Numerical Analysis of a High-Order Scheme for Space-Time Fractional Diffusion-Wave Equations with Riesz Derivatives

Abstract

In this paper, we study a class of time–space fractional partial differential equations involving Caputo time-fractional derivatives and Riesz space-fractional derivatives. A computational scheme is developed by combining a discrete approximation for the Caputo derivative in time with a modified trapezoidal method (MTM) for the Riesz derivative in space. We establish the stability and convergence of the scheme and provide detailed error analysis. The novelty of this work lies in the construction of an MTM-based spatial discretization that achieves ${\displaystyle \beta }$-order convergence in space and a ${\displaystyle (3-\alpha )}$-order convergence in time, while improving accuracy and efficiency compared to existing methods. Numerical experiments are carried out to validate the theoretical findings, confirm the stability of the proposed algorithm under perturbations, and demonstrate its superiority over a recent scheme from the literature.

1. Introduction

In the past few decades, it has often been observed that the dynamics of real-world problems of complex systems are not accurately captured by classical models based on integer-order derivatives and integrals. The relevance of fractional calculus came into play when fractional partial differential equations (FPDEs) were used to model various natural and complex phenomena, which were imperfectly described using integer-order models. The essential properties of non-integer derivatives are highlighted in [1], especially the memory and hereditary properties. Recently, FPDEs have gained increasing significance due to their wide applications in material characterization [2], control systems [1], finance [3], biological systems [4,5,6], electromagnetic wave propagation [7,8], electrochemical processes [9], quantum mechanics [10], and several other physical phenomena.

In the development of numerical solutions for FPDEs, various techniques have been proposed to obtain approximate solutions, including finite difference schemes, [11,12,13,14], finite element methods [15], finite volume algorithms [16], spectral methods [17], operational matrix schemes [18] or homotopy methods [19] as the more relevant ones. Among the aforementioned methods, the finite difference technique can be seen as one of the most well-considered methods, as it is very efficient and easy to implement. Motivated by this, we have used this kind of approach to solve a super-diffusion space–time fractional wave model.

Time-fractional differential equations (TFDEs) are widely employed to model different physical phenomena. TFDEs can be obtained by replacing the classical integer-order derivatives with non-integer derivatives [20]. Time-fractional equations with initial and boundary conditions have been developed to model various complex real-world problems, such as time-fractional diffusion-wave equations (TFDWEs). It is a well-established fact that due to localized disturbances, the wave and diffusion equations are completely different, i.e., in a phenomenon characterized by diffusion processes, there is an extremely fast speed of the disturbance propagation, whereas in comparison with wave equations, the propagation speed changes negligibly. In this context, Huang et al. [21] derived a first-order difference algorithm to provide the solution of an initial boundary value TFDWE, and Li et al. [22] proposed a higher-order convergence method involving a Caputo fractional derivative in time. The authors consider initial and boundary value problems for the diffusion-wave equation involving a Caputo fractional derivative (of order ${\displaystyle \alpha }$, with ${\displaystyle 1<\alpha <2}$) in time. A novel finite difference discrete scheme was developed using the discrete fractional time derivative, where new coefficients ${\displaystyle {(k+\frac{1}{2})}^{(2-\alpha )}-{(k-\frac{1}{2})}^{(2-\alpha )}}$ were introduced in place of the classical coefficients ${\displaystyle {(k+1)}^{(2-\alpha )}-k^{(2-\alpha )}}$.

The space-fractional derivatives are of great importance, especially in a scenario where one has to model anomalous diffusion or dispersion processes [23]. Recently, the relevance of considering the Riesz space of fractional order derivatives has gained huge attention from several scientists and engineers since it plays a vital role in scientific models embracing their physical and mathematical nature. In 2011, Rabei et al. [24] proposed the variation principle for non-integer order constrained models consisting of Riesz space fractional derivatives. Saichev and Zaslavsky [25] designed the Riesz space fractional diffusion equation (RFDE) from the kinetics of chaotic dynamics. In [26], an efficient numerical method is derived to solve RFDEs. In 2010, Yang et al. [27] provided various approximations to solve RFDEs using several methods including the fractional method of lines and the MTM approach. It establishes that the matrix transform approach produces the best outcomes among the prescribed numerical methods. In [28], space–time Riesz FDEs with periodic conditions were discussed. Further, they deduced explicit formulas for fundamental solutions of space–time Riesz FPDEs.

In recent years, it has been increasingly recognized that space–time fractional differential equations provide more accurate models for anomalous diffusion and relaxation processes [29].

In this manuscript, we consider the following continuous super-diffusion space–time fractional wave model [30]:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\mathcal{D}}_{0,t}^{\alpha }u(x,t)=-{(-\mathsf{\Delta })}^{\beta /2}u(x,t)+ð(x,t),0<x<1,0<t<T,\hfill \\ \hfill & u(0,t)=u(1,t)=0,0<t<T,\hfill \\ \hfill & u(x,0)=\varphi \left(x\right),{\displaystyle \frac{\partial u(x,0)}{\partial t}}=\psi \left(x\right),0<x<1,\hfill \end{array}\right.\end{array}$$

(1)

where ${\displaystyle {\mathcal{D}}_{0,t}^{\alpha }}$ represents the Caputo time-fractional derivative of order ${\displaystyle \alpha }$ in the temporal direction and  ${\displaystyle -{(-\mathsf{\Delta })}^{\beta /2}}$ denotes the Riesz space-fractional derivative of order ${\displaystyle \beta /2}$ in the space domain, with ${\displaystyle \alpha ,\beta \in (1,2)}$.

Purpose and Main Contribution of the Paper

• The limitations of the above-mentioned schemes motivated us to design a new algorithm with higher efficiency and simpler implementation for the proposed problem.
• Singh and Mehra [30] developed a numerical approximation for (1). They designed a finite difference scheme for the given model by replacing the Riesz space derivative with the linear sum of the left and right Riemann–Liouville derivatives. The proposed method yields first-order convergence in the spatial direction but fails to achieve a better order of convergence in the temporal direction. Moreover, their technique is computationally expensive and difficult to implement.
• A major highlight of this manuscript is that it introduces a new algorithm with ${\displaystyle \beta }$-order of convergence in space and ${\displaystyle (3-\alpha )}$-order in time. Further, the derived numerical scheme provides a more efficient and stable algorithm for solving the problem (1).
• In [31], an approximate solution of a particular space-time fractional diffusion model with a generalized Riemann–Liouville time-fractional derivative and Riesz–Feller space fractional derivative was considered. The solution was obtained using the Laplace and Fourier transform methods, and was expressed by an infinite integral involving Mittag–Leffler and the Fox H-functions.
• To construct a more accurate numerical approximation of the Riesz derivative, we employ the MTM approach [32], while in the temporal direction, we incorporated an approximation of order ${\displaystyle (3-\alpha )}$ for the Caputo fractional derivative.
• The proposed computational algorithm has several advantages: it provides an unconditionally stable numerical algorithm with ${\displaystyle \beta }$-order convergence in spatial direction and ${\displaystyle (3-\alpha )}$-order in the temporal direction, and is computationally efficient, reducing cost by delivering faster and more accurate results in half the step size compared to the scheme in [30].

Motivated by these developments, we have developed a new computational algorithm for the considered problem (1). The structure of the rest of the paper is outlined as follows. In Section 2, some basic concepts are discussed. A fully discrete scheme for the problem (1) is constructed and its matrix representation is given in Section 3. The numerical stability and convergence analysis of the developed scheme are discussed in Section 4. In Section 5, some numerical examples are provided to validate the theoretical findings. It is highlighted that the proposed scheme doubles the order of convergence, compared to the existing results in [30]. Finally, Section 6 concludes the paper with remarks and perspectives.

