\hfill \\ \hfill & -U_{M-7}^k+\mathcal{O}\left(h^8\right),\hfill \\ \hfill U_{M+2}^k=& 36U_M^k-168U_{M-1}^k+378U_{M-2}^k-504U_{M-3}^k+420U_{M-4}^k-216U_{M-5}^k\hfill \\ \hfill & +63U_{M-6}^k-8U_{M-7}^k+\mathcal{O}\left(h^8\right).\hfill \end{array}$$

3. Stability and Convergence Analyses

In this section, we conduct a comprehensive theoretical analysis to establish the stability and convergence of the proposed numerical method for solving time-fractional CDREs with variable coefficients.

3.1. Stability Analysis

In this subsection, let ${\widehat{U}}_j^n$ represent the approximate solution of Equation (5). We define the error term as

$${\epsilon }_j^n={\widehat{U}}_j^n-U_j^n,j\in [M-1],n\in \left[N\right].$$

Consequently, we have

$$\begin{array}{cc}\hfill & \left({\displaystyle \frac{\mu {\lambda }_0}{560}}-{\displaystyle \frac{q_j}{560}}-{\displaystyle \frac{\theta }{90}}\right){\epsilon }_{j+3}^n+\left(-{\displaystyle \frac{3\mu {\lambda }_0}{280}}+{\displaystyle \frac{3q_j}{280}}+{\displaystyle \frac{3\theta }{20}}\right){\epsilon }_{j+2}^n+\left({\displaystyle \frac{3\mu {\lambda }_0}{112}}-{\displaystyle \frac{3q_j}{112}}-{\displaystyle \frac{3\theta }{2}}\right){\epsilon }_{j+1}^n\hfill \\ \hfill & +\left({\displaystyle \frac{27\mu {\lambda }_0}{28}}-{\displaystyle \frac{27q_j}{28}}+{\displaystyle \frac{49\theta }{18}}\right){\epsilon }_j^n+\left({\displaystyle \frac{3\mu {\lambda }_0}{112}}-{\displaystyle \frac{3q_j}{112}}-{\displaystyle \frac{3\theta }{2}}\right){\epsilon }_{j-1}^n\hfill \\ \hfill & +\left(-{\displaystyle \frac{3\mu {\lambda }_0}{280}}+{\displaystyle \frac{3q_j}{280}}+{\displaystyle \frac{3\theta }{20}}\right){\epsilon }_{j-2}^n+\left({\displaystyle \frac{\mu {\lambda }_0}{560}}-{\displaystyle \frac{q_j}{560}}-{\displaystyle \frac{\theta }{90}}\right){\epsilon }_{j-3}^n\hfill \\ \hfill & =-\mu \sum _{k=1}^n{\lambda }_k({\displaystyle \frac{1}{560}}{\epsilon }_{j+3}^{n-k}-{\displaystyle \frac{3}{280}}{\epsilon }_{j+2}^{n-k}+{\displaystyle \frac{3}{112}}{\epsilon }_{j+1}^{n-k}+{\displaystyle \frac{27}{28}}{\epsilon }_j^{n-k}\hfill \\ \hfill & +{\displaystyle \frac{3}{112}}{\epsilon }_{j-1}^{n-k}-{\displaystyle \frac{3}{280}}{\epsilon }_{j-2}^{n-k}+{\displaystyle \frac{1}{560}}{\epsilon }_{j-3}^{n-k}).\hfill \end{array}$$

(6)

The Fourier series representation of ${\epsilon }^n\left(x\right)$ and its associated coefficients are given by

$$\begin{array}{c}\hfill {\epsilon }^n\left(x\right)=\sum _{l=-\infty }^{\infty }{\xi }_n\left(l\right)e^{i\beta x},\mathrm{where}{\xi }_n\left(l\right)={\displaystyle \frac{1}{L}}{\int }_0^L{\epsilon }^n\left(x\right)e^{-i\beta x}dx,\beta ={\displaystyle \frac{2\pi l}{L}}\mathrm{and}i=\sqrt{-1}.\end{array}$$

By leveraging the definition of the $L^2$ norm and invoking Parseval’s theorem, we derive

$$\begin{array}{c}\hfill {\parallel {\epsilon }^n\parallel }^2=h\sum _{i=1}^{M-1}|{\epsilon }_i^n{|}^2={\int }_0^L|{\epsilon }_i^n{|}^{2}dx\mathrm{and}{\int }_0^L|{\epsilon }^n{\left(x\right)|}^{2}dx=L\sum _{l=-\infty }^{\infty }{|{\xi }_n\left(l\right)|}^2,\end{array}$$

and, thus, we have

$${\parallel {\epsilon }^n\parallel }^2=L\sum _{l=-\infty }^{\infty }{\left|{\xi }_n\left(l\right)\right|}^2.$$

