Solutions of Nonlinear Fractional-Order Differential Equation Systems Using a Numerical Technique

Abstract

The primary objective of this study is to expand the application of analytical and numerical methods for solving nonlinear Systems of Fractional Differential Equations (SFDEs) with Caputo fractional derivatives (CFDs) under initial conditions. Our proposed approach, the Multistage Telescoping Decomposition Elzaki Method (MTDEM), integrates the advantages of the Elzaki transform with the Multistage Telescoping Decomposition Method (MTDM), significantly enhancing the efficiency of the solution process and improving the convergence rate. Additionally, it simplifies computational operations and reduces the computational complexity associated with solving these nonlinear systems. A comprehensive comparison is conducted to highlight the accuracy and computational advantages of our proposed method compared to existing techniques, including the exact solution and the Telescoping Decomposition Method (TDM), through numerical examples that demonstrate the effectiveness of the proposed approach. The flexibility of the MTDEM allows for its application in a wide range of nonlinear SFDEs, making it a valuable tool in various scientific and engineering fields. These systems are widely used in modeling numerous physical, biological, and economic phenomena, such as the dynamics of electrical systems, heat transfer, and population growth models, underscoring the importance of developing accurate and efficient computational methods for their solutions. Through this study, we present a novel contribution to enhancing numerical and analytical techniques, paving the way for broader applications in multiple domains that require precise and reliable solutions for complex fractional systems.

1. Introduction and Mathematical Preliminaries

Recently, fractional differential equations have attracted considerable attention, being studied and applied to various real-world phenomena across different fields. One reason for their limited popularity might be the existence of multiple, non-equivalent definitions of fractional derivatives. Additionally, their nonlocal nature makes it challenging to provide a clear geometric interpretation. However, over the recent years, scientists have increasingly focused on fractional calculus. Through the use of fractional derivatives, it has been discovered that numerous applications, particularly those spanning multiple disciplines, can be effectively described.

Various analytical and numerical techniques have emerged in recent years to tackle fractional differential equations; among the prominent techniques are the Adomian decomposition method (ADM) [1], the Homotopy perturbation method (HPM) [2,3], and the variational iteration method [4], which are frequently combined with integral transforms such as the Laplace transform [5,6,7,8], the Sumudu transform [4,9,10,11], the Elzaki transform (ET) [12,13,14,15,16,17], the ZZ transform (la transformation de Zain Ul Abadin Zafar [18,19,20,21,22,23,24]), the Aboodh transform [25], the ARA transform (Aliaa Burqan, Rania Saadeh, and Ahmad Qazza) [26,27], and the Double Kharrat–Toma transform [28] to improve their effectiveness. It is also worth noting that several modern techniques have explored the study of phenomena modeled by fractional-order equations, for example, [29,30,31].

The TDM was originally proposed in [19,21] as an analytical approach for resolving differential equations.

$$\left\{\begin{array}{cc}{u}_t\hfill & =f(t,u,u_t),t\in \mathsf{\Omega }\hfill \\ u\left(0\right)\hfill & =u_0.\hfill \end{array}\right.$$

(1)

where $\mathsf{\Omega }=[0;T]$ is a compact subset of $\mathbb{R}$. Rooted in the principles of Taylor series expansion, this method constructs solutions in the form of polynomials. A notable advantage of TDM lies in its ability to circumvent the computation of Adomian polynomials, thereby offering a more streamlined a replacement for the TDM. Subsequent advancements in the field led to the development of the MTDM, as explored in [32], where it was applied to fractional differential equations. Further extending its applicability, the authors of [33] employed this approach to solve systems of nonlinear fractional differential equations.

$$\begin{gathered} {}^{c}D^{{\alpha }_i}{y}_i\left(t\right)=f_i(t,y_1,\cdots ,y_n);y_i^{\left(k\right)}\left(0^{+}\right)=C_k^i,k=0,1,\dots ,m_i-1 \\ 1\le i\le n,m_i-1\le {\alpha }_i\le m_i \end{gathered}$$

(2)

The authors of [34] presented a new technique for addressing nonlinear fractional initial conditions problems, applying MTDM.

$${}^{c}D_{0^{+}}^{\rho }u\left(\tau \right)+Ru\left(\tau \right)+Nu\left(\tau \right)=g\left(\tau \right)$$

(3)

and the following initial condition:

$$u^{\left(k\right)}\left(0^{+}\right)=c_k,k=0,1,2,\cdots ,m-1,$$

(4)

where ${}^{c}D_{0^{+}}^{\rho }$ is the Caputo fractional derivative of the function $u\left(\tau \right)$, R is the linear differential operator of less order than $L,N$ represents the general nonlinear differential operator, and $g\left(t\right)$ is the source term.