2. Some Basic Concepts

Definition 1

([32,33]). Let us assume the Laplacian ${\displaystyle (-\mathsf{\Delta })}$ has a complete set of orthonormal eigenfunctions ${\displaystyle \left\{{\varphi }_n\right\}}$ with eigenvalues ${\displaystyle \left\{{\lambda }_n\right\}}$ on a bounded region ${\displaystyle \mathsf{\Omega }}$ with Dirichlet boundary conditions. For ${\displaystyle \beta >0}$, we define

$$F_{\beta }=\left\{f=\sum _{\begin{array}{c}n=1\end{array}}^{\infty }{f}_n{\varphi }_n,f_n=<f,{\varphi }_n>:\sum _{\begin{array}{c}n=1\end{array}}^{\infty }|f_n{|}^2{\left|{\lambda }_n\right|}^{\beta }<\infty \right\}.$$

(2)

For any ${\displaystyle f\in F_{\beta },{(-\mathsf{\Delta })}^{\beta /2}:F_{\beta }\to L_2\left(\mathsf{\Omega }\right)}$ is defined by

$${(-\mathsf{\Delta })}^{\beta /2}f=\sum _{\begin{array}{c}n=1\end{array}}^{\infty }{f}_n{\left({\lambda }_n\right)}^{\beta /2}{\varphi }_n.$$

(3)

Definition 2

([33]). For any ${\displaystyle u\in F_{\beta /2}}$ with ${\displaystyle \beta >0}$, define the norm

$${\parallel u\parallel }_{\beta /2}={({\left|u\right|}^2+{\left|u\right|}_{\beta /2}^2)}^{1/2},$$

where

$${\left|u\right|}_{\beta /2}^2=\sum _{\begin{array}{c}l=1\end{array}}^{\infty }{\left({\left(l\pi \right)}^2\right)}^{\beta /2}{\left(u_l\right)}^2,$$

and ${\displaystyle u_l}$ are the coefficients of the Fourier expansion of ${\displaystyle u}$, that is, ${\displaystyle u(x,t)=\sum _{l=1}^{\infty }{u}_l{\varphi }_l(x,t)}$.

Definition 3

([33]). For each ${\displaystyle M\in {\mathbb{N}}^{+}}$, the linear space ${\displaystyle S_M\subset L_2(0,1)}$ is defined as

$$S_M=span\left\{sin\left(l\pi x\right):x\in [0,1],1\le l\le M-1,l\in \mathbb{N}\right\}$$

and the corresponding projection operator ${\displaystyle P_M:L_2(0,1)\to S_M}$ is given by

$$P_{M}f\left(x\right)=\sum _{\begin{array}{c}l=1\end{array}}^{M-1}{f}_{l}sin\left(l\pi x\right),$$

where

$$f\left(x\right)=\sum _{\begin{array}{c}l=1\end{array}}^{\infty }{f}_{l}sin\left(l\pi x\right).$$

Also, we have ${\displaystyle {(-\mathsf{\Delta })}^{\beta /2}{P}_{M}f=P_M{(-\mathsf{\Delta })}^{\beta /2}f\forall f\in F_{\beta /2}.}$

Definition 4

([1]). Let ${\displaystyle u(x,t)}$ be a sufficiently smooth function. Then, the Riesz space fractional derivative of order ${\displaystyle \beta \in (1,2)}$ on a finite interval ${\displaystyle a\le x\le b}$ is defined as

$${\displaystyle \frac{{\partial }^{\beta }u(x,t)}{{\partial \left|x\right|}^{\beta }}}=-c_{\beta }({\mathcal{D}}_{a,x}^{\beta }+{\mathcal{D}}_{x,b}^{\beta })u(x,t),$$

(4)

where

$$c_{\beta }={\displaystyle \frac{1}{2cos\left(\frac{\pi \beta }{2}\right)}},$$

${\displaystyle {\mathcal{D}}_{a,x}^{\beta }}$ and ${\displaystyle {\mathcal{D}}_{x,b}^{\beta }}$ denote the left and right Riemann–Liouville fractional derivatives, respectively.

3. Development of the Numerical Algorithm

This section develops a computational algorithm for the concerned model (1). Let ${\displaystyle {\omega }_{\tau }=\left\{t_k\right|t_k=k\tau ,k=0,1,\cdots ,N;\tau =T/N\}}$ be a uniform grid on the given time interval ${\displaystyle [0,T]}$. Similarly, we consider ${\displaystyle {\omega }_h=\left\{x_i\right|x_i=ih,i=0,1,\cdots ,M;h=1/M\}}$, a uniform grid in the space domain ${\displaystyle [0,1]}$. Let us define the values of the functions ${\displaystyle u(x,t)}$ and ${\displaystyle ð(x,t)}$ at the grid points by ${\displaystyle u_i^k=u(x_i,t_k)}$ and ${\displaystyle {ð}_i^k=ð(x_i,t_k)}$, ${\displaystyle 0\le i\le M}$, ${\displaystyle 0\le k\le N}$, respectively. The required time-fractional Caputo derivative can be approximated as [22] follows:

$$\begin{array}{cc}\hfill {\mathcal{D}}_{0,t}^{\alpha }u(x_i,t_k)=& {\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[\sum _{\begin{array}{c}m=1\end{array}}^{k-1}(G_{k-m+1}-G_{k-m}){\delta }_t{u}_i^m+G_1{\delta }_t{u}_i^k-G_k{\psi }_i\right]\hfill \\ & +{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[{\displaystyle \frac{{\delta }_t{u}_i^k-{\delta }_t{u}_i^{k-1}}{2^{2-\alpha }}}\right]+R_1+O\left({\tau }^{3-\alpha }\right)=A_{\tau }+R_1+O\left({\tau }^{3-\alpha }\right),\hfill \end{array}$$

(5)

where

$$R_1={\displaystyle \frac{1}{\mathsf{\Gamma }(2-\alpha )}}\sum _{\begin{array}{c}m=0\end{array}}^{k-1}{\int }_{t_{m-\frac{1}{2}}}^{t_{m+\frac{1}{2}}}\left(z_s(x_i,s)-{\displaystyle \frac{z_i^{m+\frac{1}{2}}-z_i^{m-\frac{1}{2}}}{\tau }}\right){(t_k-s)}^{1-\alpha }ds,$$

(6)

$${\psi }_i=\psi \left(x_i\right),z_t(x,t)={\displaystyle \frac{{\partial }^{2}u(x,t)}{\partial t^2}},z_i^{m+{\displaystyle \frac{1}{2}}}={\displaystyle \frac{u_i^{m+1}-u_i^m}{\tau }},{\delta }_t{u}_i^k={\displaystyle \frac{u_i^k-u_i^{k-1}}{\tau }},$$

(7)

and

$$G_i={(i+1/2)}^{2-\alpha }-{(i-1/2)}^{2-\alpha }.$$

(8)

In  [32], it is stated that the Riesz space-fractional derivative is equivalent to the corresponding non-integer power of the Laplacian operator when the considered boundary conditions are of Dirichlet type. Among the available approximations, the MTM is particularly effective due to its ease of implementation and the well-defined matrix representation of fractional operators.

We will use the standard second-order central difference to approximate second-order derivatives,

$${\displaystyle \frac{{\partial }^{2}u(x_i,t_k)}{\partial x^2}}\approx {\displaystyle \frac{u_{i+1}^k-2u_i^k+u_{i-1}^k}{h^2}},i=1,\cdots ,M-1.$$

(9)

The aforementioned approximations in (9) can be expressed in the corresponding matrix form as

$${\displaystyle \frac{{\partial }^2}{\partial x^2}}{U}^k=-T_h{U}^k+b,$$

(10)

where ${\displaystyle T_h\in {\mathbb{R}}^{M-1\times M-1}}$ and ${\displaystyle U^k,b\in {\mathbb{R}}^{M-1}}$ are given by

$$\begin{array}{c}\hfill T_h={\displaystyle \frac{1}{h^2}}\left[\begin{array}{cccccc}2& -1& 0& 0& \cdots & 0\\ -1& 2& -1& 0& \cdots & 0\\ 0& -1& 2& -1& \cdots & 0\\ ⋮& & \ddots & \ddots & \ddots & ⋮\\ 0& \cdots & \cdots & -1& 2& -1\\ 0& \cdots & \cdots & 0& -1& 2\end{array}\right],U^k=\left[\begin{array}{c}{u}_1^k\\ ⋮\\ ⋮\\ u_{M-1}^k\end{array}\right],b={\displaystyle \frac{1}{h^2}}\left[\begin{array}{c}{u}_0^k\\ 0\\ 0\\ ⋮\\ u_M^k\end{array}\right].\end{array}$$

(11)

In view of the Dirichlet boundary conditions at ${\displaystyle x=0}$ and ${\displaystyle x=1}$ given in (1), (10) reduces to

$${\displaystyle \frac{{\partial }^2}{\partial x^2}}{U}^k=-T_h{U}^k.$$

(12)