(7)

We make the Fourier series assumption for ${\epsilon }_j^n={\xi }_n{e}^{i\beta jh}$, and we substitute it into Equation (6). After simplification, we obtain

$$\begin{array}{cc}\hfill & {\xi }_n[\left({\displaystyle \frac{\mu {\lambda }_0}{280}}-{\displaystyle \frac{q_j}{280}}-{\displaystyle \frac{\theta }{45}}\right)cos\left(3\beta h\right)+\left(-{\displaystyle \frac{3\mu {\lambda }_0}{140}}+{\displaystyle \frac{3q_j}{140}}+{\displaystyle \frac{3\theta }{10}}\right)cos\left(2\beta h\right)\hfill \\ \hfill & +\left({\displaystyle \frac{3\mu {\lambda }_0}{56}}-{\displaystyle \frac{3q_j}{56}}-3\theta \right)cos\left(\beta h\right)+\left({\displaystyle \frac{27\mu {\lambda }_0}{28}}-{\displaystyle \frac{27q_j}{28}}+{\displaystyle \frac{49\theta }{18}}\right)]\hfill \\ \hfill & =-\mu \sum _{k=1}^n{\lambda }_k\left({\displaystyle \frac{1}{280}}cos\left(3\beta h\right)-{\displaystyle \frac{3}{140}}cos\left(2\beta h\right)+{\displaystyle \frac{3}{56}}cos\left(\beta h\right)+{\displaystyle \frac{27}{28}}\right){\xi }_{n-k}.\hfill \end{array}$$

(8)

From Equation (8), it follows that ${\xi }_n$ can be written as

$$\begin{array}{c}\hfill {\xi }_n=\sum _{k=1}^n{\displaystyle \frac{-\mu {\lambda }_k{B}_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}{\xi }_{n-k},\end{array}$$

(9)

where $B_1$ and $B_2$ are defined as follows:

$$\begin{array}{cc}\hfill & B_1={\displaystyle \frac{1}{70}}\left({cos}^3\left(\beta h\right)-3{cos}^2\left(\beta h\right)+3cos\left(\beta h\right)+69\right),\hfill \\ \hfill & B_2=-{\displaystyle \frac{4}{45}}\left({cos}^3\left(\beta h\right)-{\displaystyle \frac{27}{4}}{cos}^2\left(\beta h\right)+{\displaystyle \frac{132}{4}}cos\left(\beta h\right)-{\displaystyle \frac{109}{4}}\right).\hfill \end{array}$$

We use C to represent various positive, bounded constants that may differ from one case to another.

Theorem 1.

The numerical scheme given by Equation (5) is unconditionally stable for $\alpha \in (0,1)$.

Proof. 

First, we apply mathematical induction to prove that

$$|{\xi }_n|\le C|{\xi }_0|,n\in \left[N\right].$$

When $n=1$, according to Equation (9), we have

$$\begin{array}{c}\hfill |{\xi }_1|=|{\displaystyle \frac{-\mu {\lambda }_1{B}_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}{\xi }_0|\le |{\displaystyle \frac{\mu {\lambda }_0{B}_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}|\left|{\xi }_0\right|,\end{array}$$

(10)

where ${\lambda }_1=-{\lambda }_0$ (refer to Kumari et al. [23] for details). Given that

$$\mu ={\displaystyle \frac{1}{\Gamma (2-\alpha )}},\theta ={\displaystyle \frac{d}{h^2}},B_1>0,B_2\ge 0,$$

and that the function

$$q\left(x\right)=p_2\left(x\right)+{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{d{p}_1}{dx}}-{\displaystyle \frac{p_1^2}{2d}}\right),$$