By integrating the MTDM with the ET, this method greatly improved computational efficiency and accuracy. Through a series of computational simulations, the recommended strategy was shown to be highly functional, trustworthy, and easy to apply, offering a solid structure for resolving complex fractional differential equations effectively.

This method is widely recognized for its precision and rapid convergence, making it particularly effective for solving intricate nonlinear equations. By decomposing the problem into successive stages, this method provides enhanced authority over the solution procedure and minimizes computational load. However, the method is not without its challenges, as it can encounter algorithmic complexity, needing a delicate equilibrium between precision and performance. Despite these challenges, it remains a powerful tool for addressing high-dimensional mathematical problems involving several variables.

This study uses the MTDEM to solve SFDEs using CFD with specified initial conditions. The following sections of this paper are structured as follows: The second section provides a comprehensive overview of the foundational concepts of fractional calculus, the ET, and the MTDEM. The third section describes how the algorithms are used, and their hybridization with the ET for solving SFDEs using Caputo derivatives, as well as analysis errors, convergence analysis, and its utilization in different examples. The fourth section evaluates the MTDEM by contrasting the results with those from previous studies to highlight its accuracy and efficiency. Finally, our conclusion and perspectives are presented.

2. Essential Definitions

This section provides essential background information on fractional calculus, specifically focusing on fractional derivatives, fractional integrals, and fundamental definitions and properties ET.

2.1. Fundamental Concepts and Key Properties

In this section, various key properties and basic definitions of fractional-order operators are introduced, which will be applied throughout the paper, including the most widely adopted definitions, defined in the context of Riemann–Liouville and Caputo, which are stated as follows.

Definition 1.

([35]). The Riemann–Liouville fractional-order integral generalizes the concept of integration by allowing for the integration of functions to non-integer orders. It is given by

$${}^{\mathit{RL}}J_{{\mathrm{a}}^{+}}^{\rho }x\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\rho \right)}}{\int }_a^t{(t-z)}^{\rho -1}x\left(z\right)dz,t>a,\rho >0,a\ge 0$$

(5)

The fractional-order derivative operators are defined accordingly within the framework of Riemann–Liouville and Caputo as follows:

$${}^{\mathit{RL}}D_{{\mathrm{a}}^{+}}^{\rho }x\left(t\right)=D^n{J}_{a^{+}}^{n-\rho }u\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(n-\rho )}}{\displaystyle \frac{d^n}{d{t}^n}}{\int }_a^t{(t-z)}^{n-\rho -1}u\left(z\right)dz,\forall t>a$$

(6)

$${}^{c}D_{{\mathrm{a}}^{+}}^{\rho }x\left(t\right){=}^{RL}{J}_{a^{+}}^{n-\rho }{D}^{n}x\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(n-\rho )}}{\int }_a^t{(t-z)}^{n-\rho -1}u\left(z\right)dz,\forall t>a$$

(7)

where $n-1<\rho <n,n\in N,$ and $\rho >0$.

Now, several important characteristics and properties of fractional-order operators are presented and demonstrated, which are crucial for understanding the core results of the present study.

Lemma 1

([35]). For $\rho \in R^{+}$, the Riemann–Liouville fractional integral of a power function is expressed as

$${}^{\mathit{RL}}J_{{\mathrm{a}}^{+}}^{\rho }{(t-a)}^s={\displaystyle \frac{\mathsf{\Gamma }(s+1)}{\mathsf{\Gamma }(s+1+\rho )}}{(t-a)}^{s+\rho }$$

(8)

Lemma 2

([35]). For $\rho \in R^{+}$, and $n-1<\rho \le n$, the CFD of a power function is expressed as

$${}^{c}D_{{\mathrm{a}}^{+}}^{\rho }{(t-a)}^s=\left\{\begin{array}{cc}{\displaystyle \frac{\mathsf{\Gamma }(s+1)}{\mathsf{\Gamma }(s+1-\rho )}}{(t-a)}^{s-\rho },\hfill & s\in {\mathbb{R}}^{+},s\ge \lceil \rho \rceil ,\hfill \\ 0,\hfill & s\in {\mathbb{R}}^{+},s<\lceil \rho \rceil ,\hfill \end{array}\right.$$

(9)

where $\lceil \rho \rceil$ denotes the integer part of ρ.