Now, following Definition 4 and [34], the fractional Laplacian can be approximated as

$${(-\mathsf{\Delta })}^{\beta /2}{U}^k={\left(-{\displaystyle \frac{{\partial }^2}{\partial x^2}}\right)}^{\beta /2}{U}^k={\left(T_h\right)}^{\beta /2}{U}^k.$$

(13)

In view of the matrix form in (11), we have ${\displaystyle T_h}$ as a positive definite symmetric matrix. Then, by a known result in the theory of linear algebra, if ${\displaystyle {\mathfrak{v}}_i,i=1,\cdots ,M-1}$, are eigenvectors of ${\displaystyle T_h}$ with eigenvalues ${\displaystyle {\lambda }_i,i=1,\cdots ,M-1}$, then there exists a non-singular matrix ${\displaystyle \mathcal{V}\in R^{M-1\times M-1}}$, represented as ${\displaystyle \mathcal{V}=[{\mathfrak{v}}_1,{\mathfrak{v}}_2,\cdots ,{\mathfrak{v}}_{M-1}]}$ such that ${\displaystyle T_h}$ can be written as follows:

$$T_h=\mathcal{V}\mathcal{L}{\mathcal{V}}^T,$$

where ${\displaystyle \mathcal{L}}$ is the diagonal matrix ${\displaystyle \mathcal{L}=diag({\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_{M-1})}$ [35]. Then, for ${\displaystyle \beta \in (1,2)}$, it is

$$T_h^{\beta /2}=\mathcal{V}{\mathcal{L}}^{\beta /2}{\mathcal{V}}^T,$$

(14)

with eigenvalues

$${\lambda }_i={\left[{\displaystyle \frac{4}{h^2}}si{n}^2\left({\displaystyle \frac{\pi i}{2M}}\right)\right]}^{\beta /2},i=1,\cdots ,M-1.$$

(15)

Moreover, ${\displaystyle T_h^{\beta /2}}$ is a real symmetric and centro-symmetric matrix of order ${\displaystyle M-1.}$ Note that, in view of Lemma 3, (13) can be expressed as

$$\begin{array}{cc}\hfill & {\left(-{\displaystyle \frac{{\partial }^2}{\partial x^2}}\right)}^{\beta /2}{U}^k=\left(\mathcal{V}{\mathcal{L}}^{\beta /2}{\mathcal{V}}^T\right)U^k.\hfill \end{array}$$

Substituting (5) and (13) into (1), we obtain a discretized form given by

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[\sum _{\begin{array}{c}m=1\end{array}}^{k-1}(G_{k-m+1}-G_{k-m}){\delta }_t{u}_i^m+G_1{\delta }_t{u}_i^k-G_k{\psi }_i\right]+{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[{\displaystyle \frac{{\delta }_t{u}_i^k-{\delta }_t{u}_i^{k-1}}{2^{2-\alpha }}}\right]\hfill \\ & +{\left(T_h\right)}_{r_i}^{\beta /2}{U}^k={ð}_i^k+R_i^k,i=1,\cdots ,M-1,\hfill \end{array}$$

(16)

where ${\displaystyle {\left(T_h\right)}_{r_i}^{\beta /2}}$ denotes the ${\displaystyle i}$-th row of the matrix ${\displaystyle {\left(T_h\right)}^{\beta /2}}$ and ${\displaystyle R_i^k}$ represents the remainder terms after using the numerical approximations of the fractional derivatives. Omitting the small term ${\displaystyle R_i^k}$ and replacing ${\displaystyle u_i^k}$ with its numerical approximation ${\displaystyle U_i^k}$, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[\sum _{\begin{array}{c}m=1\end{array}}^{k-1}(G_{k-m+1}-G_{k-m}){\delta }_t{U}_i^m+G_1{\delta }_t{U}_i^k-G_k{\psi }_i\right]+{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[{\displaystyle \frac{{\delta }_t{U}_i^k-{\delta }_t{U}_i^{k-1}}{2^{2-\alpha }}}\right]\hfill \\ & +{\left(T_h\right)}_{r_i}^{\beta /2}{U}^k={ð}_i^k,i=1,\cdots ,M-1,\hfill \end{array}$$

(17)

This can be rewritten as

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[G_1{\displaystyle \frac{U_i^k}{\tau }}\right]+{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[{\displaystyle \frac{U_i^k}{\left(\tau 2^{2-\alpha }\right)}}\right]+{\left(T_h\right)}_{r_i}^{\beta /2}{U}^k\hfill \\ & =-{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[\sum _{\begin{array}{c}m=1\end{array}}^{k-1}(G_{k-m+1}-G_{k-m}){\delta }_t{U}_i^m-G_1{\displaystyle \frac{U_i^{k-1}}{\tau }}-G_k{\psi }_i\right]\hfill \\ & -{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[{\displaystyle \frac{(-2U_i^{k-1}+U_i^{k-2})}{\left(\tau 2^{2-\alpha }\right)}}\right]+{ð}_i^k,i=1,\cdots ,M-1.\hfill \end{array}$$

(18)

Note that for ${\displaystyle k=1}$, the formula in (18) reduces to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[G_1{\displaystyle \frac{U_i^1}{\tau }}\right]+{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[{\displaystyle \frac{U_i^1}{\left(\tau 2^{2-\alpha }\right)}}\right]+{\left(T_h\right)}_{r_i}^{\beta /2}{U}^1\hfill \\ & =-{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[-G_1{\displaystyle \frac{U_i^0}{\tau }}-G_k{\psi }_i\right]-{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[{\displaystyle \frac{(-2U_i^0+U_i^{-1})}{\left(\tau 2^{2-\alpha }\right)}}\right]+{ð}_i^1,i=1,\cdots ,M-1.\hfill \end{array}$$

(19)

To obtain ${\displaystyle U_i^{-1}}$ and ${\displaystyle U_i^0}$, the initial conditions are used, which are provided as ${\displaystyle {\displaystyle \frac{\partial u(x,0)}{\partial t}}=\psi \left(x\right)}$ and ${\displaystyle u(x,0)=\varphi \left(x\right)}$. Hence, we obtain

$$u_i^0=\varphi \left(x_i\right)={\varphi }_i\mathrm{and}{u}_i^{-1}={\varphi }_i-\tau {\psi }_i.$$

Matrix Transformation of the Designed Scheme

Combining Equations (18) and (19), the matrix form of the scheme is given by

$$(W_t+T_h^{\beta /2})U^k=P$$

(20)

where ${\displaystyle \beta \in (1,2)}$, ${\displaystyle P=\left[p_{i,1}\right]}$, with 

$$\begin{array}{cc}\hfill & p_{i,1}=-{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[\sum _{\begin{array}{c}m=1\end{array}}^{k-1}(G_{k-m+1}-G_{k-m}){\delta }_t{U}_i^m-G_1{\displaystyle \frac{U_i^{k-1}}{\tau }}-G_k{\psi }_i\right]\hfill \\ \hfill & -{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[{\displaystyle \frac{(-2U_i^{k-1}+U_i^{k-2})}{\left(\tau 2^{2-\alpha }\right)}}\right]+{ð}_i^k,i=1,\cdots ,M-1,\hfill \end{array}$$

(21)

${\displaystyle W_t}$ is the ${\displaystyle (M-1)\times (M-1)}$-matrix given as ${\displaystyle W_t=aI}$ with ${\displaystyle I}$ the identity matrix of dimension ${\displaystyle M-1}$, and 

$$a={\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[{\displaystyle \frac{G_1}{\tau }}\right]+{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[{\displaystyle \frac{1}{\left(\tau 2^{2-\alpha }\right)}}\right]>1.$$

(22)

4. Theoretical Analysis

In this section, a detailed theoretical investigation is provided for the proposed scheme. The stability and convergence of the designed algorithm are established, along with some error bounds of the approximate solution.

4.1. Stability Analysis

Lemma 1.

Let ${\displaystyle \beta \in \left(1,2\right)}$, then the matrix ${\displaystyle T_h^{\beta /2}}$ defined in (14) is a diagonally dominant matrix.

Proof. 