is bounded, we have

$$\begin{array}{c}\hfill |{\xi }_1|\le C|{\xi }_0|.\end{array}$$

Next, we assume that the inequality

$$|{\xi }_n|\le C|{\xi }_0|$$

holds for $n=2,3,\dots ,N-1.$

When $n=N$, according to Equation (9) and Lemma 3, we have

$$\begin{array}{cc}\hfill |{\xi }_N|& =\left|\sum _{k=1}^N{\displaystyle \frac{-\mu {\lambda }_k{B}_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}{\xi }_{N-k}\right|\hfill \\ \hfill & \le \sum _{k=1}^N\left|{\displaystyle \frac{-\mu {\lambda }_k{B}_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}\right|\left|{\xi }_{N-k}\right|\hfill \\ \hfill & \le \sum _{k=1}^N|{\psi }_k||{\displaystyle \frac{-\mu {\tau }^{-\alpha }{B}_1}{(\mu {\tau }^{-\alpha }{\psi }_0-q_j)B_1+\theta B_2}}|C\left|{\xi }_0\right|\hfill \\ \hfill & \le C({\psi }_0+2{\psi }_2)\left|{\xi }_0\right|\hfill \\ \hfill & \le C|{\xi }_0|.\hfill \end{array}$$

(11)

By Equations (7) and (11), we obtain

$$\begin{array}{cc}\hfill \parallel {\epsilon }^n{\parallel }^2& =L\sum _{l=-\infty }^{\infty }{\left|{\xi }_n\left(l\right)\right|}^2\hfill \\ \hfill & \le L\sum _{l=-\infty }^{\infty }{C}^2{\left|{\xi }_0\left(l\right)\right|}^2\hfill \\ \hfill & =C^2{\parallel {\epsilon }^0\parallel }^2.\hfill \end{array}$$

Therefore, the numerical scheme given by Equation (5) is unconditionally stable. □

3.2. Convergence Analysis

In this subsection, we discuss the convergence of Equation (5). In a similar manner, let

$$\begin{array}{c}\hfill E_j^n=u_j^n-U_j^n,j\in [M-1],n\in \left[N\right].\end{array}$$

By subtracting Equation (4) from Equation (3) and noting that $E_j^0=0$, we can establish the following:

$$\begin{array}{cc}\hfill & \left({\displaystyle \frac{\mu {\lambda }_0}{560}}-{\displaystyle \frac{q_j}{560}}-{\displaystyle \frac{\theta }{90}}\right)E_{j+3}^n+\left(-{\displaystyle \frac{3\mu {\lambda }_0}{280}}+{\displaystyle \frac{3q_j}{280}}+{\displaystyle \frac{3\theta }{20}}\right)E_{j+2}^n+\left({\displaystyle \frac{3\mu {\lambda }_0}{112}}-{\displaystyle \frac{3q_j}{112}}-{\displaystyle \frac{3\theta }{2}}\right)E_{j+1}^n\hfill \\ \hfill & +\left({\displaystyle \frac{27\mu {\lambda }_0}{28}}-{\displaystyle \frac{27q_j}{28}}+{\displaystyle \frac{49\theta }{18}}\right)E_j^n+\left({\displaystyle \frac{3\mu {\lambda }_0}{112}}-{\displaystyle \frac{3q_j}{112}}-{\displaystyle \frac{3\theta }{2}}\right)E_{j-1}^n\hfill \\ \hfill & +\left(-{\displaystyle \frac{3\mu {\lambda }_0}{280}}+{\displaystyle \frac{3q_j}{280}}+{\displaystyle \frac{3\theta }{20}}\right)E_{j-2}^n+\left({\displaystyle \frac{\mu {\lambda }_0}{560}}-{\displaystyle \frac{q_j}{560}}-{\displaystyle \frac{\theta }{90}}\right)E_{j-3}^n\hfill \\ \hfill & =+\left({\displaystyle \frac{1}{560}}{R}_{j+3}^n-{\displaystyle \frac{3}{280}}{R}_{j+2}^n+{\displaystyle \frac{3}{112}}{R}_{j+1}^n+{\displaystyle \frac{27}{28}}{R}_j^n+{\displaystyle \frac{3}{112}}{R}_{j-1}^n-{\displaystyle \frac{3}{280}}{R}_{j-2}^n+{\displaystyle \frac{1}{560}}{R}_{j-3}^n\right)\hfill \\ \hfill & -\mu \sum _{k=1}^{n-1}{\lambda }_k({\displaystyle \frac{1}{560}}{E}_{j+3}^{n-k}-{\displaystyle \frac{3}{280}}{E}_{j+2}^{n-k}+{\displaystyle \frac{3}{112}}{E}_{j+1}^{n-k}+{\displaystyle \frac{27}{28}}{E}_j^{n-k}+{\displaystyle \frac{3}{112}}{E}_{j-1}^{n-k}\hfill \\ \hfill & -{\displaystyle \frac{3}{280}}{E}_{j-2}^{n-k}+{\displaystyle \frac{1}{560}}{E}_{j-3}^{n-k})\hfill \end{array}$$

(12)

where $R_j^n=\mathcal{O}({\tau }^{4-\alpha }+h^8)$.