2.2. Basic Concepts of ET

The ET is an advanced integral transform constructed for functions of exponential order. It is particularly useful in solving fractional differential equations by simplifying their complexity [14]. The function space considered for this transform is

Definition 2.

Given a set of functions,

$$X=\left\{x\left(t\right):\exists M,k_1,k_2>0,\left|x\left(t\right)\right|<M{e}^{{\displaystyle \frac{\left|t\right|}{k_j}}},\mathrm{for}t\in {(-1)}^j[0,\infty )\right\}.$$

(10)

$k_1,k_2$ can be finite or infinite, and M is a real finite number. The ET of $x\left(t\right)\in X$, $\forall t\ge 0$, as

$$\mathcal{E}\left[x\left(t\right)\right]=T\left(s\right)=s{\int }_0^{\infty }x\left(t\right)e^{-{\displaystyle \frac{t}{s}}}dt,k_1\le s\le k_2.$$

(11)

Next, the inverse of the ET is expressed as ${\mathcal{E}}^{-1}\left[T\left(s\right)\right]=x\left(t\right),t\ge 0$, then $x\left(t\right)$ is called the inverse transform of Elzaki of $T\left(s\right)$, where ${\mathcal{E}}^{-1}$ is the operator of the inverse Elzaki transform.

When $\rho$ is a fractional number, there is

$$\mathcal{E}\left[t^{\rho }\right]=\mathsf{\Gamma }(\rho +1)s^{\rho +2}.$$

(12)

Definition 3.

([14]). The ET of CFD is given by

$$\mathcal{E}\left\{\left({}^{c}D_{0^{+}}^{\rho }x\right)\left(t\right),s\right\}={\displaystyle \frac{T\left(s\right)}{s^{\rho }}}-\sum _{i=0}^{n-1}{s}^{i-\rho +2}{x}^{\left(i\right)}\left(0\right).$$

(13)

2.3. Investigation of MTDM

Consider generalized nonlinear fractional equation

$$\mathcal{L}x+\mathcal{R}x+\mathcal{N}x=f$$

(14)

Here, $\mathcal{L}$ is a linear differential operator of higher order which is easily invertible, and $\mathcal{N}$ signifies the nonlinear operators, while $\mathcal{R}$ is the remaining linear part, f is a prescribed function, and x is the unknown function.

The fundamental concept of TDM [28] is used to represent the nonlinear term $\mathcal{N}x$ as an infinite series within the Banach space,

$$\mathcal{N}x\left(t\right)=\sum _{r=0}^{\infty }{\mathcal{N}}_{r}x\left(t\right),$$

(15)

where ${\mathcal{N}}_r$ is determined by

$$\left\{\begin{array}{cc}{\displaystyle {\mathcal{N}}_{0}x\left(t\right)=\mathcal{N}{x}_0\left(t\right),}& \\ & & \\ {\mathcal{N}}_{r}x\left(t\right)=\mathcal{N}{U}_r\left(t\right)-\mathcal{N}{U}_{r-1}\left(t\right)\end{array}\right.$$

(16)

with

$$U_r\left(t\right)=\sum _{k=0}^r{x}_k\left(t\right),U_{r-1}\left(t\right)=\sum _{k=0}^{r-1}{x}_k\left(t\right),r=1,2,3,\cdots$$

(17)

where $x\left(t\right)$ is the solution of Equation (14) and the residual linear part $\mathcal{R}u$ is expressed as

$$\mathcal{R}x\left(t\right)=\sum _{r=0}^{\infty }\mathcal{R}{x}_r\left(t\right).$$

(18)

Principle of the MTDM

The principle of the MTDM relies on dividing the time interval $[0,T]$ into subintervals, $[t_0,t_1],[t_1,t_2],\cdots ,[t_{M-1},t_M]$, such that $t_0=0,t_M=T$ and ${\bigcup }_{i=0}^{M-1}[t_i,t_{i+1}]=[0,T]$. The subintervals may be of uniform length $\mathsf{\Delta }$, i.e., $\mathsf{\Delta }{t}_i=\mathsf{\Delta }t$ for $i=0,\cdots ,M-1$.