We begin by expressing ${\displaystyle T_h}$ as ${\displaystyle T_h=\frac{2}{h^2}(I+\mathcal{Z})}$, where ${\displaystyle I}$ is the identity matrix of dimension ${\displaystyle M-1}$ and ${\displaystyle \mathcal{Z}}$ is a tridiagonal non-positive matrix given by ${\displaystyle \mathcal{Z}=\mathrm{tri}{\left[\frac{-1}{2},0,\frac{-1}{2}\right]}_{M-1\times M-1}}$. We obtain

$${\parallel \mathcal{Z}\parallel }_{max}=\underset{1\le k\le M-1}{max}\sum _{\begin{array}{c}j=1\end{array}}^{M-1}\left|{\left[Z\right]}_{k,j}\right|\le 1.$$

(23)

By utilizing the binomial expansion for ${\displaystyle {\displaystyle \frac{1}{2}}\le \beta /2\le 1}$, we obtain

$${(I+\mathcal{Z})}^{\beta /2}=\sum _{\begin{array}{c}k=0\end{array}}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\beta /2}{k}}\right){\mathcal{Z}}^k=I+\sum _{\begin{array}{c}k=1\end{array}}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\beta /2}{k}}\right){\mathcal{Z}}^k.$$

(24)

Now, combining (23) and [36] together, we obtain the inequality given below:

$${∥\sum _{\begin{array}{c}k=1\end{array}}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{\beta /2}{k}}\right){\mathcal{Z}}^k∥}_{max}\le \sum _{\begin{array}{c}k=1\end{array}}^{\infty }\left|\left({\displaystyle \genfrac{}{}{0pt}{}{\beta /2}{k}}\right)\right|{\parallel {\mathcal{Z}}^k\parallel }_{max}\le \sum _{\begin{array}{c}k=1\end{array}}^{\infty }\left|\left({\displaystyle \genfrac{}{}{0pt}{}{\beta /2}{k}}\right)\right|=1-\sum _{\begin{array}{c}k=0\end{array}}^{\infty }{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\beta /2}{k}}\right)=1.$$

(25)

We adopt for convention that ${\displaystyle q_{jk}}$ represent the entries of ${\displaystyle \sum _{\begin{array}{c}k=1\end{array}}^{\infty }\left(\genfrac{}{}{0pt}{}{\beta /2}{k}\right){\mathcal{Z}}^k}$. Therefore, we can express the entries of ${\displaystyle {(I+\mathcal{Z})}^{\beta /2}}$ as follows:

$$\begin{array}{c}\hfill {(I+\mathcal{Z})}^{\beta /2}=\left[a_{jk}\right]=\left\{\begin{array}{c}{q}_{jk}\mathrm{if}j\ne k,j,k=1,2,\cdots ,M-1,\hfill \\ 1+q_{jj}\mathrm{if}j=k,j,k=1,2,\cdots ,M-1.\hfill \end{array}\right.\end{array}$$

(26)

To prove that ${\displaystyle {(I+\mathcal{Z})}^{\beta /2}}$ is a diagonally dominant matrix, we must show that

$$\left|a_{jj}\right|-\sum _{\begin{array}{c}j=1,j\ne k\end{array}}^{M-1}\left|a_{jk}\right|\ge 0\forall j,j=1,\cdots ,M-1.$$

Considering the ${\displaystyle j}$-th row of ${\displaystyle {(I+\mathcal{Z})}^{\beta /2}}$ and using (25), we arrive at

$$\begin{array}{cc}\hfill |a_{jj}|-\sum _{j=1,j\ne k}^{M-1}\left|a_{jk}\right|& =|1+q_{jj}|-\sum _{j=1,j\ne k}^{M-1}\left|q_{jk}\right|\hfill \\ \hfill & \ge 1-\sum _{j=1,j\ne k}^{M-1}\left|q_{jk}\right|\hfill \\ \hfill & \ge 0.\hfill \end{array}$$

This proves that ${\displaystyle {(I+\mathcal{Z})}^{\beta /2}={\displaystyle \frac{h^{\beta }}{{\left(2\right)}^{\beta /2}}}{T}_h^{\beta /2}}$ is diagonally dominant. Hence, ${\displaystyle T_h^{\beta /2}}$ is also diagonally dominant.    ${\displaystyle \square }$

The following lemma is essential to obtain the desired result.

Lemma 2

([37]). Let ${\displaystyle \mathcal{J}}$ be a diagonally dominant matrix such that ${\displaystyle \xi =\underset{k}{min}\left\{|{\mathcal{J}}_{kk}|-\sum _{\begin{array}{c}j=1,j\ne k\end{array}}^{M-1}\left|{\mathcal{J}}_{kj}\right|\right\}}$. Then, ${\displaystyle \mathcal{J}}$ is an invertible matrix, with  ${\displaystyle \parallel {\mathcal{J}}^{-1}{\parallel }_{max}\le {\displaystyle \frac{1}{\xi }}}$.

Theorem 1.

For ${\displaystyle \beta \in (1,2)}$, the developed numerical algorithm (20) is unconditionally stable.

Proof. 

According to Lemma 1, ${\displaystyle T_h^{\beta /2}}$ is diagonally dominant with positive diagonal entries. Considering the ${\displaystyle k}$-th row of ${\displaystyle W_t+T_h^{\beta /2}}$, we have

$$a+{\left[T_h^{\beta /2}\right]}_{kk}-\sum _{\begin{array}{c}j=1,j\ne k\end{array}}^{M-1}\left|{\left[T_h^{\beta /2}\right]}_{kj}\right|\ge a>1,$$

(27)

and thus, ${\displaystyle W_t+T_h^{\beta /2}}$ is also diagonally dominant.

Let us denote ${\displaystyle \xi =\underset{k}{min}\left\{|{[W_t+T_h^{\beta /2}]}_{kk}|-\sum _{j=1,j\ne k}^{M-1}|{[W_t+T_h^{\beta /2}]}_{kj}|\right\}}$. Therefore, applying Lemma 2 to ${\displaystyle W_t+T_h^{\beta /2}}$, we observe that ${\displaystyle \parallel W_t+T_h^{\beta /2}{\parallel }^{-1}\le {\displaystyle \frac{1}{\xi }}\le 1}$, which satisfies the stability condition in the ${\displaystyle {\parallel ·\parallel }_{max}}$-norm.    ${\displaystyle \square }$

4.2. Optimal Error Bounds and Convergence Analysis

Lemma 3.

Given ${\displaystyle \beta \in (1,2)}$ and ${\displaystyle l\in \mathbb{Z},1\le l\le M-1}$, there exists a real constant ${\displaystyle M_{\beta }}$ such that

$${\left(l\pi \right)}^{\beta }-{\left({\displaystyle \frac{4}{h^2}}{sin}^2\left({\displaystyle \frac{l\pi h}{2}}\right)\right)}^{\beta /2}\le M_{\beta }{h}^{\beta }{\left(l\pi \right)}^{2\beta }.$$

(28)

Proof. 

Firstly, since ${\displaystyle sin\left(x\right)\le x}$ for all ${\displaystyle x\ge 0}$, we have

$$B={\displaystyle \frac{4}{h^2}}{sin}^2\left({\displaystyle \frac{l\pi h}{2}}\right)\le {\left(l\pi \right)}^2=A.$$

(29)

Secondly, from the Taylor expansion of ${\displaystyle {sin}^2(l\pi h/2)}$ around ${\displaystyle h=0}$, we have

$${\displaystyle \frac{4}{h^2}}{sin}^2\left({\displaystyle \frac{l\pi h}{2}}\right)={\left(l\pi \right)}^2-h^2{\displaystyle \frac{{\left(l\pi \right)}^4}{12}}cos\left({\theta }_l\right),$$

where ${\displaystyle {\theta }_l}$ is in a neighborhood of the origin. Note that in view of the inequality in (29) and the even symmetry of the cosine function, we may assume that ${\displaystyle {\theta }_l\in [0,\pi /2]}$. Now, for ${\displaystyle \beta \in \left(1,2\right)}$, using the inequality ${\displaystyle A^{\beta /2}-B^{\beta /2}\le {(A-B)}^{\beta /2}}$ for ${\displaystyle A\ge B>0}$, with the values in (29), we obtain

