A Fourier Series Technique for Approximate Solutions of Modified Anomalous Time-Fractional Sub-Diffusion Equations

Abstract

This article presents an innovative approximating technique for addressing modified anomalous time-fractional sub-diffusion equations (MAFSDEs) of the Caputo type. These equations generalize classical diffusion equations, which involve fractional derivatives with respect to time, capturing the non-local and history-dependent behavior typical in sub-diffusion processes. In such a model, the particle transports slower than in a standard diffusion, often due to obstacles or memory effects in the medium. The core of the proposed technique involves transforming the original problem into a family of independent fractional-order ordinary differential equations (FODEs). This transformation is achieved using the Fourier expansion method. Each of these resulting FODEs is defined under initial value conditions which are derived from the initial condition of the original problem. To solve them, for each resulting FODE, some secondary initial value problems are introduced. By solving these secondary initial value problems, some particular solutions are obtained and then we combine them linearly in an optimal manner. This combination is essential to estimate the solution of the original problem. To evaluate the accuracy and effectiveness of the proposed scheme, we conduct a various test problem. For each problem, we analyze the observed convergence order indicators and compare them with those from other methods. Our comparison demonstrates that the proposed technique provides enhanced precision and reliability in respect with the current numerical approaches in the literature.

1. Introduction

Real-world models demonstrate that a single parameter, such as the diffusion coefficient, is insufficient to fully describe the wide range of diffusion rates encountered in both physical and biological systems [1,2]. In contrast, anomalous diffusion can be effectively characterized using three key elements: the diffusion coefficient $\kappa$, the scaling exponent $\lambda$, and the second-order moment of the particle’s displacement, $\langle x^2\left(t\right)\rangle$. The relevant relationship is typically expressed by:

$$\langle x^2\left(t\right)\rangle \sim {\displaystyle \frac{2{\kappa }_{\lambda }}{\Gamma (1+\lambda )}}{t}^{\lambda },t\to \infty ,0<\lambda <1,$$

(1)

where $\lambda$ quantifies the extent of anomaly, and ${\kappa }_{\lambda }$ represents the generalized diffusion coefficient. In the special case where $\lambda =1$, the diffusion coefficient reduces to ${\kappa }_1=D$, corresponding to classical Brownian diffusion. The diffusion dynamics governed by the relation (1) exhibit sub-diffusive behavior when $0<\lambda <1$, while for $\lambda >1$ the process becomes super-diffusive. Brown et al. [3], in their study, utilized multiphoton fluorescence recovery after photobleaching (FRAP) techniques to quantify the diffusion of membrane proteins across different cellular environments under sub-diffusive conditions $(0<\lambda <1)$. Furthermore, a comprehensive analysis of anomalous protein diffusion mechanisms within biological cells conducted by Malchus and Weiss [4] provides an extensive insight into this phenomenon. In environments that are characterized by barriers or binding sites, the phenomenon of anomalous sub-diffusion, typically observed for $0<\lambda <1$, necessitates a departure from classical diffusion models [5,6]. To accurately represent the dynamics of sub-diffusive processes, it is essential to employ a fractional-order diffusion equation rather than the conventional diffusion equation. Metzler [7] introduced a fractional-order sub-diffusion equation to model such systems

$$v_t(x,t)={}_{0^{+}}{}^{RL}D_t^{1-\lambda }\left({\kappa }_{\lambda }{v}_{xx}(x,t)\right),\mathrm{for}(x,t)\in [0,a]\times [0,T],\mathrm{and}0<\lambda <1.$$

In this formulation, $v(x,t)$ denotes the function of density probability, and ${}_{0^{+}}{}^{RL}D_t^{\lambda }$ represents the Riemann–Liouville time-fractional derivative of order $\lambda$.

Yuste et al. [8] explored the dynamics of particles in one dimension, specifically focusing on those displaying sub-diffusive behavior. Their research delved into how these particles’ motion differs from the classical diffusion processes, focusing on the characteristic features and implications of sub-diffusion in a linear spatial domain. Their research encompassed coagulation dynamics, represented by the reactions $A+A\to A$ and $A+A\rightleftharpoons A$, as well as annihilation dynamics, characterized by $A+A\to 0$, where these reactions are based on a known process denoted by A. Furthermore, their work delved into the interplay between anomalous diffusion and anomalous kinetics, suggesting that combining these phenomena yields more complex dynamics than each problem considered in isolation. Yuste et al. [9] focused their investigation on the dynamics of $A+A$ reactions, exploring the effects of sub-diffusive behavior within a one-dimensional linear system. Their study aimed to understand how particles exhibit sub-diffusion influence reaction processes and overall dynamics. By examining these reactions in a one-dimensional context, they analyzed how deviations from classical diffusion impact reaction rates and system behavior, offering important insights into the role of sub-diffusion in reaction kinetics. This further analysis enhanced the understanding of the interactions among these combined effects. The $A+B$ problem poses a notable challenge, as explicit formulations or exact solutions remain undeveloped. To address this, Yuste et al. [10] proposed a set of sub-diffusion equations for the $A+B$ reaction

$$\begin{array}{c}\begin{array}{c}\hfill \left\{\begin{array}{c}{a}_t(x,t)={\kappa }_{\lambda }{}_{0^{+}}{}^{RL}D_t^{1-\lambda }{a}_{xx}(x,t)-{\mathcal{R}}_{\lambda }(x,t),\hfill \\ b_t(x,t)={\kappa }_{\lambda }{}_{0^{+}}{}^{RL}D_t^{1-\lambda }{b}_{xx}(x,t)-{\mathcal{R}}_{\lambda }(x,t),\hfill \end{array}\right.\end{array}\end{array}$$

where ${\mathcal{R}}_{\lambda }(x,t)$ represents the reaction term, wherein certain properties of the system remain invariant with respect to its specific form. This reaction term can take various forms. For example, the following formulation for ${\mathcal{R}}_{\lambda }(x,t)$ was proposed by Seki et al. [11]:

$${\mathcal{R}}_{\lambda }(x,t)=\kappa {}_{0^{+}}{}^{RL}D_t^{1-\lambda }a(x,t)b(x,t).$$

Another example is the fractional extension of Stokes’ first problem, focusing on its application to a heated fluid described by a generalized second-grade model that was examined by Chen et al. [12]. They developed both implicit and explicit methods for approximating solutions to the following extended problem:

$$v_t(x,t)={}_{0^{+}}{}^{RL}D_t^{1-\lambda }\left({\kappa }_1{v}_{xx}(x,t)\right)+{\kappa }_2{v}_{xx}(x,t)+\mathcal{F}(x,t),$$

wherein $\mathcal{F}(x,t)$ is the source function. Further, to model the anomalous diffusion property, a range of methodologies has been employed, including continuous-time random walks and the Monte Carlo method [5], along with additional approaches [13,14,15]. Sokolov et al. [1] proposed a model to address the anomalous diffusion phenomena observed over long time scales. The dynamics of the model are expressed by the following governing equation:

$$v_t(x,t)=A{}_{0^{+}}{}^{RL}D_t^{1-\alpha }\kappa v_{xx}(x,t)+B{}_{0^{+}}{}^{RL}D_t^{1-\beta }\kappa v_{xx}(x,t),\mathrm{for}0<\alpha <\beta \le 1\mathrm{and}A,B\in {\mathbb{R}}^{+}.$$

Additionally, the dynamics of a sub-diffusive particle can be quantified through the second-order moment of its displacement, which is represented as follows:

$$\langle x^2\left(t\right)\rangle ={\displaystyle \frac{2A}{\Gamma (1+\alpha )}}{t}^{\alpha }+{\displaystyle \frac{2B}{\Gamma (1+\beta )}}{t}^{\beta },t\to \infty .$$

This equation is highly relevant for simulating real-world phenomena, particularly within the field of econophysics. This domain encompasses a range of sophisticated models that employ time random walks [16,17]. Notably, empirical observations have identified the interaction between less and more anomalous behaviors in the oscillations of stock prices [18].