Similarly to the stability analysis, we define the grid functions $E^n\left(x\right)$ and $R^n\left(x\right)$:

$$\begin{array}{c}\hfill E^n\left(x\right)=\sum _{l=-\infty }^{\infty }{e}_n\left(l\right)e^{i\beta x},\mathrm{where}{e}_n\left(l\right)={\displaystyle \frac{1}{L}}{\int }_0^L{E}^n\left(x\right)e^{-i\beta x}dx,\beta ={\displaystyle \frac{2\pi l}{L}}\mathrm{and}i=\sqrt{-1},\\ \hfill R^n\left(x\right)=\sum _{l=-\infty }^{\infty }{r}_n\left(l\right)e^{i\beta x},\mathrm{where}{r}_n\left(l\right)={\displaystyle \frac{1}{L}}{\int }_0^L{R}^n\left(x\right)e^{-i\beta x}dx,\beta ={\displaystyle \frac{2\pi l}{L}}\mathrm{and}i=\sqrt{-1}.\end{array}$$

Assuming that $E_j^n=e_n{e}^{i\beta jh}$ and $R_j^n=r_n{e}^{i\beta jh}$ and subtracting them from Equation (12), we derive

$$\begin{array}{cc}\hfill & e_n[\left({\displaystyle \frac{\mu {\lambda }_0}{280}}-{\displaystyle \frac{q_j}{280}}-{\displaystyle \frac{\theta }{45}}\right)cos\left(3\beta h\right)+\left(-{\displaystyle \frac{3\mu {\lambda }_0}{140}}+{\displaystyle \frac{3q_j}{140}}+{\displaystyle \frac{3\theta }{10}}\right)cos\left(2\beta h\right)\hfill \\ \hfill & +\left({\displaystyle \frac{3\mu {\lambda }_0}{56}}-{\displaystyle \frac{3q_j}{56}}-3\theta \right)cos\left(\beta h\right)+\left({\displaystyle \frac{27\mu {\lambda }_0}{28}}-{\displaystyle \frac{27q_j}{28}}+{\displaystyle \frac{49\theta }{18}}\right)]\hfill \\ \hfill & =-\mu \sum _{k=1}^{n-1}{\lambda }_k\left({\displaystyle \frac{1}{280}}cos\left(3\beta h\right)-{\displaystyle \frac{3}{140}}cos\left(2\beta h\right)+{\displaystyle \frac{3}{56}}cos\left(\beta h\right)+{\displaystyle \frac{27}{28}}\right)e_{n-k}\hfill \\ \hfill & +\left({\displaystyle \frac{1}{280}}cos\left(3\beta h\right)-{\displaystyle \frac{3}{140}}cos\left(2\beta h\right)+{\displaystyle \frac{3}{56}}cos\left(\beta h\right)+{\displaystyle \frac{27}{28}}\right)r_n.\hfill \end{array}$$

It follows that

$$\begin{array}{c}\hfill e_n=\sum _{k=1}^{n-1}{\displaystyle \frac{-\mu {\lambda }_k{B}_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}{e}_{n-k}+{\displaystyle \frac{-\mu {\lambda }_k{B}_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}{r}_n.\end{array}$$

(13)

In preparation for the proof of Theorem 2, we first introduce some necessary definitions:

$${\parallel E^n\parallel }^2=h\sum _{j=1}^{M-1}|E_j^n{|}^2=L\sum _{l=-\infty }^{\infty }{|e_n\left(l\right)|}^2,$$

(14)

$${\parallel R^n\parallel }^2=h\sum _{j=1}^{M-1}|R_j^n{|}^2=L\sum _{l=-\infty }^{\infty }{|r_n\left(l\right)|}^2.$$

(15)

Theorem 2.

The numerical scheme given by Equation (5) converges with order $\mathcal{O}({\tau }^{4-\alpha }+h^8)$.

Proof. 