${\displaystyle {\left(l\pi \right)}^{\beta }-{\left({\displaystyle \frac{4}{h^2}}{sin}^2\left({\displaystyle \frac{l\pi h}{2}}\right)\right)}^{\beta /2}\le {\left(h^2{\displaystyle \frac{{\left(l\pi \right)}^4}{12}}cos{\theta }_l\right)}^{\beta /2}={\displaystyle \frac{1}{{12}^{\beta /2}}}{h}^{\beta }{\left(l\pi \right)}^{2\beta },\forall 1\le l\le M-1}$.    ${\displaystyle \square }$

Now, we introduce the operator ${\displaystyle {\widehat{T}}_h^{\beta /2}}$: ${\displaystyle S_M\to S_M}$ defined as

$${\widehat{T}}_h^{\beta /2}f\left(x\right)=\sum _{l=1}^{M-1}{f}_l{\left({\displaystyle \frac{4}{h^2}}{sin}^2\left({\displaystyle \frac{l\pi h}{2}}\right)\right)}^{\beta /2}sin\left(l\pi x\right).$$

The operators ${\displaystyle {\widehat{T}}_h^{\beta /2}}$ and ${\displaystyle T_h^{\beta /2}}$ are equivalent in the sense that

$$\left[{\widehat{T}}_h^{\beta /2}f\left(x\right)\right]{|}_{{\mathsf{\Omega }}_h}=T_h^{\beta /2}\left[f\left(x\right){|}_{{\mathsf{\Omega }}_h}\right],$$

with ${\displaystyle {\mathsf{\Omega }}_h=\left\{x_i=ih,1\le i\le M-1\right\}}$.

Theorem 2.

Given ${\displaystyle \beta \in (1,2)}$ and ${\displaystyle u\in S_M}$, there exists a positive real constant ${\displaystyle C_m}$ such that

$${\parallel (-\mathsf{\Delta })}^{\beta /2}u-{\widehat{T}}_h^{\beta /2}u\parallel \le C_m{h}^{\beta }{\left|u\right|}_{\beta +1}.$$

Proof. 

Since ${\displaystyle u\in S_M}$, it can be expressed as ${\displaystyle u\left(x\right)=\sum _{l=1}^{M-1}{u}_{l}sin\left(l\pi x\right)}$. Thus, we can write

$$\begin{array}{c}\hfill {(-\mathsf{\Delta })}^{\beta /2}u\left(x\right)=\sum _{l=1}^{M-1}{u}_l{\left[{\left(l\pi \right)}^2\right]}^{\beta /2}sin\left(l\pi x\right),\mathrm{and}\\ \hfill {\widehat{T}}_h^{\beta /2}u\left(x\right)=\sum _{l=1}^{M-1}{u}_l{\left({\displaystyle \frac{4}{h^2}}{sin}^2\left({\displaystyle \frac{l\pi h}{2}}\right)\right)}^{\beta /2}sin\left(l\pi x\right).\end{array}$$

Now, using (28) and Definition 2 we have

$$\begin{array}{cc}\hfill {\left||{(-\mathsf{\Delta })}^{\alpha /2}u-T_h^{\beta /2}u|\right|}^2& =\sum _{l=1}^{M-1}{u}_l^2\left({\left[{\left(l\pi \right)}^2\right]}^{\beta /2}-{\left({\displaystyle \frac{4}{h^2}}{sin}^2\left({\displaystyle \frac{l\pi h}{2}}\right)\right)}^{\beta /2}\right)\hfill \\ \hfill & \le M_{\beta }{h}^{\beta }\sum _{l=1}^{M-1}{u}_l^2{\left[{\left(l\pi \right)}^2\right]}^{\beta }\le M_{\beta }{h}^{\beta }\sum _{l=1}^{M-1}{u}_l^2{\left[{\left(l\pi \right)}^2\right]}^{(\beta +1)}\hfill \\ \hfill & =M_{\beta }{h}^{\beta }{\left|u\right|}_{\beta +1}^2.\hfill \end{array}$$

(30)

The required result follows immediately.    ${\displaystyle \square }$

Lemma 4.

Assuming that ${\displaystyle u_{ttt}}$ is bounded on ${\displaystyle (0,1)\times (-t_1,T)}$, for the discrete approximation of the Caputo fractional derivative,

$$\begin{array}{cc}\hfill {\mathcal{D}}_{0,t}^{\alpha }U(x_i,t_k)& \simeq {\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[\sum _{\begin{array}{c}m=1\end{array}}^{k-1}(G_{k-m+1}-G_{k-m}){\delta }_t{U}_i^m+G_1{\delta }_t{U}_i^k-G_k{\psi }_i\right]\hfill \\ & +{\displaystyle \frac{{\tau }^{1-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}\left[{\displaystyle \frac{{\delta }_t{U}_i^k-U_i^{k-1}}{2^{2-\alpha }}}\right]+R_1,\hfill \end{array}$$

there exists a positive real constant ${\displaystyle C}$ such that ${\displaystyle \left|R_1\right|\le C{\tau }^{3-\alpha }.}$

Proof. 

See [22].    ${\displaystyle \square }$

Lemma 5.

The error estimates of the remainder terms in (16) are given by

$$\parallel R_i^k\parallel \le M_{\alpha ,\beta }({\tau }^{3-\alpha }+h^{\beta }),$$

(31)

where ${\displaystyle M_{\alpha ,\beta }\in {\mathbb{R}}^{+}}$.

Proof. 

Since ${\displaystyle \parallel R_i^k\parallel }$ represents the remainder terms of the numerical approximations of the fractional derivatives, we have

$$\begin{array}{cc}\hfill & \parallel R_i^k\parallel =\parallel {\mathcal{D}}_{0,t}^{\alpha }u(x_i,t_k)+{(-\mathsf{\Delta })}^{\beta /2}u(x_i,t_k)-A_{\tau }-{\widehat{T}}_h^{\beta /2}u(x_i,t_k)\parallel .\hfill \end{array}$$

By the triangle inequality we observe that

$$\begin{array}{cc}\hfill \parallel {\mathcal{D}}_{0,t}^{\alpha }u(x_i,t_k)+& {(-\mathsf{\Delta })}^{\beta /2}u(x_i,t_k)-A_{\tau }-{\widehat{T}}_h^{\beta /2}u(x_i,t_k)\parallel \hfill \\ \hfill & \le \parallel {\mathcal{D}}_{0,t}^{\alpha }u(x_i,t_k)-A_{\tau }\parallel +\parallel {(-\mathsf{\Delta })}^{\beta /2}u(x_i,t_k)-{\widehat{T}}_h^{\beta /2}u(x_i,t_k)\parallel .\hfill \end{array}$$

Now, from Lemma 4 and Theorem 2, we obtain

$$\parallel {\mathcal{D}}_{0,t}^{\alpha }u(x_i,t_k)-A_{\tau }\parallel \le C{\tau }^{3-\alpha },$$

and

$${\parallel (-\mathsf{\Delta })}^{\beta /2}u(x_i,t_k)-{\widehat{T}}_h^{\beta /2}u(x_i,t_k)\parallel \le C_m{h}^{\beta }{\left|u\right|}_{\beta +1}.$$

Thus, combining these estimates gives (31).

$$\parallel R_i^k\parallel \le \parallel {\mathcal{D}}_{0,t}^{\alpha }u(x_i,t_k)-A_{\tau }\parallel +\parallel {(-\mathsf{\Delta })}^{\beta /2}u(x_i,t_k)-{\widehat{T}}_h^{\beta /2}u(x_i,t_k)\parallel \le M_{\alpha ,\beta }({\tau }^{3-\alpha }+h^{\beta }).$$

   ${\displaystyle \square }$

Theorem 3.

The computational algorithm (20) is convergent for any ${\displaystyle \beta \in \left(1,2\right)}$.

Proof. 

Let us define

$$e_i^k=u_i^k-U_i^k,0\le i\le M,0\le k\le N,$$

where ${\displaystyle u_k^i}$ is the exact solution and ${\displaystyle U_k^i}$ its numerical approximation.