More specifically, Equation (15) is resolved by TDM in each subinterval ${\mathsf{\Omega }}_i(i=0,1,\cdots ,M-1)$. The solution of Equation (14) is obtained piecewise. The iterative sequence follows: for ${\mathsf{\Omega }}_0=\left[t_0,t_1\right]$,

$${\mathsf{\Psi }}_{M_0}^0\left(t\right)=\sum _{k=0}^{M_0}{x}_k^0\left(t\right)=x_0^0\left(t\right)+x_1^0\left(t\right)+\cdots +x_{M_0}^0\left(t\right),M_0\in {\mathbb{N}}^{\ast }$$

(19)

and in every subsequent interval, ${\mathsf{\Omega }}_i=\left[t_i,t_{i+1}\right]$ for $(i=1,2,\cdots ,M-1)$, the team set

$$\left\{\begin{array}{cc}{x}_0^i\left(t\right)\hfill & ={\mathsf{\Psi }}_{M_0}^{i-1}\left(t_i\right)\hfill \\ {\mathsf{\Psi }}_{M_0}^i\left(t\right)\hfill & =\sum _{k=0}^{M_0}{x}_k^i\left(t\right).\hfill \end{array}\right.$$

(20)

3. Basic Idea of MTDEM

Assume a general SFDE of the form

$$\left\{\begin{array}{c}{}^c\mathcal{D}_{0^{+}}^{{\rho }_1}{x}_1\left(t\right)+{\mathcal{N}}_1(t,x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right))+{\mathcal{R}}_1(t,x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right))=g_1\left(t\right)\hfill \\ {}^c\mathcal{D}_{0^{+}}^{{\rho }_2}{x}_2\left(t\right)+{\mathcal{N}}_2(t,x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right))+{\mathcal{R}}_2(t,x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right))=g_2\left(t\right)\hfill \\ ⋮\hfill \\ {}^c\mathcal{D}_{0^{+}}^{{\rho }_n}{x}_n\left(t\right)+{\mathcal{N}}_n(t,x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right))+{\mathcal{R}}_n(t,x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right))=g_n\left(t\right)\hfill \end{array}\right.$$

(21)

where $m_i-1<{\rho }_i\le m_i,i=1,\cdots ,n,m_i=1,2,\cdots$, and the initial conditions are

$$x_i^{\left(k\right)}\left(0^{+}\right)={\mathcal{C}}_{i,k},i=1,\cdots ,n,k=0,1,\cdots ,m_i-1,$$

(22)

where $D^{{\rho }_i}$ are the CFD of the functions ${\mathcal{R}}_i$ are the linear differential operator, $N_i$ stand for the general nonlinear differential operator, and $g_i$ are the source terms, ${\mathcal{C}}_{i,k}\in \mathbb{R}$ end the solutions $x_i,i=1,\cdots ,n$ in the Banach space.

Theorem 1.

The solution of a SFDE with CFDs is provided as an infinite series that converges to the exact solution of the problem (21) and (22).

Proof. 

By applying the ET to both sides of Equations (21) and (22), it is shown that

$$\mathcal{E}\left[{}^c\mathcal{D}_{0^{+}}^{{\rho }_i}{x}_i\left(t\right)\right]+\mathcal{E}\left[{\mathcal{N}}_i(t,x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right))\right]+\mathcal{E}\left[{\mathcal{R}}_i(t,x_1\left(t\right),x_2\left(t\right),\cdots ,x_n\left(t\right))\right]=\mathcal{E}\left[g_i\left(t\right)\right]$$

(23)

Exploiting a property of the ET, from this, the following form can be derived

$$\mathcal{E}\left[x_i\left(t\right)\right]=\sum _{k=0}^{m_i-1}{x}_i^{\left(k\right)}\left(0\right)s^{k+2}+V^{{\rho }_i}\mathcal{E}\left[g_i\left(t\right)\right]-s^{{\rho }_i}\left[\mathcal{E}\left[{\mathcal{R}}_i(t,x_1\left(t\right),\cdots ,x_n\left(t\right))\right]+\mathcal{E}\left[{\mathcal{N}}_i(t,x_1\left(t\right),\cdots ,x_n\left(t\right))\right]\right]$$

(24)

Performing the inverse ET on both sides of Equation (24).