We begin by employing mathematical induction to demonstrate the existence of a positive constant C such that the following inequality holds for all $n\in \left[N\right]$:

$$\begin{array}{c}\hfill |e_n|\le C|r_1|.\end{array}$$

Since $R_j^n=\mathcal{O}({\tau }^{4-\alpha }+h^8)$, by Equation (15), we can assert the existence of a positive constant $C_n$ for each $n\in \left[N\right]$ such that

$$\begin{array}{c}\hfill |r_n|\le C_n\left|r_1\right|.\end{array}$$

When $n=1$, according to Equation (13), we obtain

$$\begin{array}{cc}\hfill |e_1|& =\left|{\displaystyle \frac{-\mu {\lambda }_1{B}_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}{r}_1\right|\hfill \\ \hfill & \le \left|{\displaystyle \frac{\mu {\lambda }_0{B}_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}\right|\left|r_1\right|,\hfill \end{array}$$

and, similar to Equation (10), we can derive

$$\begin{array}{c}\hfill |e_1|\le C_1\left|r_1\right|.\end{array}$$

Now, we assume that the inequality

$$\begin{array}{c}\hfill |e_n|\le C_n\left|r_1\right|\end{array}$$

holds for $n=2,3,\dots ,N-1$.

We define $C^{\prime }$ as the maximum value among the constants $\{C_1,C_2,\dots ,C_N\}$.

When $n=N$, according to Equation (13) and Lemma 3, we derive

$$\begin{array}{cc}\hfill |e_N|& =|\sum _{k=1}^{N-1}{\displaystyle \frac{-\mu {\lambda }_k{B}_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}{e}_{N-k}+{\displaystyle \frac{B_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}{r}_N|\hfill \\ \hfill & \le \sum _{k=1}^{N-1}\left|{\displaystyle \frac{-\mu {\lambda }_k{B}_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}\right||e_{N-k}|+|{\displaystyle \frac{B_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}|\left|r_N\right|\hfill \\ \hfill & \le \sum _{k=1}^{N-1}\left|{\displaystyle \frac{-\mu {\lambda }_k{B}_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}\right|C_{N-k}|r_1|+|{\displaystyle \frac{B_1}{(\mu {\lambda }_0-q_j)B_1+\theta B_2}}|C_N\left|r_1\right|\hfill \\ \hfill & \le C^{\prime }\left|{\displaystyle \frac{\mu {\tau }^{-\alpha }{B}_1}{(\mu {\tau }^{-\alpha }{\psi }_0-q_j)B_1+\theta B_2}}\right|({\psi }_0+2{\psi }_2)|r_1|+C^{\prime }|{\displaystyle \frac{B_1}{(\mu {\tau }^{-\alpha }{\psi }_0-q_j)B_1+\theta B_2}}|\left|r_1\right|\hfill \\ \hfill & \le C^{\prime }(1+{\psi }_0+2{\psi }_2)\left|r_1\right|\hfill \\ \hfill & \le C|r_1|.\hfill \end{array}$$

Then, by Equation (14) and (15), we obtain

$$\begin{array}{cc}\hfill \parallel E^n{\parallel }^2& =L\sum _{l=-\infty }^{\infty }{\left|e_n\left(l\right)\right|}^2\hfill \\ \hfill & \le L\sum _{l=-\infty }^{\infty }{C}^2{\left|r_1\left(l\right)\right|}^2\hfill \\ \hfill & =C^2{\parallel R^1\parallel }^2\hfill \\ \hfill & \le C^2{({\tau }^{4-\alpha }+h^8)}^2.\hfill \end{array}$$

Therefore, the numerical scheme given by Equation (5) converges with order $\mathcal{O}({\tau }^{4-\alpha }+h^8)$. □

4. Numerical Results

Our innovative high-order scheme for time-fractional CDREs with variable coefficients was rigorously validated through computational experiments, which revealed its convergence order and confirmed its effectiveness. All numerical experiments were conducted using MATLAB 2020b. The errors $E_1$ and $E_2$ were computed with mesh sizes $h_1$ and $h_2$, respectively, and the errors and convergence rates were quantified as follows:

$$L^{\infty }-\mathrm{Error}=\underset{1\le n\le N}{max}|u^n-U^n|,L^{\infty }-\mathrm{Rate}=log(E_1/E_2)/log(h_1/h_2).$$

To rigorously validate the accuracy and robustness of our proposed method, we compared its results against known analytical solutions under controlled conditions in Examples 1–3.

Example 1.

We examine Equation (2) in the domain $[0,2]\times [0,1]$ with $d=1$, $p_1\left(x\right)={\displaystyle \frac{1}{1+x}}$, $p_2\left(x\right)=x^2$, and $f(x,t)=\left({\displaystyle \frac{\Gamma (5+\alpha )}{24}}-t^{\alpha }{x}^2\right)t^4{(1+x)}^2$. The exact solution is $v(x,t)=t^{4+\alpha }{(1+x)}^2$.

To validate our approach, we compare our results with those in [21].

Table 1. Comparison of error and temporal convergence rates for Example 1 at $M=100$.