Recent developments have established several frameworks for fractional derivatives, notably including the Caputo and Riemann–Liouville fractional derivatives which are particularly valued for their practical applications. These derivative types are closely related and it is possible to transform one into the other under certain regularity conditions [19]. However, the fractional derivative in the Riemann–Liouville type requires initial value conditions that involve the limit values at $t=0$, a concept whose physical interpretation can be unclear. In contrast, the Caputo fractional derivative is favored because its initial value conditions are analogous to those of classical ordinary differential equations at $t=0$; for a comprehensive discussion, see [19,20].

This study proposes a novel methodology for obtaining approximate solutions to the fractional partial differential equation, which is characterized by:

$$\begin{array}{c}\begin{array}{cc}& v_t(x,t)-\left({\lambda }_1{}_{0^{+}}{}^C\mathcal{D}_t^{\alpha }{v}_{xx}(x,t)+{\lambda }_2{}_{0^{+}}{}^C\mathcal{D}_t^{\beta }{v}_{xx}(x,t)\right)=s(x,t),(x,t)\in [0,\ell ]\times [0,T],\hfill \\ & v(x,0)=f\left(x\right),x\in [0,\ell ],\hfill \\ & v(0,t)=v_0\left(t\right),v(\ell ,t)=v_{\ell }\left(t\right),t\in [0,T].\hfill \end{array}\end{array}$$

(2)

Here, $0<\alpha ,\beta \le 1$, where $\alpha$ and $\beta$ typically represent fractional orders associated with temporal derivatives. The parameters ${\lambda }_1,{\lambda }_2\in {\mathbb{R}}^{+}$ exhibit diffusion rates. It is assumed that $s(x,t)$ and $f\left(x\right)$ adhere to Dirichlet’s boundary conditions, which impose certain values on these functions within the boundaries of the spatial and temporal domains, ensuring a well-posedness of the problem. Furthermore, ${}_{0^{+}}{}^C\mathcal{D}_t^{\mu }$ represents the Caputo-type time-fractional derivative operator of order $\mu$. This derivative is commonly used in fractional calculus because it allows for the inclusion of memory effects, making it well suited for modeling physical phenomena where the system’s current state depends on its past behavior.

Luchko [21] conducted a thorough investigation into the existence and uniqueness of solutions for a more generalized form of diffusion equations of distributed order, significantly contributing to the theoretical understanding of fractional differential equations (FDEs). In a related study, Ma et al. [22] focused on the asymptotic behavior of solutions to anomalous diffusion equations, contributing valuable knowledge to the long-term behavior of such systems. Despite these advances, obtaining closed-form solutions for FDEs remains highly challenging in practical applications due to the complex nature of these equations. As a result, research efforts have increasingly prioritized the development of sophisticated numerical and analytical techniques designed to generate accurate approximate solutions. These methods, which form the cornerstone of contemporary FDE research, are continuously refined to improve their precision and computational efficiency. The following provides a concise overview of some prominent approaches which are currently employed to address the complexities of solving FDEs.

Liu et al. [2] introduced a new implicit difference method to enhance MAFSDEs with known nonlinear source terms, incorporating a second-order time-fractional derivative into the diffusion term. Lin and his coauthor Xu [23] investigated a numerical method with high-order accuracy to approximate solutions for FDEs. The temporal discretization was achieved using a finite difference scheme, while the spatial variable was discretized through the Legendre spectral method. Cui [24] developed a compact finite difference method to solve one-dimensional fractional diffusion equations and combine it with the Grünwald–Letnikov discretization scheme to create a fully discrete implicit method. Wang and Vong [25] improved temporal accuracy in solving MAFSDEs by employing compact finite difference schemes with a weighted and shifted Grünwald difference operator.

Dehghan et al. [26] applied Liu and Xu’s approach for solving MAFSDEs by utilizing a Legendre spectral element scheme to handle spatial discretization, in contrast to Liu and Xu, who applied a Legendre finite difference scheme for the same purpose. A semi-analytical method was employed by Reutskiy [27] to solve MAFSDEs. A new numerical scheme was developed by Cao et al. [28] applies the implicit midpoint approach for solving semi-discrete MAFSDEs with Grünwald–Letnikov and compact difference operators being employed to obtain an approximate solution for the fractional derivative with respect to spatial partial derivatives. For a comprehensive perspective on approximate solutions for fractional differential equations, researchers are advised to review the recent publications [29,30,31] and their associated references.

The structure of the present study is as follows: Essential definitions and properties of fractional calculus necessary for understanding the subsequent sections are provided in Section 2. Section 3 presents the main algorithm for solving MAFSDEs, divided into two sub-sections: the first sub-section applies the Fourier expansion method to decompose the problem into independent time-fractional ODEs, while the second introduces a novel technique for solving these ODEs. Section 4 discusses the application of the proposed technique, including solutions of some test problems. Finally, Section 5 offers a concise discussion of the methodology and the obtained results.

2. Preliminaries

This section provides the fundamental definitions and properties of fractional-order derivatives and integrals. For a more in-depth exploration of these concepts, readers are advised to consult references [32,33,34].

Definition 1

([32]). The fractional integral of order $\mu >0$ for the function $v:(t_0,+\infty )\mapsto \mathbb{R}$ is expressed as:

$${\mathcal{I}}_{t_0^{+}}^{\mu }v\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{t_0}^t{(t-s)}^{\mu -1}v\left(s\right)ds,$$

whereby the integration on right is defined point-wise over the interval $(t_0,+\infty )$.

Definition 2

([32]). Consider a known real-valued function $v\left(t\right)$ on domain $D=(t_0,+\infty )$. The fractional derivative of order $\mu >0$ in the Caputo sense is

$${}_{t_0^{+}}{}^C\mathcal{D}_t^{\mu }v\left(t\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\Gamma (m-\mu )}}{\int }_{t_0}^t{(t-s)}^{m-\mu -1}{v}^{\left(m\right)}\left(s\right)ds,\hfill & \mu \notin \mathbb{N},m=\lceil \mu \rceil ,\hfill \\ {\displaystyle \frac{d^m}{d{t}^m}}v\left(t\right),\hfill & \mu \in \mathbb{N},m=\mu ,\hfill \end{array}\right.$$

where $\lceil ·\rceil$ denotes the ceiling function, and $v^{\left(m\right)}$ shows the m-th order derivative of function v. This fractional operator was first formulated by the Italian mathematician Caputo in 1967 [35].

By Definition 2, the fractional derivative of ${(t-t_0)}^k$ is:

$${}_{t_0^{+}}{}^C\mathcal{D}_t^{\mu }{(t-t_0)}^k=\left\{\begin{array}{cc}{\displaystyle \frac{\Gamma (k+1)}{\Gamma (k-\mu +1)}}{(t-t_0)}^{k-\mu },\hfill & \left(k\in {\mathbb{N}}_0,k\ge \lceil \mu \rceil \right)\mathrm{or}\left(k\notin \mathbb{N},k>\lfloor \mu \rfloor \right),\hfill \\ 0,\hfill & \left(k\in {\mathbb{N}}_0,k<\lceil \mu \rceil \right),\hfill \end{array}\right.$$

where ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$ and $\lfloor ·\rfloor$ denotes the flooring function. It is clear that when $\mu \in \mathbb{N}$, Caputo’s derivative coincides with the standard derivative.