With the initial conditions, (22) gives

$$x_i\left(t\right)=G_i\left(t\right)-{\mathcal{E}}^{-1}\left[s^{{\rho }_i}\left[\mathcal{E}\left[{\mathcal{R}}_i(t,x_1\left(t\right),\cdots ,x_n\left(t\right))\right]+\mathcal{E}\left[{\mathcal{N}}_i(t,x_1\left(t\right),\cdots ,x_n\left(t\right))\right]\right]\right]$$

(25)

where $G_i\left(t\right)$ denotes the contributions from the nonhomogeneous terms and the given initial conditions.

On the other hand, the solutions are represented as follows:

$$u_i\left(t\right)=\sum _{j=0}^{\infty }{x}_{ij}\left(t\right),i=1,\cdots ,n,$$

(26)

and nonlinear terms can be decomposed:

$${\mathcal{N}}_i(t,u_1\left(t\right),\cdots ,x_n\left(t\right))=\sum _{j=0}^{\infty }{\mathcal{N}}_{ij}(t,x_1\left(t\right),\cdots ,x_n\left(t\right)),i=1,\cdots ,n.$$

(27)

where

$$\left\{\begin{array}{c}{\mathcal{N}}_{i,j}\left(t,x_1\left(t\right)\cdots ,u_n\left(t\right)\right)={\mathcal{N}}_i(t,U_{1j}\left(t\right),\cdots ,U_{nj}\left(t\right))-{\mathcal{N}}_i(t,U_{1j-1}\left(t\right),\cdots ,U_{nj-1}\left(t\right)),i=1,\cdots ,n.\hfill \\ \\ U_{ij}\left(t\right)=\sum _{k=0}^j{u}_{ik}\left(t\right),U_{i(j-1)}\left(t\right)=\sum _{k=0}^{j-1}{x}_{ik}\left(t\right),i=1,\cdots ,n.\hfill \end{array}\right.$$

(28)

By using (26) and (27), it can be rewritten (25) as

$$\sum _{j=0}^{\infty }{x}_{ij}\left(t\right)=G_i\left(t\right)-{\mathcal{E}}^{-1}\left\{s^{{\rho }_i}\left[\mathcal{E}\left(\sum _{j=0}^{\infty }{\mathcal{R}}_i(t,u_{1j}\left(t\right),\cdots ,x_{nj}\left(t\right))\right)+\mathcal{E}\left(\sum _{j=0}^{\infty }{\mathcal{N}}_i(t,u_1\left(t\right),\cdots ,u_n\left(t\right))\right)\right]\right\}$$

(29)

The two sides of Equation (29) are examined and compared, yielding the first term of the solution

$$x_{00}\left(t\right)=G_0\left(t\right),x_{10}\left(t\right)=G_1\left(t\right),\cdots ,x_{n0}\left(t\right)=G_n\left(t\right)$$

By following the same method, the general recursive relation is obtained

$$\left\{\begin{array}{c}{x}_{i0}\left(t\right)=G_i\left(t\right)\hfill \\ x_{ij}\left(t\right)=-{\mathcal{E}}^{-1}\left\{s^{{\rho }_i}\left[\mathcal{E}\left({\mathcal{R}}_i(t,x_{1j}\left(t\right),\cdots ,x_{nj}\left(t\right))\right)+\mathcal{E}\left({\mathcal{N}}_i(t,u_1\left(t\right),\cdots ,x_n\left(t\right))\right)\right]\right\}.\hfill \end{array}\right.$$

(30)

Ultimately, the approximate solution is obtained through

$$x_i\left(t\right)=\underset{N\to +\infty }{lim}\sum _{j=0}^N{x}_{ij}\left(t\right),i=1,\cdots ,n.$$

(31)

 □

3.1. Convergence Analysis

Theorem 2.

The series solution of a SFDE with CFDs (21) and (22) using MTDLM converges if $|x_{i1}|<\infty$ and $0<\beta <1$, and $\beta =\left(L_1+L_2\right){\displaystyle \frac{C^{{\rho }_i}}{\mathsf{\Gamma }({\rho }_i+1)}}$ where $L_1$ and $L_2$ are Lipschitz constants.

Proof. 

Define a sequence $\left\{S_{in}\right\}$ such that $S_{in}\left(t\right)={\sum }_{j=0}^n{x}_{ij}\left(t\right),i=1,\cdots ,n$ represents the sequence of partial sums derived from the series solution ${\sum }_{j=0}^{\infty }{x}_{ij}\left(t\right),i=1,\cdots ,n$.