	N	$\mathit{\alpha }=0.25$		$\mathit{\alpha }=0.5$		$\mathit{\alpha }=0.75$
		${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate	${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate	${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate
[21]	40	2.5734 × ${10}^{-5}$	—	8.3385 × ${10}^{-5}$	—	1.9864 × ${10}^{-4}$	—
	80	3.2434 × ${10}^{-6}$	2.9881	1.0540 × ${10}^{-5}$	2.9839	2.5199 × ${10}^{-5}$	2.9787
	160	4.0709 × ${10}^{-7}$	2.9941	1.3249 × ${10}^{-6}$	2.9920	3.1730 × ${10}^{-6}$	2.9894
	320	5.0989 × ${10}^{-8}$	2.9971	1.6607 × ${10}^{-7}$	2.9960	3.9807 × ${10}^{-7}$	2.9947
Our	40	3.8755 × ${10}^{-6}$	—	3.0304 × ${10}^{-5}$	—	1.7518 × ${10}^{-4}$	—
	80	2.9991 × ${10}^{-7}$	3.6918	2.7402 × ${10}^{-6}$	3.4672	1.8793 × ${10}^{-5}$	3.2206
	160	2.2996 × ${10}^{-8}$	3.7051	2.4561 × ${10}^{-7}$	3.4798	1.9964 × ${10}^{-6}$	3.2347
	320	1.7106 × ${10}^{-9}$	3.7488	2.1851 × ${10}^{-8}$	3.4906	2.1095 × ${10}^{-7}$	3.2424

shows a comparison of the error and temporal convergence rates between our method and that in [21] under different N values and $\alpha$ values with $M=100$. Similarly,

Table 2. Comparison of error and spatial convergence rates for Example 1.

	M	$\mathit{\alpha }=0.25$		$\mathit{\alpha }=0.5$		$\mathit{\alpha }=0.75$
		${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate	${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate	${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate
[21]	8	7.1868 × ${10}^{-8}$	—	5.6976 × ${10}^{-8}$	—	4.4166 × ${10}^{-8}$	—
	16	4.5156 × ${10}^{-9}$	3.9924	3.5806 × ${10}^{-9}$	3.9921	2.7763 × ${10}^{-9}$	3.9917
	32	2.8239 × ${10}^{-10}$	3.9992	2.2351 × ${10}^{-10}$	4.0018	1.7308 × ${10}^{-10}$	4.0036
	64	1.7339 × ${10}^{-11}$	4.0255	1.3457 × ${10}^{-11}$	4.0539	1.0351 × ${10}^{-11}$	4.0635
Our	8	1.0969 × ${10}^{-2}$	—	3.4659 × ${10}^{-2}$	—	6.6072 × ${10}^{-2}$	—
	16	5.9367 × ${10}^{-5}$	7.5295	1.7548 × ${10}^{-4}$	7.6258	4.3821 × ${10}^{-4}$	7.2363
	32	2.5068 × ${10}^{-7}$	7.8877	7.5292 × ${10}^{-7}$	7.8646	1.7378 × ${10}^{-6}$	7.9782
	64	1.0027 × ${10}^{-9}$	7.9658	3.0181 × ${10}^{-9}$	7.9627	6.2153 × ${10}^{-9}$	8.1273

compares the error and spatial convergence rates between our method and that in [21] under different M values and $\alpha$ values (where N = 15,000 in [21], and $N={\displaystyle \lfloor }{({\displaystyle \frac{M}{2}})}^{{\displaystyle \frac{8}{4-\alpha }}}{\displaystyle \rfloor }$ in our method). These comparisons clearly demonstrate the advantages of our approach. Figure 1 visually presents the exact solution, numerical solution, absolute error, and error contour plot when $\alpha =0.01$, $M=100$, and $N=100$, further validating the accuracy and convergence of our method. Together, these results provide a solid foundation for validating our approach and demonstrate its effectiveness and reliability under various parameter settings.

Example 2.

We examine Equation (2) with the following parameter settings: $d=1$, $p_1\left(x\right)=-sinx$, $p_2\left(x\right)=-cosx$, and $f(x,t)={\displaystyle \frac{\Gamma (5+\alpha )}{24}}{t}^{4}cosx+t^{4+\alpha }(1+cosx)$. The exact solution for validation purposes is given by $v(x,t)=t^{4+\alpha }cosx$.

Table 3. Comparison of error and temporal convergence rates for Example 2 with different $\alpha$ values.