Now subtracting (17) from (16), we obtain

$${\mathsf{\Pi }}_1\left(e_i^k\right)={\mathsf{\Pi }}_2\left(e_i^{k-1}\right)+R_i^k,1\le i\le M-1,1\le k\le N,$$

(32)

with boundary and initial conditions

$$e_0^k=e_M^k=0,1\le k\le N,$$

(33)

$$e_i^0=0,0\le i\le M.$$

(34)

Here,

$${\mathsf{\Pi }}_1\left(e_i^k\right)=d\left(G_1+{\displaystyle \frac{1}{2^{2-\alpha }}}\right)e_i^k+{\left(T_h\right)}_{r_i}^{\beta /2}{e}_i^k,$$

(35)

$${\mathsf{\Pi }}_2\left(e_i^{k-1}\right)=-d\left[\sum _{\begin{array}{c}m=1\end{array}}^{k-1}(G_{k-m+1}-G_{k-m})(e_i^m-e_i^{m-1})-G_1{e}_i^{k-1}+{\displaystyle \frac{-2e_i^{k-1}+e_i^{k-2}}{2^{2-\alpha }}}\right],$$

(36)

where ${\displaystyle d={\displaystyle \frac{{\tau }^{-\alpha }}{\mathsf{\Gamma }(3-\alpha )}}}$.

Taking norms, we obtain

$$\begin{array}{c}\hfill \parallel {\mathsf{\Pi }}_1\left(e_i^k\right)\parallel =\left|d\right|∥\left(G_1+{\displaystyle \frac{1}{2^{2-\alpha }}}\right)e_i^k+{\left(T_h\right)}_{r_i}^{\beta /2}{e}_i^k∥\end{array}$$

(37)

and

$$\begin{array}{cc}\hfill \parallel {\mathsf{\Pi }}_2\left(e_i^{k-1}\right)\parallel & =|-d|∥\left[\sum _{\begin{array}{c}m=1\end{array}}^{k-1}(G_{k-m+1}-G_{k-m})(e_i^m-e_i^{m-1})-G_1{e}_i^{k-1}+{\displaystyle \frac{-2e_i^{k-1}+e_i^{k-2}}{2^{2-\alpha }}}\right]∥\hfill \\ \hfill & \le \left|d\right|\left[∥\sum _{\begin{array}{c}m=1\end{array}}^{k-1}(G_{k-m+1}-G_{k-m})(e_i^m-e_i^{m-1})∥+\parallel G_1{e}_i^{k-1}\parallel +∥{\displaystyle \frac{-2e_i^{k-1}+e_i^{k-2}}{2^{2-\alpha }}}∥\right]\hfill \\ \hfill & \le C\parallel e_i^{k-1}\parallel ,\hfill \end{array}$$

(38)

where ${\displaystyle C}$ is a positive real constant. Now, using (32) and (38), we obtain

$$\begin{array}{c}\hfill \parallel e_i^k\parallel \le \parallel {\mathsf{\Pi }}_1\left(e_i^k\right)\parallel =\parallel {\mathsf{\Pi }}_2\left(e_i^{k-1}\right)+R_i^k\parallel \le C\parallel e_i^{k-1}\parallel +\parallel R_i^k\parallel .\end{array}$$

(39)

For ${\displaystyle k=1}$, we have

$$\parallel e_i^1\parallel \le C\parallel e_i^0\parallel +\parallel R_i^1\parallel .$$

(40)

Using Lemma 5 together with (34) and (40) in (39), we obtain

$$\parallel e_i^k\parallel \le C({\tau }^{3-\alpha }+h^{\beta }),1\le k\le N,1\le i\le M-1,$$

which demonstrates the convergence of the proposed scheme, with ${\displaystyle (3-\alpha )}$-th order of convergence in time and ${\displaystyle \beta }$-th order of convergence in space.    ${\displaystyle \square }$

5. Numerical Experiments

This section validates the theoretical results using numerical experiments based on the proposed algorithm for problem (1). Several test functions are considered. A pseudocode of the algoritm is given in Algorithm 1. We report a detailed analysis of ${\displaystyle L_2}$-errors and convergence rates. The errors are compared against those obtained by the scheme of Singh and Mehra [30]. The error metrics are defined as follows [35]:

$$\begin{array}{c}\hfill E^M=\left\{\begin{array}{c}{\left({\displaystyle \sum _{\begin{array}{c}i=1\end{array}}^{M-1}}h{|U(x_i,T)-u(x_i,T)|}^2\right)}^{1/2},\mathrm{if}\mathrm{the}\mathrm{exact}\mathrm{solution}\mathrm{is}\mathrm{known},\hfill \\ \\ {\left({\displaystyle \sum _{\begin{array}{c}i=1\end{array}}^{M-1}}h{|U_{2i}^{2M,N}-U_i^{M,N}|}^2\right)}^{1/2},\mathrm{if}\mathrm{the}\mathrm{exact}\mathrm{solution}\mathrm{is}\mathrm{unknown},\hfill \end{array}\right.\end{array}$$

where ${\displaystyle U^{2M,N}}$ is the numerical solution when the number of subintervals is doubled in space, whereas ${\displaystyle U^{M,N}}$ is a solution on a mesh with ${\displaystyle M}$ spatial and ${\displaystyle N}$ temporal subintervals, respectively. The convergence rates have been obtained using the formula

$$\mathrm{Rate}={log}_{{\displaystyle \frac{M_1}{M_2}}}{\displaystyle \frac{E^{M_1}}{E^{M_2}}},$$

where ${\displaystyle E^{M_i}}$ denotes the error corresponding to a spatial grid with ${\displaystyle M_i}$ subintervals.

This section also serves to corroborate the numerical stability of the designed scheme. To do this, we first add some noisy data to the initial parameters of the subjected test function as prescribed below. Given the initial values ${\displaystyle \varphi \left(x_i\right)}$ for (1), let ${\displaystyle {\varphi }^{ϵ}\left(x_i\right)}$ be the initial values with an additive noise ${\displaystyle ϵ}$, that is,

$${\varphi }^{ϵ}\left(x_i\right)=\varphi \left(x_i\right)+ϵ.$$

We have taken the value of ${\displaystyle ϵ=\omega \%}$ of the maximum point-wise errors. To examine the numerical stability we have taken Example 1 into account with ${\displaystyle M=512,ϵ=20\%}$. In 

Table 1. ${\displaystyle L_2}$ errors and convergence rates of Example 1 for ${\displaystyle \alpha =1.5}$, ${\displaystyle \beta =1.2}$ and ${\displaystyle N=1000}$ via scheme-I [30] and the proposed algorithm.

${\displaystyle \mathit{M}}$	Scheme-I [30]		Proposed Scheme
	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error with ${\displaystyle \mathit{ϵ}}$
8	${\displaystyle 5.900\times {10}^{-3}}$	-	${\displaystyle 1.112\times {10}^{-4}}$	-	${\displaystyle 1.112\times {10}^{-4}}$
16	${\displaystyle 3.800\times {10}^{-3}}$	0.63	${\displaystyle 2.627\times {10}^{-5}}$	2.08	${\displaystyle 2.627\times {10}^{-5}}$
32	${\displaystyle 2.200\times {10}^{-3}}$	0.78	${\displaystyle 6.524\times {10}^{-6}}$	2.06	${\displaystyle 6.524\times {10}^{-6}}$
64	${\displaystyle 1.200\times {10}^{-3}}$	0.87	${\displaystyle 1.639\times {10}^{-6}}$	2.00	${\displaystyle 1.639\times {10}^{-6}}$
128	${\displaystyle 6.170\times {10}^{-4}}$	0.95	${\displaystyle 4.140\times {10}^{-7}}$	2.00	${\displaystyle 4.140\times {10}^{-7}}$
256	${\displaystyle 3.140\times {10}^{-4}}$	0.97	${\displaystyle 1.051\times {10}^{-7}}$	1.99	${\displaystyle 1.051\times {10}^{-7}}$
512	${\displaystyle 1.570\times {10}^{-4}}$	0.99	${\displaystyle 2.690\times {10}^{-8}}$	1.99	${\displaystyle 2.690\times {10}^{-8}}$

, we have shown that there is a negligible change in the solution when noise is added for ${\displaystyle \alpha =1.5,\beta =1.2}$. Similar results can be seen in

Table 2. ${\displaystyle L_2}$ errors and convergence rates of Example 1 for ${\displaystyle \alpha =1.5}$, ${\displaystyle \beta =1.5}$ and ${\displaystyle N=1000}$ via scheme-I [30] and the proposed algorithm.