Remark 1.

The following properties hold for ${\mathcal{I}}_{t_0^{+}}^{\mu }$ and ${}_{t_0^{+}}{}^C\mathcal{D}_t^{\mu }$:

• ${\mathcal{I}}_{t_0^{+}}^{\mu }{(t-t_0)}^k={\displaystyle \frac{\Gamma (k+1)}{\Gamma (\mu +k+1)}}{(t-t_0)}^{\mu +k},k>-1.$
• ${\mathcal{I}}_{t_0^{+}}^{{\mu }_1}{\mathcal{I}}_{t_0^{+}}^{{\mu }_2}v\left(t\right)={\mathcal{I}}_{t_0^{+}}^{{\mu }_2}{\mathcal{I}}_{t_0^{+}}^{{\mu }_1}v\left(t\right)={\mathcal{I}}_{t_0^{+}}^{{\mu }_1+{\mu }_2}v\left(t\right),{\mu }_1,{\mu }_2\ge 0.$
• ${}_{t_0^{+}}{}^C\mathcal{D}_t^{\mu }\kappa =0,\kappa =\mathit{constant}.$
• ${}_{t_0^{+}}{}^C\mathcal{D}_t^{\mu }{\mathcal{I}}_{t_0^{+}}^{\mu }v\left(t\right)=v\left(t\right).$
• ${\mathcal{I}}_{t_0^{+}}^{\mu }{}_{t_0^{+}}{}^C\mathcal{D}_t^{\mu }v\left(t\right)=v\left(t\right)-{\sum }_{j=0}^{\lceil \mu \rceil -1}{\displaystyle \frac{{(t-t_0^{+})}^j}{j!}}{v}^{\left(j\right)}\left(t_0^{+}\right),t>t_0^{+}.$
• ${}_{t_0^{+}}{}^C\mathcal{D}_t^{\mu }\left({\sum }_{m=1}^M{\alpha }_m{v}_m\left(t\right)\right)={\sum }_{m=1}^M\left({}_{t_0^{+}}{}^C\mathcal{D}_t^{\mu }{\alpha }_m{v}_m\left(t\right)\right).$
• ${\mathcal{I}}_{t_0^{+}}^{\mu }\left({\sum }_{m=1}^M{\alpha }_m{v}_m\left(t\right)\right)={\sum }_{m=1}^M\left({\mathcal{I}}_{t_0^{+}}^{\mu }{\alpha }_m{v}_m\left(t\right)\right).$

3. Main Method

This section is divided into three parts. In the first part, we use the Fourier expansion method to decompose the main equation to a set of independent time-fractional ODEs, which are subject to specified initial boundary conditions. In the second part, we introduce a novel technique to solve these initial value problems. This technique offers a semi-analytical approximate solution for each equation by solving secondary problems and constructing residual error functions. To minimize these residual error functions, we employ the collocation method. Finally, in the third part of this section, we will present an algorithm for implementing the proposed method.

3.1. Reducing the Main Problem to a Set of Independent Initial Value Problems (IVPs)

The main object of this part is decomposing the governing Equation (2) into a family of independent time-fractional ODEs subject to initial value conditions. First, we convert the nonhomogeneous spatial boundary conditions into homogeneous ones. This step is necessary because solving time-fractional equations is generally more tractable when homogeneous boundary conditions are imposed. After that, we apply the following transform:

$$v(x,t)=u(x,t)+{\displaystyle \frac{x}{\ell }}{v}_{\ell }\left(t\right)+{\displaystyle \frac{\ell -x}{\ell }}{v}_0\left(t\right).$$

(3)

Substituting the transformation (3) into the main Equation (2) results in:

$$\begin{array}{c}\begin{array}{cc}& {\displaystyle \frac{\partial u(x,t)}{\partial t}}-\left({\lambda }_1{}_{0^{+}}{}^C\mathcal{D}_t^{\alpha }+{\lambda }_2{}_{0^{+}}{}^C\mathcal{D}_t^{\beta }\right){\displaystyle \frac{{\partial }^{2}u(x,t)}{\partial x^2}}=\mathcal{F}(x,t),(x,t)\in [0,\ell ]\times [0,T],\hfill \\ & u(x,0)=u_0\left(x\right),x\in [0,\ell ],\hfill \\ & u(0,t)=u(\ell ,t)=0,t\in [0,T],\hfill \end{array}\end{array}$$

(4)

where $\mathcal{F}(x,t)=s(x,t)-{\displaystyle \frac{x}{\ell }}{\displaystyle \frac{d{v}_{\ell }\left(t\right)}{dt}}+{\displaystyle \frac{x-\ell }{\ell }}{\displaystyle \frac{d{v}_0\left(t\right)}{dt}}$ and $u_0\left(x\right)=f\left(x\right)-{\displaystyle \frac{x}{\ell }}{v}_{\ell }\left(0\right)+{\displaystyle \frac{x-\ell }{\ell }}{v}_0\left(0\right)$. Both $\mathcal{F}(x,t)$ and $u_0\left(x\right)$ satisfy Dirichlet boundary conditions. We look for a solution to problem (4) using the sine-Fourier expansion

$$u(x,t)={\displaystyle \sum _{k=1}^{\infty }}{b}_k\left(t\right)sin{\displaystyle \frac{k\pi x}{\ell }}.$$

(5)

It is assumed that $\mathcal{F}(x,t)$ can be presented as:

$$\mathcal{F}(x,t)={\displaystyle \sum _{k=1}^{\infty }}{F}_k\left(t\right)sin{\displaystyle \frac{k\pi x}{\ell }},$$

(6)

where $F_k\left(t\right)={\displaystyle \frac{2}{\ell }}{\int }_0^{\ell }\mathcal{F}(x,t)sin\left({\displaystyle \frac{k\pi x}{\ell }}\right)dx$. From (5) and (6), we can reformulate (4) as follows:

$$\begin{array}{c}\begin{array}{c}\hfill {\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{d{b}_k\left(t\right)}{dt}}sin{\displaystyle \frac{k\pi x}{\ell }}+{\displaystyle \sum _{k=1}^{\infty }}{\displaystyle \frac{k^2{\pi }^2}{{\ell }^2}}\left({\lambda }_1{}_{0^{+}}{}^C\mathcal{D}_t^{\alpha }{b}_k\left(t\right)+{\lambda }_2{}_{0^{+}}{}^C\mathcal{D}_t^{\beta }{b}_k\left(t\right)\right)sin{\displaystyle \frac{k\pi x}{\ell }}={\displaystyle \sum _{k=1}^{\infty }}{F}_k\left(t\right)sin{\displaystyle \frac{k\pi x}{\ell }},\end{array}\end{array}$$

(7)

which can be rewritten as:

$$\begin{array}{c}\begin{array}{c}\hfill {\displaystyle \sum _{k=1}^{\infty }}\left({\displaystyle \frac{d{b}_k\left(t\right)}{dt}}+{\displaystyle \frac{k^2{\pi }^2}{{\ell }^2}}\left({\lambda }_1{}_{0^{+}}{}^C\mathcal{D}_t^{\alpha }{b}_k\left(t\right)+{\lambda }_2{}_{0^{+}}{}^C\mathcal{D}_t^{\beta }{b}_k\left(t\right)\right)\right)sin{\displaystyle \frac{k\pi x}{\ell }}={\displaystyle \sum _{k=1}^{\infty }}{F}_k\left(t\right)sin{\displaystyle \frac{k\pi x}{\ell }}.\end{array}\end{array}$$

(8)