Assume that ${\mathcal{R}}_{i}x\left(t\right)$ and ${\mathcal{N}}_{ij}x\left(t\right)$ are Lipschitzian with $|{\mathcal{R}}_{i}x\left(t\right)-{\mathcal{R}}_i\check{x}\left(t\right)|<L_1|x\left(t\right)-\check{x}\left(t\right)|$ and $|{\mathcal{N}}_{i}x\left(t\right)-{\mathcal{N}}_i\check{x}\left(t\right)|<L_2|u\left(t\right)-\check{x}\left(x\right)|,i=1,\cdots ,n$.

Let $\left\{S_{in}\right\}$ and $\left\{S_{im}\right\}$ be two arbitrary partial sums with $n>m,i=1,\cdots ,n$. It will now be proven that $\left\{S_{in}\right\}$ is a Cauchy sequence in this Banach space.

$$\begin{array}{cc}\hfill \parallel S_{in}-S_{im}\parallel & =\underset{t\in I}{max}|S_{in}-S_{im}|=\underset{t\in I}{max}\left|\sum _{j=m+1}^n{x}_{ij}\left(t\right)\right|\hfill \\ & =\underset{t\in I}{max}\left|{\mathcal{E}}^{-1}\left\{v^{{\rho }_i}\left(\mathcal{E}\left[\sum _{j=m+1}^n{\mathcal{R}}_i(x_{1(j-1)},\cdots ,x_{n(j-1)})\right]+\mathcal{E}\left[\sum _{j=m+1}^n{\mathcal{N}}_{i(j-1)}(x_1,\cdots ,x_n)\right]\right)\right\}\right|\hfill \\ & =\underset{t\in I}{max}\left|{\mathcal{E}}^{-1}\left\{s^{{\rho }_i}\left(\mathcal{E}\left[\sum _{j=m}^{n-1}{\mathcal{R}}_i(x_{1j},\cdots ,x_{nj})\right]+\mathcal{E}\left[\sum _{j=m}^{n-1}{\mathcal{N}}_{ij}(x_1,\cdots ,x_n)\right]\right)\right\}\right|\hfill \\ & =\underset{\tau \in I}{max}\left|{\mathcal{E}}^{-1}\left\{s^{{\rho }_i}\left(\mathcal{E}\left[{\mathcal{R}}_i\left(S_{i(n-1)}\right)-{\mathcal{R}}_i\left(S_{i(m-1)}\right)\right]+\mathcal{E}\left[{\mathcal{N}}_{ij}\left(S_{i(n-1)}\right)-{\mathcal{N}}_{ij}\left(S_{i(m-1)}\right)\right]\right)\right\}\right|\hfill \\ & \le \underset{t\in I}{max}\left|{\mathcal{E}}^{-1}\left\{s^{{\rho }_i}\left(\mathcal{E}\left[{\mathcal{R}}_i\left(S_{i(n-1)}\right)-{\mathcal{R}}_i\left(S_{i(m-1)}\right)\right]\right)\right\}\right|\hfill \\ & +\underset{t\in I}{max}\left|E^{-1}\left\{s^{{\rho }_i}\left(E\left[{\mathcal{N}}_{ij}\left(S_{i(n-1)}\right)-{\mathcal{N}}_{ij}\left(S_{i(m-1)}\right)\right]\right)\right\}\right|.\hfill \\ & \le \underset{t\in I}{max}{\mathcal{E}}^{-1}\left\{s^{{\rho }_i}\left(\mathcal{E}{L}_1\left|S_{i(n-1)}-S_{i(m-1)}\right|\right)\right\}+\underset{t\in I}{max}{\mathcal{E}}^{-1}\left\{s^{{\rho }_i}\left(\mathcal{E}{L}_2\left|S_{i(n-1)}-S_{i(m-1)}\right|\right)\right\}\hfill \\ & \le L_1{\mathcal{E}}^{-1}\left\{s^{{\rho }_i}\left(\mathcal{E}∥S_{i(n-1)}-S_{i(m-1)}∥\right)\right\}+L_2{\mathcal{E}}^{-1}\left\{v^{{\rho }_i}\left(\mathcal{E}∥S_{i(n-1)}-S_{i(m-1)}∥\right)\right\}\hfill \\ & \le \left(L_1+L_2\right){\mathcal{E}}^{-1}\left\{s^{{\rho }_i}\left(\mathcal{E}∥S_{i(n-1)}-S_{i(m-1)}∥\right)\right\}\hfill \\ & \le \left(L_1+L_2\right){\displaystyle \frac{C^{{\rho }_i}}{\mathsf{\Gamma }({\rho }_i+1)}}∥S_{i(n-1)}-S_{i(m-1)}∥\hfill \\ & \le \beta ∥S_{i(n-1)}-S_{i(m-1)}∥\hfill \end{array}$$