${\displaystyle \mathit{M}}$	Scheme-I [30]		Proposed Scheme
	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error with ${\displaystyle \mathit{ϵ}}$
8	${\displaystyle 1.500\times {10}^{-3}}$	-	${\displaystyle 1.479\times {10}^{-4}}$	-	${\displaystyle 1.479\times {10}^{-4}}$
16	${\displaystyle 9.470\times {10}^{-4}}$	0.66	${\displaystyle 3.487\times {10}^{-5}}$	2.08	${\displaystyle 3.487\times {10}^{-5}}$
32	${\displaystyle 5.290\times {10}^{-4}}$	0.84	${\displaystyle 8.619\times {10}^{-6}}$	2.02	${\displaystyle 8.619\times {10}^{-6}}$
64	${\displaystyle 2.780\times {10}^{-4}}$	0.92	${\displaystyle 2.149\times {10}^{-6}}$	2.01	${\displaystyle 2.149\times {10}^{-6}}$
128	${\displaystyle 1.410\times {10}^{-4}}$	0.97	${\displaystyle 5.371\times {10}^{-7}}$	2.00	${\displaystyle 5.371\times {10}^{-7}}$
256	${\displaystyle 7.000\times {10}^{-5}}$	1.01	${\displaystyle 1.343\times {10}^{-7}}$	2.00	${\displaystyle 1.343\times {10}^{-7}}$
512	${\displaystyle 3.400\times {10}^{-5}}$	1.05	${\displaystyle 3.358\times {10}^{-8}}$	1.99	${\displaystyle 3.358\times {10}^{-8}}$

and 

Table 3. ${\displaystyle L_2}$ errors and convergence rates of Example 1 for ${\displaystyle \alpha =1.5}$, ${\displaystyle \beta =1.8}$ and ${\displaystyle N=1000}$ via scheme-I [30] and the proposed algorithm.

${\displaystyle \mathit{M}}$	Scheme-I [30]		Proposed Scheme
	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error with ${\displaystyle \mathit{ϵ}}$
8	${\displaystyle 3.520\times {10}^{-4}}$	-	${\displaystyle 1.866\times {10}^{-4}}$	-	${\displaystyle 1.866\times {10}^{-4}}$
16	${\displaystyle 7.300\times {10}^{-5}}$	2.27	${\displaystyle 4.291\times {10}^{-5}}$	2.12	${\displaystyle 4.291\times {10}^{-5}}$
32	${\displaystyle 4.200\times {10}^{-5}}$	0.81	${\displaystyle 1.056\times {10}^{-6}}$	2.02	${\displaystyle 1.056\times {10}^{-6}}$
64	${\displaystyle 2.900\times {10}^{-5}}$	0.51	${\displaystyle 2.631\times {10}^{-6}}$	2.01	${\displaystyle 2.631\times {10}^{-6}}$
128	${\displaystyle 1.600\times {10}^{-5}}$	0.87	${\displaystyle 6.572\times {10}^{-7}}$	2.00	${\displaystyle 6.572\times {10}^{-7}}$
256	${\displaystyle 7.000\times {10}^{-6}}$	1.10	${\displaystyle 1.642\times {10}^{-7}}$	2.00	${\displaystyle 1.642\times {10}^{-7}}$
512	${\displaystyle 3.000\times {10}^{-6}}$	1.22	${\displaystyle 4.107\times {10}^{-8}}$	2.00	${\displaystyle 4.107\times {10}^{-8}}$

for ${\displaystyle \alpha =1.5,}$ ${\displaystyle \beta =1.5,1.8}$.

Algorithm 1: Numerical solution of problem (1) using the proposed MTM/time discretization

Example 1

([30]). Let us consider

$$\begin{array}{cc}\hfill & u(x,t)=(2+t+t^{(\alpha +2)})x^4{(1-x)}^4,\hfill \end{array}$$

to be the exact solution of (1), where

$$T=1,\varphi \left(x\right)=2x^4{(1-x)}^4,\psi \left(x\right)=x^4{(1-x)}^4,ð(x,t)=P+Q+R,$$

with

$$\begin{array}{cc}\hfill & P=x^4{(1-x)}^4\left[{\displaystyle \frac{\mathsf{\Gamma }(\alpha +3)}{\mathsf{\Gamma }\left(3\right)}}{t}^2\right],\hfill \\ \hfill & Q={\displaystyle \frac{(t^{\alpha +2}+t+2)}{2cos(\beta \pi /2)}}\left[{\displaystyle \frac{\mathsf{\Gamma }\left(9\right)}{\mathsf{\Gamma }(9-\beta )}}{x}^{8-\beta }-{\displaystyle \frac{4\mathsf{\Gamma }\left(8\right)}{\mathsf{\Gamma }(8-\beta )}}{x}^{7-\beta }+{\displaystyle \frac{6\mathsf{\Gamma }\left(7\right)}{\mathsf{\Gamma }(7-\beta )}}{x}^{6-\beta }\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{4\mathsf{\Gamma }\left(6\right)}{\mathsf{\Gamma }(6-\beta )}}{x}^{5-\beta }+{\displaystyle \frac{\mathsf{\Gamma }\left(5\right)}{\mathsf{\Gamma }(5-\beta )}}{x}^{4-\beta }\right],\hfill \\ \hfill & R={\displaystyle \frac{(t^{\alpha +2}+t+2)}{2cos(\beta \pi /2)}}\left[{\displaystyle \frac{\mathsf{\Gamma }\left(9\right)}{\mathsf{\Gamma }(9-\beta )}}{(1-x)}^{8-\beta }-{\displaystyle \frac{4\mathsf{\Gamma }\left(8\right)}{\mathsf{\Gamma }(8-\beta )}}{(1-x)}^{7-\beta }+{\displaystyle \frac{6\mathsf{\Gamma }\left(7\right)}{\mathsf{\Gamma }(7-\beta )}}{(1-x)}^{6-\beta }\right]\hfill \\ \hfill & +{\displaystyle \frac{(t^{\alpha +2}+t+2)}{2cos(\beta \pi /2)}}\left[-{\displaystyle \frac{4\mathsf{\Gamma }\left(6\right)}{\mathsf{\Gamma }(6-\beta )}}{(1-x)}^{5-\beta }+{\displaystyle \frac{\mathsf{\Gamma }\left(5\right)}{\mathsf{\Gamma }(5-\beta )}}{(1-x)}^{4-\beta }\right].\hfill \end{array}$$

Discussion of results for Example 1:

• Table 1 shows the detailed numerical results corresponding to Example 1 after applying Algorithm 1. This table highlights the comparison drawn for ${\displaystyle L_2}$ errors and spatial convergence rates among the scheme in [30] and the proposed algorithm for ${\displaystyle \alpha =1.5}$ and ${\displaystyle \beta =1.2}$. It can be seen that the proposed algorithm produces higher accuracy and ${\displaystyle \beta }$-order convergence in the space domain.
• Table 2 shows ${\displaystyle L_2}$ errors and spatial convergence rates for ${\displaystyle \alpha =1.5}$ and ${\displaystyle \beta =1.5}$. Table 3 shows the ${\displaystyle L_2}$ errors and convergence rates for ${\displaystyle \alpha =1.5}$ and ${\displaystyle \beta =1.8}$. Note that the convergence rates in Table 1, Table 2 and Table 3 agree with the theoretical findings.
• In Table 3, it is observed that when the values of ${\displaystyle \beta }$ are taken close to the final value ${\displaystyle \beta =2}$, the given numerical scheme achieves second-order convergence in the spatial direction.
• Along with the above findings, Table 1, Table 2 and Table 3 demonstrate the numerical stability of the proposed method: errors with added noise match those without perturbations.
• Table 4. ${\displaystyle L_2}$ errors and convergence rates of Example 1 for ${\displaystyle \beta =1.5}$ and ${\displaystyle M=1000}$ via the proposed scheme.