Assume that

$$\begin{array}{c}\begin{array}{cc}& u_0\left(x\right)=u(x,0)={\displaystyle \sum _{k=1}^{\infty }}{b}_k\left(0\right)sin{\displaystyle \frac{k\pi x}{\ell }},\hfill \\ & \mathrm{where}{b}_k\left(0\right)={\displaystyle \frac{2}{\ell }}{\int }_0^{\ell }{u}_0\left(x\right)sin{\displaystyle \frac{k\pi x}{\ell }}dx=b_{k0},k=1,2,3,\dots .\hfill \end{array}\end{array}$$

(9)

Consequently, from (8) and (9), we obtain the following set of independent initial value ordinary differential equations:

$$\left\{\begin{array}{cc}{\displaystyle \frac{d{b}_k\left(t\right)}{dt}}+{\displaystyle \frac{k^2{\pi }^2}{{\ell }^2}}\left({\lambda }_1{}_{0^{+}}{}^C\mathcal{D}_t^{\alpha }{b}_k\left(t\right)+{\lambda }_2{}_{0^{+}}{}^C\mathcal{D}_t^{\beta }{b}_k\left(t\right)\right)=F_k\left(t\right),\hfill & \hfill \\ b_k\left(0\right)=b_{k0},\hfill \end{array}\right.k=1,2,3,\dots .$$

(10)

Remark 2.

It must be noted that a function, say g, has a convergent Fourier series, including its $\mathrm{sine}-$ and $\mathrm{cosine}-$ expansions, if Dirichlet conditions hold for it [36]. That is if the following properties hold for g:

• ${\int }_{-\ell }^{\ell }\left|g\left(x\right)\right|$ exists and is finite;
• $g\left(x\right)$ must have finite number of extreme points;
• $g\left(x\right)$ has a finite number of finite discontinuities.

It is well known that, these conditions are not very restrictive. Hence, hereafter using these conditions, we can write $\mathrm{sine}-$ expansion for our functions under study.

3.2. Solving the Obtained IVPs

In this part, we derive approximate solutions for the initial value problems given by (10) using a notable semi-analytical technique. This approach starts by formulating secondary IVPs derived from (10). For each k, the approximate solution of (10) is represented as a specific linear combination of the solutions to these secondary equations, configured to satisfy the initial boundary conditions imposed by (10). This linear combination involves unknown coefficients. To determine these coefficients, we construct a residual function by substituting the approximate solution back into (10). We then enforce that the residual function equals zero across the problem’s domain. The unknown coefficients are subsequently extracted from the resulting system. Notably, this technique provides an approximate solution without the need for discretization or linearization.

To solve each IVP in (10), we consider the following function as its approximate solution:

$$b_k\left(t\right)\simeq {\zeta }_k\left(t\right)+{\displaystyle \sum _{j=0}^J}{\gamma }_j{\eta }_j\left(t\right)\equiv {\widehat{b}}_k^J\left(t\right),$$

(11)

where $J\in \mathbb{N}$ is an arbitrary integer, and ${\gamma }_j\in \mathbb{R}$ for $j=0,1,\cdots ,J$ are unknown coefficients whose values will be determined later. Additionaly, ${\zeta }_k\left(t\right)$ and ${\eta }_j\left(t\right)$ for $j=1,\cdots ,J$ will be chosen such that ${\widehat{b}}_k^J\left(t\right)$ satisfies the initial condition (10), that is,

$${\widehat{b}}_k^J\left(0\right)={\zeta }_k\left(0\right)+{\displaystyle \sum _{j=0}^J}{\gamma }_j{\eta }_j\left(0\right)=b_{k0}.$$

(12)

Clearly, ${\widehat{b}}_k^J\left(t\right)$ satisfies the initial condition (10) if the following conditions hold:

$$\left\{\begin{array}{cc}{\zeta }_k\left(0\right)=b_{k0},\hfill & \\ {\eta }_j\left(0\right)=0,\hfill & j=0,1,\cdots ,J.\hfill \end{array}\right.$$

(13)

Therefore, our goal is to ensure that ${\zeta }_k\left(t\right)$ and ${\eta }_j\left(t\right)$ for $j=0,1,\cdots ,J$ satisfy the following secondary IVPs:

$$\begin{array}{c}\begin{array}{cc}& \left\{\begin{array}{cc}{\displaystyle \frac{d{\zeta }_k\left(t\right)}{dt}}=F_k\left(t\right),\hfill & {\zeta }_k\left(0\right)=b_{k0},k\in \mathbb{N},\hfill \\ {\displaystyle \frac{d{\eta }_j\left(t\right)}{dt}}=t^{j\delta },\hfill & {\eta }_j\left(0\right)=0,j=0,1,\cdots ,J,\hfill \end{array}\right.0\le t\le T,\hfill \end{array}\end{array}$$

where $t^{j\delta }$ denotes the Müntz polynomial function defined over the interval $[0,T]$, $j=0,1,\cdots$, and $0<\delta \le 1$. It should be noted that the accuracy of the method exhibits negligible dependence on the choice of different values for $\delta$. For more details, interested readers are referred to [37,38]. By integrating both sides of the above ODEs, we obtain:

$$\left\{\begin{array}{cc}{\zeta }_k\left(t\right)={\int }_0^t{F}_k\left(s\right)ds+c_k,\hfill & k\in \mathbb{N}\hfill \\ {\eta }_j\left(t\right)={\displaystyle \frac{1}{j\delta +1}}{t}^{j\delta +1}+c_j^{\prime },\hfill & j=0,1,\dots ,J,\hfill \end{array}\right.0\le t\le T.$$

(14)

Using the initial conditions (13), the unknown parameters $c_k$ and $c_j^{\prime }$ can be immediately determined, yielding $c_k=b_{k0}$ and $c_j^{\prime }=0$. Furthermore, from (11) we have:

$${\widehat{b}}_k^J\left(t\right)={\int }_0^t{F}_k\left(s\right)ds+{\displaystyle \sum _{j=0}^J}{\displaystyle \frac{{\gamma }_j}{j\delta +1}}{t}^{j\delta +1}+b_{k0},k\in \mathbb{N}.$$

(15)

Taking the Caputo derivative of order $\mu$ of ${\widehat{b}}_k^J\left(t\right)$ for $0<\mu \le 1$ yields:

$${}_{0^{+}}{}^C\mathcal{D}_t^{\mu }{\widehat{b}}_k^J\left(t\right)=\left\{\begin{array}{cc}{\int }_0^t{\displaystyle \frac{{(t-s)}^{-\mu }}{\Gamma (1-\mu )}}{F}_k\left(s\right)ds+{\displaystyle \sum _{j=0}^J}\left[{\gamma }_j{\displaystyle \frac{\Gamma (j\delta +1)}{\Gamma (j\delta -\mu +2)}}{t}^{j\delta -\mu +1}\right],\hfill & 0<\mu <1,\hfill \\ F_k\left(t\right)+{\displaystyle {\sum }_{j=0}^J}{\gamma }_j{t}^{j\delta },\hfill & \mu =1.\hfill \end{array}\right.$$

(16)

To determine the unknown parameters ${\left\{{\gamma }_j\right\}}_{j=0}^J$, we construct a residual error function ${\mathcal{R}}_k^J(t;\mathbf{\gamma })$ by substituting ${\widehat{b}}_k^J\left(t\right)$ into (10), i.e.,

$${\mathcal{R}}_k^J(t;\mathbf{\gamma })={\displaystyle \frac{d{\widehat{b}}_k^J\left(t\right)}{dt}}+{\displaystyle \frac{k^2{\pi }^2}{{\ell }^2}}\left({\lambda }_1{}_{0^{+}}{}^C\mathcal{D}_t^{\alpha }{\widehat{b}}_k^J\left(t\right)+{\lambda }_2{}_{0^{+}}{}^C\mathcal{D}_t^{\beta }{\widehat{b}}_k^J\left(t\right)\right)-F_k\left(t\right),$$