	$\mathit{M}\times \mathit{N}$	$\mathit{\alpha }=0.25$		$\mathit{\alpha }=0.5$		$\mathit{\alpha }=0.75$
		${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate	${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate	${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate
[21]	1000 × 40	2.7763 × ${10}^{-6}$	—	9.0510 × ${10}^{-6}$	—	2.1821 × ${10}^{-5}$	—
	1000 × 80	3.4991 × ${10}^{-7}$	2.9881	1.1439 × ${10}^{-6}$	2.9841	2.7671 × ${10}^{-6}$	2.9793
	1000 × 160	4.3916 × ${10}^{-8}$	2.9941	1.4378 × ${10}^{-7}$	2.9921	3.4835 × ${10}^{-7}$	2.9897
	1000 × 320	5.4995 × ${10}^{-9}$	2.9974	1.8021 × ${10}^{-8}$	2.9961	4.3698 × ${10}^{-8}$	2.9949
Our	200 × 40	4.1605 × ${10}^{-7}$	—	3.2513 × ${10}^{-6}$	—	1.8881 × ${10}^{-5}$	—
	200 × 80	3.2193 × ${10}^{-8}$	3.6920	2.9393 × ${10}^{-7}$	3.4675	2.0243 × ${10}^{-6}$	3.2215
	200 × 160	2.4677 × ${10}^{-9}$	3.7055	2.6341 × ${10}^{-8}$	3.4801	2.1497 × ${10}^{-7}$	3.2352
	200 × 320	1.8287 × ${10}^{-10}$	3.7543	2.3426 × ${10}^{-9}$	3.4911	2.2711 × ${10}^{-8}$	3.2427

presents a detailed comparison of the error and temporal convergence rates for Example 2 with different $\alpha$ values, directly comparing our results with those in [21].

Table 4. Error and spatial convergence rates for Example 2 with different $\alpha$ values and $N={\displaystyle \lfloor }{({\displaystyle \frac{M}{2}})}^{{\displaystyle \frac{8}{4-\alpha }}}{\displaystyle \rfloor }$.

M	$\mathit{\alpha }=0.25$		$\mathit{\alpha }=0.5$		$\mathit{\alpha }=0.75$
	${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate	${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate	${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate
16	2.2560 × ${10}^{-4}$	—	7.8067 × ${10}^{-4}$	—	1.5142 × ${10}^{-3}$	—
24	9.3857 × ${10}^{-6}$	7.8418	2.9720 × ${10}^{-5}$	8.0607	6.6303 × ${10}^{-5}$	7.7157
32	9.3441 × ${10}^{-7}$	8.0193	2.9686 × ${10}^{-6}$	8.0079	6.7649 × ${10}^{-6}$	7.9340
40	1.5472 × ${10}^{-7}$	8.0589	4.8711 × ${10}^{-7}$	8.0995	1.1578 × ${10}^{-6}$	7.9106

presents the error and spatial convergence rates for Example 2 under different M values and $\alpha$ values with $N={\displaystyle \lfloor }{({\displaystyle \frac{M}{2}})}^{{\displaystyle \frac{8}{4-\alpha }}}{\displaystyle \rfloor }$. This analysis is conducted over the domain $[0,\pi ]\times [0,1]$, highlighting the effectiveness of our method against established results. For plotting and visualization purposes, we employ a different region, $[0,6\pi ]\times [0,1]$, to better highlight the specific features of the solution. This extended domain allows us to observe the periodic nature of the solution more clearly by capturing multiple periods, providing valuable insights into the long-term behavior and periodicity. Additionally, it helps in understanding the boundary effects and their influence on the solution, which is particularly important when boundary conditions significantly impact the overall behavior.

Figure 2 uses the larger domain $[0,6\pi ]\times [0,1]$ to illustrate the exact solution, numerical solution, absolute error, and error contour plot for $\alpha =0.99$, $M=200$, and $N=100$. These visualizations further validate the accuracy and convergence of our method under these conditions.

Example 3.

We examine Equation (2) in the domain $[0,\pi ]\times [0,2]$ with $d=1$, $p_1\left(x\right)=-1$, $p_2\left(x\right)=1$, and $f(x,t)={\displaystyle \frac{\Gamma (5+\alpha )}{24}}{t}^{4}cosx+t^{4+\alpha }sinx$. The exact solution for validation purposes is given by $v(x,t)=t^{4+\alpha }cosx$.

In this final example, we switch to constant coefficients to consolidate our findings from Examples 1 and 2, which deal with variable coefficients. This allows us to reinforce the reliability and robustness of our method across different types of coefficient settings.