  ${\displaystyle \mathit{N}}$	${\displaystyle (\mathit{\alpha }=1.1)}$		${\displaystyle (\mathit{\alpha }=1.5)}$		${\displaystyle (\mathit{\alpha }=1.8)}$
  	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate
  16	${\displaystyle 2.982\times {10}^{-6}}$	-	${\displaystyle 3.476\times {10}^{-5}}$	-	${\displaystyle 1.178\times {10}^{-4}}$	-
  32	${\displaystyle 8.083\times {10}^{-7}}$	1.88	${\displaystyle 1.246\times {10}^{-5}}$	1.47	${\displaystyle 5.504\times {10}^{-5}}$	1.09
  64	${\displaystyle 2.115\times {10}^{-7}}$	1.93	${\displaystyle 4.377\times {10}^{-6}}$	1.50	${\displaystyle 2.496\times {10}^{-5}}$	1.14
  128	${\displaystyle 5.240\times {10}^{-8}}$	2.01	${\displaystyle 1.519\times {10}^{-6}}$	1.52	${\displaystyle 1.113\times {10}^{-5}}$	1.16
  256	${\displaystyle 1.159\times {10}^{-8}}$	2.17	${\displaystyle 5.217\times {10}^{-7}}$	1.54	${\displaystyle 4.925\times {10}^{-6}}$	1.17

  shows that the derived scheme provides ${\displaystyle (3-\alpha )}$-th order of convergence in the temporal direction, which agrees with the theoretical analysis.
• Figure 1 and Figure 2 illustrate the agreement between the exact solution and the error plot for Example 1 when ${\displaystyle \alpha =1.5,\beta =1.5}$ and taking ${\displaystyle M=1000}$.

Example 2.

Let us consider problem (1), where

$$\begin{array}{cc}\hfill & T=1,\varphi \left(x\right)=2x^4{(1-x)}^4,\psi \left(x\right)=x^4{(1-x)}^4,\hfill \\ \hfill & ð(x,t)={\displaystyle \frac{2x^2{(1-x)}^2\left(t^{2-\left(\alpha \right)}\right)}{\mathsf{\Gamma }(3-\alpha )}}\hfill \\ \hfill & +{\displaystyle \frac{t^2\left(((3-\beta )(4-\beta ){(1-x)}^{2-\beta }+x^{2-\beta }))\right)}{\mathsf{\Gamma }(5-\beta )cos(\beta \pi /2)}}\hfill \\ \hfill & +{\displaystyle \frac{t^2\left(-6(4-\beta )\left((1-x)\right(3-\beta )+x^{3-\beta })+12({(1-x)}^{3-\beta }+x^{3-\beta })\right)}{\mathsf{\Gamma }(5-\beta )cos(\beta \pi /2)}}.\hfill \end{array}$$

Discussion of results for Example 2:

• ${\displaystyle L_2}$ errors and convergence rates for Example 2 via the derived scheme are reflected in

  Table 5. ${\displaystyle L_2}$ errors and convergence rates of Example 2 for ${\displaystyle \alpha =1.5}$ and ${\displaystyle N=1000}$ via the proposed algorithm.

  ${\displaystyle \mathit{M}}$	${\displaystyle (\mathit{\beta }=1.2)}$		${\displaystyle (\mathit{\beta }=1.5)}$		${\displaystyle (\mathit{\beta }=1.8)}$
  	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate
  8	${\displaystyle 1.011\times {10}^{-3}}$	-	${\displaystyle 2.282\times {10}^{-4}}$	-	${\displaystyle 6.121\times {10}^{-4}}$	-
  16	${\displaystyle 3.692\times {10}^{-4}}$	1.57	${\displaystyle 6.553\times {10}^{-5}}$	1.80	${\displaystyle 1.464\times {10}^{-4}}$	2.06
  32	${\displaystyle 1.218\times {10}^{-4}}$	1.60	${\displaystyle 1.887\times {10}^{-5}}$	1.79	${\displaystyle 3.524\times {10}^{-5}}$	2.05
  64	${\displaystyle 3.854\times {10}^{-5}}$	1.65	${\displaystyle 5.302\times {10}^{-6}}$	1.83	${\displaystyle 8.519\times {10}^{-6}}$	2.04
  128	${\displaystyle 1.208\times {10}^{-5}}$	1.69	${\displaystyle 1.454\times {10}^{-6}}$	1.86	${\displaystyle 2.070\times {10}^{-6}}$	2.04

  for the space domain. The results were obtained for ${\displaystyle \alpha =1.5}$ and ${\displaystyle \beta =1.2,1.5}$, 1.8.
• The scheme consistently achieves ${\displaystyle \beta }$-order convergence in space, matching theoretical predictions.

Example 3.

Let us consider problem (1), where

$$T=1,\varphi \left(x\right)=sin\left(x\right),\psi \left(x\right)=0,$$

${\displaystyle ð(x,t)=P+Q+R,}$ with

$$\begin{array}{cc}\hfill & P={\displaystyle \frac{2x^3{(1-x)}^2\left(t^{2-\left(\alpha \right)}\right)}{\mathsf{\Gamma }(3-\alpha )}}+{\displaystyle \frac{(1+t^2)\left(((3-\beta )(4-\beta )(5-\beta ){(1-x)}^{2-\beta }+x^{2-\beta })\right)}{\mathsf{\Gamma }(6-\beta )cos(\beta \pi /2)}}\hfill \\ \hfill & Q={\displaystyle \frac{(1+t^2)\left((3(4-\beta )(5-\beta )(-3{(1-x)}^{3-\beta }+x^{3-\beta })-12(5-\beta )(-3{(1-x)}^{4-\beta }+2x^{4-\beta }))\right)}{\mathsf{\Gamma }(6-\beta )cos(\beta \pi /2)}}\hfill \\ \hfill & R={\displaystyle \frac{(1+t^2)\left(60(-{(1-x)}^{5-\beta }+x^{5-\beta })\right)}{\mathsf{\Gamma }(6-\beta )cos(\beta \pi /2)}}.\hfill \end{array}$$

Discussion of results for Example 3:

• ${\displaystyle L_2}$ errors and convergence rates via derived numerical techniques are provided in

  Table 6. ${\displaystyle L_2}$ errors and convergence rates of Example 3 for ${\displaystyle \alpha =1.5,N=1000}$, and different values of ${\displaystyle \beta }$ via the proposed algorithm.

  ${\displaystyle \mathit{M}}$	${\displaystyle (\mathit{\beta }=1.2)}$		${\displaystyle (\mathit{\beta }=1.5)}$		${\displaystyle (\mathit{\beta }=1.8)}$
  	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate	${\displaystyle {\mathit{L}}_{\mathbf{2}}}$ Error	Rate
  8	${\displaystyle 5.041\times {10}^{-2}}$	-	${\displaystyle 1.443\times {10}^{-2}}$	-	${\displaystyle 5.121\times {10}^{-3}}$	-
  16	${\displaystyle 1.692\times {10}^{-2}}$	1.50	${\displaystyle 4.447\times {10}^{-3}}$	1.73	${\displaystyle 1.323\times {10}^{-3}}$	1.90
  32	${\displaystyle 5.882\times {10}^{-3}}$	1.60	${\displaystyle 1.265\times {10}^{-3}}$	1.81	${\displaystyle 3.598\times {10}^{-4}}$	1.93
  64	${\displaystyle 1.863\times {10}^{-3}}$	1.66	${\displaystyle 3.453\times {10}^{-4}}$	1.85	${\displaystyle 9.223\times {10}^{-5}}$	1.96
  128	${\displaystyle 5.802\times {10}^{-4}}$	1.68	${\displaystyle 9.345\times {10}^{-5}}$	1.88	${\displaystyle 2.346\times {10}^{-5}}$	1.97

  . The data was computed for fixed ${\displaystyle \alpha =1.5}$ and ${\displaystyle \beta =1.2,1.5,}$ 1.8.
• As expected, the scheme achieves ${\displaystyle \beta }$-order spatial convergence.

6. Conclusions

A well-posed, efficient, and coherent computational algorithm is presented for space–time FDWEs by embedding two different approximations for time and space, respectively. In the spatial direction, the MTM approach is used to approximate the Riesz derivative and a finite difference technique is utilized to formulate the time-fractional Caputo derivative. Unconditional stability and ${\displaystyle \beta }$-th order of convergent in the space domain are demonstrated with the help of optimal error bounds. Additionally, it is shown that the proposed scheme is numerically stable when noise disturbances are added to the initial values of the considered problem. The efficiency and solvability of the scheme are tested upon numerical experiments and the obtained results show a better performance of the proposed scheme compared with the scheme in [30]. Finally, some directions for future research based on the proposed numerical algorithm are as follows:

• The numerical method proposed in this manuscript can be extended to more general and complex boundary conditions, including time-dependent and mixed-type constraints.
• Furthermore, the method can be implemented on nonlinear models and extended to higher-dimensional problems, where stability and efficiency will be key challenges.
• Another potential direction is the development of adaptive mesh refinement strategies to improve accuracy while reducing computational cost.