Let $n=m+1$; then,

$$∥S_{i(m+1)}-S_{im}∥\le \beta ∥S_{im}-S_{i(m-1)}∥\le \cdots \le {\beta }^m∥S_{i1}-S_{i0}∥$$

(32)

According to the triangle inequality, it can be show that

$$\begin{array}{cc}\hfill ∥S_{in}-S_{im}∥& =∥S_{in}-S_{i(n-1)}+S_{i(n-1)}-S_{i(n-2)}+\cdots +S_{i(m+1)}-S_{im}∥\hfill \\ & \le ∥S_{in}-S_{i(n-1)}∥+∥S_{i(n-1)}-S_{i(n-2)}∥+\cdots +∥S_{i(m+1)}-S_{im}∥\hfill \\ & \le \left({\beta }^n+{\beta }^{n-1}+\cdots +{\beta }^{m+1}\right)∥S_{i1}-S_{i0}∥\hfill \\ & \le {\beta }^{m+1}\left({\beta }^{n-m-1}+{\beta }^{n-m-2}+\cdots +\beta +1\right)∥S_{i1}-S_{i0}∥\hfill \\ & \le {\beta }^{m+1}\left({\displaystyle \frac{1-{\beta }^{n-m}}{1-\beta }}\right)∥x_{i1}\left(t\right)∥\hfill \end{array}$$

Since, $0<\beta <1$ and $n>m$, then $(1-{\beta }^{n-m})\le 1$. As a result

$$\begin{array}{c}\hfill ∥S_{in}-S_{im}∥\le \left({\displaystyle \frac{{\beta }^{m+1}}{1-\beta }}\right)∥x_{i1}\left(t\right)∥i=1\cdots ,n\end{array}$$

However, $|x_{i1}\left(t\right)|<\infty$ and as $m\to \infty$, then $\parallel S_{in}-S_{im}\parallel \to 0$, and hence $\left\{S_{in}\right\}$ forms a Cauchy sequence within this Banach space. So, the series ${\sum }_{j=0}^{\infty }{x}_{ij}\left(t\right),i=1,\cdots ,n$ converges, and this completes the proof. □

3.2. Error Analysis

For MTDEM, the maximum absolute truncated error of the series solution can be evaluated using the following theorem.

Theorem 3.

The upper bound of the absolute truncation error for the series solution ${\sum }_{j=0}^{\infty }{x}_{ij}\left(t\right),i=1,\cdots ,n$ to the system (21) and (22) is calculated to be

$$\underset{t\in I}{max}\left|x_i\left(t\right)-\sum _{j=0}^m{x}_{ij}\left(t\right)\right|\le {\displaystyle \frac{{\beta }^{m+1}}{1-\beta }}\underset{t\in I}{max}\left|x_{i1}\left(t\right)\right|.$$

(33)

Proof. 

From Theorem (37), it follows that equation $\parallel S_{in}-S_{im}\parallel \le {\displaystyle \frac{{\beta }^{m+1}}{1-\beta }}{max}_{t\in I}\left|y_{i1}\left(t\right)\right|.$ But $S_{in}={\sum }_{j=0}^n{x}_{ij}\left(t\right)$, and as $n\to \infty$, then $S_{in}\to u_i\left(t\right)$, so

$$\parallel x_i\left(t\right)-S_{im}\parallel \le {\displaystyle \frac{{\beta }^{m+1}}{1-\beta }}\underset{t\in I}{max}\left|x_{i1}\left(t\right)\right|.$$

(34)

Thus, the upper bound of the absolute truncation error in the interval I is

$$\underset{t\in I}{max}\left|x_i\left(t\right)-\sum _{j=0}^m{x}_{ij}\left(t\right)\right|\le {\displaystyle \frac{{\beta }^{m+1}}{1-\beta }}\underset{t\in I}{max}\left|x_{i1}\left(t\right)\right|.$$

(35)

Thus, the desired result follows. □

3.3. Demonstrative Example

This part employs the TDETM to resolve SFDEs. By analyzing this numerical example, readers will gain a clearer understanding of how the scheme can be implemented in SFDEs.