(17)

where $\mathbf{\gamma }=({\gamma }_0,{\gamma }_1,\cdots ,{\gamma }_J)$. Using (16), we can equivalently rewrite (17) as

$$\begin{array}{c}\begin{array}{cc}\hfill {\mathcal{R}}_k^J(t;\mathbf{\gamma })& ={\displaystyle \sum _{j=0}^J}{\gamma }_j{t}^{j\delta }+{\displaystyle \frac{k^2{\pi }^2}{{\ell }^2}}[{\int }_0^t\left({\lambda }_1{\displaystyle \frac{{(t-s)}^{-\alpha }}{\Gamma (1-\alpha )}}+{\lambda }_2{\displaystyle \frac{{(t-s)}^{-\beta }}{\Gamma (1-\beta )}}\right)F_k\left(s\right)ds\hfill \\ & +{\displaystyle \sum _{j=0}^J}{\gamma }_j\Gamma (j\delta +1)\left({\displaystyle \frac{{\lambda }_1{t}^{j\delta -\alpha +1}}{\Gamma (j\delta -\alpha +2)}}+{\displaystyle \frac{{\lambda }_1{t}^{j\delta -\beta +1}}{\Gamma (j\delta -\beta +2)}}\right)].\hfill \end{array}\end{array}$$

We then set ${\mathcal{R}}_k^J(t;\mathbf{\gamma })$ to zero at $J+1$ collocation points $t_n={\displaystyle \frac{T}{2}}\left(1+cos{\displaystyle \frac{n\pi }{J}}\right),n=0,\cdots ,J,$ to determine the unknown parameters ${\left\{{\gamma }_j\right\}}_{j=0}^J$.

After evaluating the values of ${\gamma }_j$, we can construct ${\widehat{b}}_k^J\left(t\right)$ using (15). Additionally, we use ${\left\{{\widehat{b}}_k^J\left(t\right)\right\}}_{k=1}^K$ to construct the K-term truncated approximate solution $u_{app}^{K,J}(x,t)={\sum }_{k=1}^K{\widehat{b}}_k^J\left(t\right)sin\left({\displaystyle \frac{k\pi x}{\ell }}\right)$. Finally, using (3) we obtain the approximate solution of the main Equation (2) as

$$\begin{array}{c}\begin{array}{c}\hfill v_{app}^{K,J}(x,t)={\displaystyle \sum _{k=1}^K}{\widehat{b}}_k^J\left(t\right)sin{\displaystyle \frac{k\pi x}{\ell }}+{\displaystyle \frac{x}{\ell }}{v}_{\ell }\left(t\right)+{\displaystyle \frac{\ell -x}{\ell }}{v}_0\left(t\right).\end{array}\end{array}$$

(18)

3.3. Algorithm

This sub-section provides a comprehensive display of Algorithm 1 developed for the established scheme, which is utilized to solve Equation (2).

Algorithm 1: Provided Algorithm

Require:  $K,J\in \mathbb{N}$, $\delta \in (0,1]$.

$\mathcal{F}(x,t)=s(x,t)-{\displaystyle \frac{x}{\ell }}{\displaystyle \frac{d{v}_{\ell }\left(t\right)}{dt}}+{\displaystyle \frac{x-\ell }{\ell }}{\displaystyle \frac{d{v}_0\left(t\right)}{dt}};$

$u_0\left(x\right)=f\left(x\right)-{\displaystyle \frac{x}{\ell }}{v}_{\ell }\left(0\right)+{\displaystyle \frac{x-\ell }{\ell }}{v}_0\left(0\right)$;

for  $n=0:J$ do

$t_n={\displaystyle \frac{T}{2}}\left(1+cos{\displaystyle \frac{n\pi }{J}}\right);$

end for

for  $k=1:K$ do

$F_k\left(t\right)={\displaystyle \frac{2}{\ell }}{\int }_0^{\ell }\mathcal{F}(x,t)sin\left({\displaystyle \frac{k\pi x}{\ell }}\right)dx;$

$b_{k0}={\displaystyle \frac{2}{\ell }}{\int }_0^{\ell }{u}_0\left(x\right)sin{\displaystyle \frac{k\pi x}{\ell }}dx;$

for  $n=0:J$ do

${\mathcal{R}}_k^J(t_n;\mathbf{\gamma })={\sum }_{j=0}^J{\gamma }_j{t}_n^{j\delta }+{\displaystyle \frac{k^2{\pi }^2}{{\ell }^2}}[{\int }_0^{t_n}\left({\lambda }_1{\displaystyle \frac{{(t_n-s)}^{-\alpha }}{\Gamma (1-\alpha )}}+{\lambda }_2{\displaystyle \frac{{(t_n-s)}^{-\beta }}{\Gamma (1-\beta )}}\right)F_k\left(s\right)ds$

$+{\sum }_{j=0}^J{\gamma }_j\Gamma (j\delta +1)\left({\displaystyle \frac{{\lambda }_1{t}_n^{j\delta -\alpha +1}}{\Gamma (j\delta -\alpha +2)}}+{\displaystyle \frac{{\lambda }_1{t}_n^{j\delta -\beta +1}}{\Gamma (j\delta -\beta +2)}}\right)];$

end for

$\mathrm{Solve}\left(\left\{{\mathcal{R}}_k^J(t_0;\mathbf{\gamma })=0,{\mathcal{R}}_k^J(t_1;\mathbf{\gamma })=0,\cdots ,{\mathcal{R}}_k^J(t_J;\mathbf{\gamma })=0\right\},\left\{{\gamma }_0,{\gamma }_1,\cdots ,{\gamma }_J\right\}\right);$

${\widehat{b}}_k^J\left(t\right)={\int }_0^t{F}_k\left(s\right)ds+{\sum }_{j=0}^J{\displaystyle \frac{{\gamma }_j}{j\delta +1}}{t}^{j\delta +1}+b_{k0};$

end for

$v(x,t)\simeq v_{app}^{K,J}(x,t)={\sum }_{k=1}^K{\widehat{b}}_k^J\left(t\right)sin{\displaystyle \frac{k\pi x}{\ell }}+{\displaystyle \frac{x}{\ell }}{v}_{\ell }\left(t\right)+{\displaystyle \frac{\ell -x}{\ell }}{v}_0\left(t\right).$

4. Numerical Results

In this section, we apply our method to a series of test problems. The obtained numerical results are compared with exact solutions for each problem, which demonstrate the effectiveness of our approach. All computations were performed using Maple 2024 on a computer with an Intel Core i9-13900HX processor, with 16 GB RAM.

To perform a detailed analysis of the occurring error and assess the performance of the presented algorithm, we consider the following convergence criteria:

•

Max absolute error: $M.A.E.(K,J)=\underset{\begin{array}{c}0\le x\le \ell \\ 0\le t\le T\end{array}}{max}\left|v_{app}^{K,J}(x,t)-v(x,t)\right|$.

•

Relative error: $R.E.(K,J)={\displaystyle \frac{{\displaystyle \sum _{k=1}^N}{\displaystyle \sum _{j=1}^N}{\left(v_{app}^{K,J}(x_k,t_j)-v(x_k,t_j)\right)}^2}{{\displaystyle \sum _{k=1}^N}{\displaystyle \sum _{j=1}^N}v{(x_k,t_j)}^2}},\mathrm{for}\left(x_k,t_j\right)=\left({\displaystyle \frac{k\ell }{N}},{\displaystyle \frac{jT}{N}}\right)$.