Table 5. ${\mathit{L}}^{\mathbf{\infty }}$-error and temporal/spatial convergence rates for Example 3 at $\alpha =0.5$.

Temporal Convergence			Spatial Convergence
$\mathbf{M}\times \mathbf{N}$	${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate	$\mathbf{M}\times \mathbf{N}$	${\mathit{L}}^{\mathbf{\infty }}$ -Error	${\mathit{L}}^{\mathbf{\infty }}$ -Rate
$100\times 10$	7.4329 × ${10}^{-3}$	—	$10\times 1000$	3.4979 × ${10}^{-4}$	—
$100\times 20$	7.0324 × ${10}^{-4}$	3.4018	$15\times 1000$	1.4734 × ${10}^{-5}$	7.8112
$100\times 40$	6.4632 × ${10}^{-5}$	3.4437	$20\times 1000$	1.4621 × ${10}^{-6}$	8.0308
$100\times 80$	5.8474 × ${10}^{-6}$	3.4664	$25\times 1000$	2.4045 × ${10}^{-7}$	8.0893

presents the $L^{\infty }$-error and convergence rates for temporal and spatial discretizations at $\alpha =0.5$. For brevity, we focus on $\alpha =0.5$, though our findings generalize to other $\alpha$ values. This table illustrates the method’s performance under different grid resolutions, demonstrating the expected convergence behavior.

Table 6. Comparison of numerical solutions ($v_n$) and exact solutions ($v_e$) for Example 3.

x	$\mathit{\alpha }=0.01$		$\mathit{\alpha }=0.5$		$\mathit{\alpha }=0.99$
	$v_n$	$v_e$	$v_n$	$v_e$	$v_n$	$v_e$
${\displaystyle \frac{\pi }{8}}$	14.88488997	14.88488997	20.90500831	20.90500744	29.35998710	29.35993057
${\displaystyle \frac{2\pi }{8}}$	11.39240157	11.39240156	16.00000041	16.00000000	22.47115654	22.47111801
${\displaystyle \frac{3\pi }{8}}$	6.16552330	6.16552330	8.65913689	8.65913760	12.16126336	12.16128143
${\displaystyle \frac{4\pi }{8}}$	0.00000000	0.00000000	−0.00000188	0.00000000	−0.00008026	0.00000000
${\displaystyle \frac{5\pi }{8}}$	−6.16552330	−6.16552330	−8.65914020	−8.65913760	−12.16140299	−12.16128143
${\displaystyle \frac{6\pi }{8}}$	−11.39240157	−11.39240156	−16.00000257	−16.00000000	−22.47124269	−22.47111801
${\displaystyle \frac{7\pi }{8}}$	−14.88488997	−14.88488997	−20.90500912	−20.90500744	−29.36001345	−29.35993057

compares the numerical solutions $\left(v_n\right)$ and exact solutions $\left(v_e\right)$ for different $\alpha$ values and grid points at $T=2$. This comparison provides a direct assessment of the numerical solution’s accuracy across various parameter settings. Figure 3 provides a visual convergence analysis with different $\alpha$ values: (a) $log\left(Error\right)$ vs. $log\left(\tau \right)$ for temporal convergence and (b) $log\left(Error\right)$ vs. $log\left(h\right)$ for spatial convergence. These plots further validate the method’s effectiveness and convergence properties for a range of $\alpha$ values, serving as a solid conclusion to our series of examples.

5. Conclusions

In conclusion, we develop a novel numerical scheme for time-fractional CDREs with variable coefficients that combines Lagrange polynomial interpolation for time discretization with a compact finite difference approximation for spatial derivatives. The proposed method exhibits unconditional stability and achieves high-order convergence rates, reaching $4-\alpha$-order accuracy in time and eighth-order accuracy in space, as rigorously verified through a Fourier analysis. These results represent a substantial improvement in the numerical solution of fractional partial differential equations. Numerical results validate our theoretical analysis, demonstrating the superior accuracy and efficiency of our method over existing solutions. The present study focuses on 1D problems to establish the foundational accuracy and convergence properties of our method. While this simplifies validation, we acknowledge that extensions to 2D/3D or more complex geometries will require further development, particularly in computational efficiency and boundary condition handling. Additionally, our tests primarily involve linear problems. Application to nonlinear CDREs or real-world scenarios remains a challenge for us. Future work may also explore adaptive mesh strategies and the incorporation of nonlinearity in spatial operators for more realistic modeling.