In this paper, we assume that $N=10$. Finally, if $\mathcal{E}$ is a representative of one of the above criteria, then

$${\rho }_{k,j}^{\mathcal{E}(K,J)}=ln{\displaystyle \frac{\mathcal{E}(K,J)}{\mathcal{E}(K+k,J+j)}}.$$

shows the observed convergence rate.

In practical terms, ${\rho }_{k,j}^{\mathcal{E}(K,J)}=r$ indicates that increasing the number of sine-Fourier expansion terms in (18) from K to $K+k$, and the count of approximation components ${\eta }_j\left(t\right)$ in (11) from J to $J+j$, leads to a reduction by a factor of $e^{-r}$ in the maximum value of $\mathcal{E}$.

Problem 1.

In this test problem, the following MAFSDE is considered:

$$\begin{array}{c}\begin{array}{cc}& v_t(x,t)-\left({}_{0^{+}}{}^C\mathcal{D}_t^{\alpha }{v}_{xx}(x,t)+{}_{0^{+}}{}^C\mathcal{D}_t^{\beta }{v}_{xx}(x,t)\right)=s(x,t),\mathrm{for}(x,t)\in [0,1]\times [0,1],\hfill \\ & v(0,t)=0,v(1,t)=sin\left(1\right)t^{3-\alpha -\beta },\mathrm{for}0\le t\le 1,\hfill \\ & v(x,0)=0,\mathrm{for}0\le x\le 1,\hfill \end{array}\end{array}$$

(19)

where

$$s(x,t)=\left({\displaystyle \frac{\Gamma (4-\alpha -\beta )t^{3-\alpha -2\beta }}{\Gamma (4-\alpha -2\beta )}}+{\displaystyle \frac{\Gamma (4-\alpha -\beta )t^{3-2\alpha -\beta }}{\Gamma (4-2\alpha -\beta )}}+(3-\alpha -\beta )t^{2-\alpha -\beta }\right)sinx.$$

The exact solution of this equation, in a closed form, is $v(x,t)=t^{3-\alpha -\beta }sin\left(x\right).$

The approximate solution of (19) has been addressed in earlier works by Wang and Vong [25] and Mohebbi et al. [39]. We compare the maximum absolute error from our approximation method with the results reported in [25,39] at the boundary $t=1$, as shown in

Table 1. Evaluation of maximum absolute error measurements at $t=1$ and computation time for Problem 1, $\alpha =0.35$, and $\beta =0.05$.

Our Results for $\mathit{\delta }=0.15$				Reported Results in [39]				Reported Results in [25]
$\mathit{K}$	$\mathit{J}$	$\mathit{M}.\mathit{A}.\mathit{E}.$	CPU Time (s)	$\mathit{h}$	$\mathit{\tau }$	$\mathit{M}.\mathit{A}.\mathit{E}.$	CPU Time (s)	$\mathit{h}$	$\mathit{\tau }$	$\mathit{M}.\mathit{A}.\mathit{E}.$
16	16	$1.0266\times {10}^{-5}$	7.487	$1/32$	$1/10$	$1.7440\times {10}^{-2}$	0.1570	$1/30$	$1/5$	$8.2083\times {10}^{-4}$
24	16	$2.1401\times {10}^{-6}$	12.581	$1/32$	$1/20$	$9.0091\times {10}^{-3}$	0.2349	$1/30$	$1/10$	$1.9894\times {10}^{-4}$
32	16	$4.8259\times {10}^{-7}$	17.835	$1/32$	$1/40$	$4.5776\times {10}^{-3}$	0.4839	$1/30$	$1/20$	$5.0072\times {10}^{-5}$
40	16	$1.1062\times {10}^{-7}$	23.641	$1/32$	$1/80$	$2.3071\times {10}^{-3}$	1.3439	$1/30$	$1/40$	$1.2567\times {10}^{-5}$
48	16	$2.4653\times {10}^{-8}$	29.783	$1/32$	$1/160$	$1.1582\times {10}^{-3}$	4.4690	$1/30$	$1/80$	$3.1502\times {10}^{-6}$
56	16	$5.0402\times {10}^{-9}$	36.310	$1/32$	$1/320$	$5.8025\times {10}^{-4}$	16.4840	$1/30$	$1/160$	$7.8883\times {10}^{-7}$
64	16	$8.2116\times {10}^{-10}$	43.310	$1/32$	$1/640$	$2.9041\times {10}^{-4}$	63.3280

. The numerical computations were performed using the parameter values $(\alpha ,\beta )=(0.35,0.05)$. Table 1 illustrates that the presented algorithm achieves higher precision in results than the other methods. Furthermore,

Table 2. CPU time and the max absolute errors, $M.A.E.(K,J)$, and the relative errors, $R.E.(K,J)$, along with the corresponding convergence rates using our computed results for Problem 1 with $(\alpha ,\beta )=(0.35,0.05)$.

J	K	$\mathit{M}.\mathit{A}.\mathit{E}.(\mathit{K},\mathit{J})$	${\mathit{\rho }}_{8,0}^{\mathit{M}.\mathit{A}.\mathit{E}.(\mathit{K},\mathit{J})}$	$\mathit{R}.\mathit{E}.(\mathit{K},\mathit{J})$	${\mathit{\rho }}_{8,0}^{\mathit{R}.\mathit{E}.(\mathit{K},\mathit{J})}$	CPU Time (s)
16	8	$6.26520\times {10}^{-5}$	1.80881	$8.58237\times {10}^{-10}$	5.32615	2.918
	16	$1.02655\times {10}^{-5}$	1.56795	$4.17340\times {10}^{-12}$	3.73332	7.487
	24	$2.14006\times {10}^{-6}$	1.48942	$9.97994\times {10}^{-14}$	3.0481	12.581
	32	$4.82591\times {10}^{-7}$	1.47311	$4.73540\times {10}^{-15}$	2.71235	17.835
	40	$1.10615\times {10}^{-7}$	1.50115	$3.14338\times {10}^{-16}$	3.47158	23.641
	48	$2.46532\times {10}^{-8}$	1.58745	$9.76585\times {10}^{-18}$	3.38432	29.783
	56	$5.04024\times {10}^{-9}$	1.81449	$3.31069\times {10}^{-19}$	2.50484	36.310
	64	$8.21158\times {10}^{-10}$	2.17612	$2.70447\times {10}^{-20}$	1.9722	43.431
	72	$9.31860\times {10}^{-11}$	0.57701	$3.76327\times {10}^{-21}$	1.91675	50.919
	80	$5.23311\times {10}^{-11}$	0.40594	$5.53516\times {10}^{-22}$	2.86123	58.848
	88	$3.48707\times {10}^{-11}$	0.77341	$3.16603\times {10}^{-23}$	3.03152	67.200
	96	$1.60907\times {10}^{-11}$	0.92099	$1.52737\times {10}^{-24}$	2.32035	76.073
	104	$6.40610\times {10}^{-12}$	1.01064	$1.50047\times {10}^{-25}$	1.70813	85.546
	112	$2.33174\times {10}^{-12}$	1.08048	$2.71891\times {10}^{-26}$	1.66315	95.438
	120	$7.91468\times {10}^{-13}$	1.14695	$5.15344\times {10}^{-27}$	2.63196	105.852
	128	$2.51373\times {10}^{-13}$		$3.70726\times {10}^{-28}$		116.853

demonstrates the maximum absolute errors, $M.A.E.(K,J)$, and the relative errors $R.E.(K,J)$ of our results along with the corresponding observed convergence rates for this problem. As shown in the table, by increasing the number of sine-Fourier expansion terms, K, the error indicators exhibit a decreasing trend. Additionally, the computation time is detailed in the final column of Table 2. Figure 1 illustrates the error function $v_{app}^{K,J}(x,t)-v(x,t)$ in $3D$ along with the contour plot of the absolute error function for $K=128$ and $J=16$, obtained using our proposed method.

Problem 2.

In this test problem, the following MAFSDE is considered:

$$\begin{array}{c}\begin{array}{cc}& v_t(x,t)-\left({\displaystyle \frac{1}{2}}{}_{0^{+}}{}^C\mathcal{D}_t^{\alpha }{v}_{xx}(x,t)+{\displaystyle \frac{1}{2}}{}_{0^{+}}{}^C\mathcal{D}_t^{\beta }{v}_{xx}(x,t)\right)=s(x,t),\mathrm{for}(x,t)\in [0,1]\times [0,1],\hfill \\ & v(x,0)=0,x\in [0,1],\hfill \\ & v(0,t)=t^{2-\alpha }+t^{2-\beta },t\in [0,1],\hfill \\ & v(1,t)=exp\left(1\right)\left(t^{2-\alpha }+t^{2-\beta }\right),t\in [0,1],\hfill \end{array}\end{array}$$

where

$$\begin{array}{c}\begin{array}{cc}\hfill s(x,t)=& -[(\alpha -2)t^{1-\alpha }+{\displaystyle \frac{\Gamma (3-\alpha )}{2}}\left({\displaystyle \frac{t^{2-2\alpha }}{\Gamma (3-2\alpha )}}+{\displaystyle \frac{t^{2-\alpha -\beta }}{\Gamma (3-\alpha -\beta )}}\right)\hfill \\ & +(\beta -2)t^{1-\beta }+{\displaystyle \frac{\Gamma (3-\beta )}{2}}\left({\displaystyle \frac{t^{2-2\beta }}{\Gamma (3-2\beta )}}+{\displaystyle \frac{t^{2-\alpha -\beta }}{\Gamma (3-\alpha -\beta )}}\right)]exp\left(x\right).\hfill \end{array}\end{array}$$

The exact solution of this problem is $v(x,t)=\left(t^{2-\alpha }+t^{2-\beta }\right)exp\left(x\right)$.

Liu et al. [2] and also Dehghan et al. [26] previously investigated this test problem.

Table 3. Evaluation of absolute error measurements at $t=1$ for Problem 2, $\alpha =0.5$, and $\beta =0.8$.

x	Our Results for $(\mathit{\delta }=0.1,\mathit{J}=16)$			Reported Results in [2]	Reported Results in [26]
	$\mathit{K}=\mathbf{16}$	$\mathit{K}=\mathbf{24}$	$\mathit{K}=\mathbf{32}$
0.1	$2.6059\times {10}^{-6}$	$5.1546\times {10}^{-7}$	$8.0429\times {10}^{-8}$	$2.045\times {10}^{-3}$	$3.1829\times {10}^{-5}$
0.2	$5.2143\times {10}^{-6}$	$1.8225\times {10}^{-7}$	$3.6830\times {10}^{-8}$	$3.730\times {10}^{-3}$	$1.1293\times {10}^{-5}$
0.3	$1.8701\times {10}^{-6}$	$3.7198\times {10}^{-7}$	$7.8454\times {10}^{-9}$	$5.039\times {10}^{-3}$	$8.7748\times {10}^{-6}$
0.4	$3.0722\times {10}^{-6}$	$5.3332\times {10}^{-8}$	$2.0156\times {10}^{-8}$	$5.947\times {10}^{-3}$	$1.2005\times {10}^{-5}$
0.5	$4.4153\times {10}^{-6}$	$3.6801\times {10}^{-7}$	$4.3321\times {10}^{-8}$	$6.417\times {10}^{-3}$	$1.3001\times {10}^{-5}$
0.6	$2.9825\times {10}^{-7}$	$2.6272\times {10}^{-7}$	$5.1846\times {10}^{-8}$	$6.398\times {10}^{-3}$	$6.5135\times {10}^{-5}$
0.7	$5.9596\times {10}^{-6}$	$4.0341\times {10}^{-7}$	$3.3943\times {10}^{-8}$	$5.833\times {10}^{-3}$	$1.2941\times {10}^{-4}$
0.8	$8.0670\times {10}^{-6}$	$6.7808\times {10}^{-7}$	$2.5095\times {10}^{-8}$	$4.642\times {10}^{-3}$	$1.7532\times {10}^{-4}$
0.9	$7.8264\times {10}^{-7}$	$8.0127\times {10}^{-7}$	$1.6717\times {10}^{-7}$	$2.734\times {10}^{-3}$	$1.5478\times {10}^{-4}$

shows the absolute errors of the presented method in [2], for $(\tau ,h)=(0.01,0.1)$, and the method in [26], for $N_e=2$, $N=8$, and $\tau =1/16$. Additionally, we include results for our method, evaluated at the final time $t=1$, using varying values of K and J. The numerical computations were performed using the parameter values $(\alpha ,\beta )=(0.5,0.8)$. As demonstrated in Table 3, our method yields higher accuracy than the other methods.

Moreover,

Table 4. CPU time and the max absolute errors, $M.A.E.(K,J)$, and the relative errors, $R.E.(K,J)$, along with the corresponding convergence rates using our computed results for Problem 2, with $(\alpha ,\beta )=(0.5,0.8)$.

J	K	$\mathit{M}.\mathit{A}.\mathit{E}.(\mathit{K},\mathit{J})$	${\mathit{\rho }}_{8,0}^{\mathit{M}.\mathit{A}.\mathit{E}.(\mathit{K},\mathit{J})}$	$\mathit{R}.\mathit{E}.(\mathit{K},\mathit{J})$	${\mathit{\rho }}_{8,0}^{\mathit{R}.\mathit{E}.(\mathit{K},\mathit{J})}$	CPU Time (s)
16	8	$2.91840\times {10}^{-4}$	2.18220	$4.63258\times {10}^{-10}$	5.32615	2.939
	16	$3.29174\times {10}^{-5}$	1.93531	$1.15670\times {10}^{-12}$	3.73332	7.478
	24	$4.75260\times {10}^{-6}$	1.85493	$1.31679\times {10}^{-14}$	3.04810	12.417
	32	$7.43614\times {10}^{-7}$	1.83758	$2.93714\times {10}^{-16}$	2.71235	17.677
	40	$1.18385\times {10}^{-7}$	1.86460	$9.19137\times {10}^{-18}$	3.47158	23.480
	48	$1.83447\times {10}^{-8}$	1.94912	$1.10745\times {10}^{-19}$	3.38432	29.827
	56	$2.61228\times {10}^{-9}$	2.17051	$1.98267\times {10}^{-21}$	2.50484	36.634
	64	$2.98113\times {10}^{-10}$	2.52752	$8.84114\times {10}^{-23}$	1.97220	44.029
	72	$2.38063\times {10}^{-11}$	1.00220	$6.01974\times {10}^{-24}$	1.91675	52.009
	80	$8.73863\times {10}^{-12}$	0.75487	$4.20637\times {10}^{-25}$	2.86123	60.292
	88	$4.10779\times {10}^{-12}$	1.13331	$9.34180\times {10}^{-27}$	3.03152	68.570
	96	$1.32257\times {10}^{-12}$	1.28313	$2.30048\times {10}^{-28}$	2.32035	77.756
	104	$3.66573\times {10}^{-13}$	1.37353	$1.25108\times {10}^{-29}$	1.70813	87.249
	112	$9.28205\times {10}^{-14}$	1.44361	$1.12900\times {10}^{-30}$	1.66315	97.394
	120	$2.19124\times {10}^{-14}$	1.51002	$1.02442\times {10}^{-31}$	2.63196	107.755
	128	$4.84057\times {10}^{-15}$		$2.84612\times {10}^{-33}$		118.732

demonstrates the maximum absolute errors, $M.A.E.(K,J)$, and relative errors, $R.E.(K,J)$, for our results, along with the corresponding observed convergence rates for this problem. The table indicates that increasing the number of sine-Fourier expansion terms, K, results in a decreasing trend in error indicators. Additionally, the computation time is detailed in the final column of Table 4.

Figure 2 illustrates the error function $v_{app}^{K,J}(x,t)-v(x,t)$ in 3D, accompanied by a contour plot of the absolute error function for $K=128$ and $J=16$, obtained using our proposed method.

Problem 3.

In this test problem, the following MAFSDE is considered:

$$\begin{array}{c}\begin{array}{cc}& v_t(x,t)-\left({}_{0^{+}}{}^C\mathcal{D}_t^{\alpha }{v}_{xx}(x,t)+{}_{0^{+}}{}^C\mathcal{D}_t^{\beta }{v}_{xx}(x,t)\right)=s(x,t),\mathrm{for}(x,t)\in [0,1]\times [0,1],\hfill \\ & v(x,0)=0,x\in [0,1],\hfill \\ & v(0,t)=t^{3-\alpha -\beta },t\in [0,1],\hfill \\ & v(1,t)=t^{3-\alpha -\beta }exp\left(1\right),t\in [0,1],\hfill \end{array}\end{array}$$

where

$$\begin{array}{c}\begin{array}{cc}\hfill s(x,t)=& \left[(3-\alpha -\beta )t^{2-\alpha -\beta }-{\displaystyle \frac{\Gamma (4-\alpha -\beta )}{\Gamma (4-2\alpha -\beta )}}{t}^{3-2\alpha -\beta }-{\displaystyle \frac{\Gamma (4-\alpha -\beta )}{\Gamma (4-\alpha -2\beta )}}{t}^{3-\alpha -2\beta }\right]exp\left(x\right).\hfill \end{array}\end{array}$$

The exact solution of this problem is $v(x,t)=t^{3-\alpha -\beta }exp\left(x\right)$.

Dehghan et al. [26] previously analyzed this test problem by employing the Legendre spectral element method. In

Table 5. CPU time and evaluation of different error measurements, along with the corresponding convergence rates, using our obtained results and the reported results in [26] for Problem 3, with $\alpha =0.9$ and $\beta =0.7$.

Our Results for $\mathit{\delta }=0.15$							Reported Results in [26]
$\mathit{J}$	$\mathit{K}$	$\mathit{M}.\mathit{A}.\mathit{E}.(\mathit{K},\mathit{J})$	${\mathit{\rho }}_{\mathbf{8},\mathbf{0}}^{\mathit{M}.\mathit{A}.\mathit{E}.(\mathit{K},\mathit{J})}$	$\mathit{R}.\mathit{E}.(\mathit{K},\mathit{J})$	${\mathit{\rho }}_{\mathbf{8},\mathbf{0}}^{\mathit{R}.\mathit{E}.(\mathit{K},\mathit{J})}$	CPU Time (s)	$\mathit{\tau }$	$\mathit{M}.\mathit{A}.\mathit{E}.\left(\mathit{\tau }\right)$	${\mathit{\rho }}^{\mathit{M}.\mathit{A}.\mathit{E}.}$
8	24	$7.086\times {10}^{-6}$	1.4912	$1.173\times {10}^{-13}$	3.0430	2.653	${\displaystyle \frac{1}{100}}$	$1.791\times {10}^{-4}$	0.4561
	32	$1.595\times {10}^{-6}$	1.4750	$5.594\times {10}^{-15}$	2.8655	6.631	${\displaystyle \frac{1}{200}}$	$1.135\times {10}^{-4}$	0.5209
	40	$3.649\times {10}^{-7}$	1.5042	$3.186\times {10}^{-16}$	3.4574	11.036	${\displaystyle \frac{1}{400}}$	$6.742\times {10}^{-5}$	0.5611
	48	$8.108\times {10}^{-8}$	1.5939	$1.004\times {10}^{-17}$	3.1578	15.837	${\displaystyle \frac{1}{800}}$	$3.847\times {10}^{-5}$	0.5888
	56	$1.647\times {10}^{-8}$	1.8357	$4.269\times {10}^{-19}$	2.3677	20.801	${\displaystyle \frac{1}{1600}}$	$2.135\times {10}^{-5}$	0.6083
	64	$2.627\times {10}^{-9}$	1.9745	$4.000\times {10}^{-20}$	1.7463	26.456	${\displaystyle \frac{1}{3200}}$	$1.162\times {10}^{-5}$	0.6237
	72	$3.647\times {10}^{-10}$	0.6521	$6.977\times {10}^{-21}$	3.0176	32.567	${\displaystyle \frac{1}{6400}}$	$6.228\times {10}^{-6}$	0.6345
	80	$1.900\times {10}^{-10}$	0.4381	$3.413\times {10}^{-22}$	1.4481	38.924	${\displaystyle \frac{1}{\mathrm{12,800}}}$	$3.302\times {10}^{-6}$	0.6435
	88	$1.226\times {10}^{-10}$	0.7784	$8.021\times {10}^{-23}$	2.3326	45.849	${\displaystyle \frac{1}{\mathrm{25,600}}}$	$1.735\times {10}^{-6}$	0.7135
	96	$5.629\times {10}^{-11}$		$7.784\times {10}^{-24}$		53.098	${\displaystyle \frac{1}{\mathrm{51,200}}}$	$8.500\times {10}^{-7}$

, the maximum absolute errors from the method proposed in [26] are reported for $N_e=10$, $N=4$, and multiple values of τ. Additionally, we present our results, including maximum absolute errors, relative errors, associated convergence rates, and CPU time of computations, computed under various values of K and J. The numerical computations were performed using the parameter values $(\alpha ,\beta )=(0.9,0.7)$. As observed in Table 5, our method exhibits superior accuracy relative to the method presented in [26].

It should be noted that the convergence rates in the last column of Table 5 are computed by ${\rho }^{M.A.E.}=ln{\displaystyle \frac{M.A.E.\left(\tau \right)}{M.A.E.\left(2\tau \right)}}$.

Figure 3 illustrates the error function $v_{app}^{K,J}(x,t)-v(x,t)$ in 3D, accompanied by a contour plot of the absolute error function for $K=128$ and $J=8$, obtained using our proposed method.

5. Conclusions

We have developed a novel and efficient algorithm to address modified anomalous time-fractional sub-diffusion equations, incorporating a generalized framework for these equations. The core of the method has involved transforming the original problem into a series of fractional ordinary differential equations (FODEs), such that any of them can be solved independently. This transformation has been achieved using the Fourier expansion technique. For each resulting FODE, initial value conditions have been derived from the original problem’s initial conditions. Additionally, we have introduced secondary initial value problems for each FODE, whose solutions have been combined linearly in an optimal manner. This linear combination has been essential for approximating the solution of original problem. Numerical experiments have validated the efficiency and accuracy of our algorithm. Consequently, this method serves as a robust approach to handle various kinds of differential equations, including time-fractional sub-diffusion problems, time-fractional telegraph equations, and diffusion-wave ones. Further, the following items can be performed with the method in future works:

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Solving variable-order fractional differential equations.

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Solving FDEs with other types of fractional differentiation.

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Solving (2+1)- and (3+1)-dimensional FDEs.

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As the obtained systems of ODEs are independent equations, we can consider a parallel algorithm to solve them.