Example 1.

The first step, SFDEs, involves analyzing time-fractional derivatives.

$$\left\{\begin{array}{c}{}^c\mathcal{D}^{{\rho }_1}{x}_1\left(t\right)=2x_2^2\left(t\right),0<{\rho }_1\le 1,x_1\left(0\right)=0\hfill \\ {}^c\mathcal{D}^{{\rho }_2}{x}_2\left(t\right)=t{x}_1\left(t\right),0<{\rho }_2\le 1,x_2\left(0\right)=1\hfill \\ {}^c\mathcal{D}^{{\rho }_3}{x}_3\left(t\right)=x_2\left(t\right)x_3\left(t\right),0<{\rho }_3\le 1,x_3\left(0\right)=1\hfill \end{array}\right.$$

(36)

By applying the ET on either side of Equation (36) and employing the differentiation property, it can be derived that

$$\left\{\begin{array}{c}{x}_1\left(t\right)=2{\mathcal{E}}^{-1}\left[s^{{\rho }_1}\mathcal{E}\left[x_2^2\left(t\right)\right]\right]\hfill \\ x_2\left(t\right)={\mathcal{E}}^{-1}\left[s^2\right]+{\mathcal{E}}^{-1}[s^{{\rho }_2}\mathcal{E}\left[t{x}_1\left(t\right)\right]\hfill \\ x_3\left(t\right)={\mathcal{E}}^{-1}\left[s^2\right]+{\mathcal{E}}^{-1}\left[s^{{\rho }_3}{x}_2\left(t\right)x_3\left(t\right)\right].\hfill \end{array}\right.$$

(37)

In the interval ${\mathsf{\Omega }}_0=\left[t_0,t_1\right]$, the solution using the MTDEM method is given as

$${\mathsf{\Phi }}_{N_0}^0\left(t\right)=\left\{\begin{array}{c}{\mathsf{\Psi }}_{1,N_0}^0\left(t\right)=\sum _{j=0}^{N_0}{x}_{1j}^0\left(t\right)\hfill \\ {\mathsf{\Psi }}_{2,N_0}^0\left(t\right)=\sum _{j=0}^{N_0}{x}_{2j}^0\left(t\right)\hfill \\ {\mathsf{\Psi }}_{3,N_0}^0\left(t\right)=\sum _{j=0}^{N_0}{x}_{3j}^0\left(t\right)\hfill \end{array}\right.$$

(38)

• For each interval ${\mathsf{\Omega }}_k=\left[t_k,t_{k+1}\right]$

The initial conditions are

$$\left\{\begin{array}{c}{x}_{10}^k\left(t\right)={\mathsf{\Psi }}_{1,N_0}^{k-1}\left(t_k\right),\hfill \\ x_{20}^k\left(t\right)={\mathsf{\Psi }}_{2,N_0}^{k-1}\left(t_k\right),\hfill \\ x_{30}^k\left(t\right)={\mathsf{\Psi }}_{3,N_0}^{k-1}\left(t_k\right).\hfill \end{array}\right.$$

(39)

and

$${\mathsf{\Phi }}_{N_0}^k\left(t\right)=\left\{\begin{array}{c}{\mathsf{\Psi }}_{1,N_0}^k\left(t\right)=\sum _{j=0}^{N_0}{x}_{1j}^k\left(t\right)\hfill \\ {\mathsf{\Psi }}_{2,N_0}^k\left(t\right)=\sum _{j=0}^{N_0}{x}_{2j}^k\left(t\right)\hfill \\ {\mathsf{\Psi }}_{3,N_0}^k\left(t\right)=\sum _{j=0}^{N_0}{x}_{3j}^k\left(t\right)\hfill \end{array}\right.$$

(40)

Now, the solution of the system (36) is provided in the piecewise form as

$${\mathsf{\Phi }}_{N_0}\left(t\right)=\left\{\begin{array}{c}{\mathsf{\Psi }}_{N_0}^0\left(t\right),0\le t\le t_1,\hfill \\ {\mathsf{\Psi }}_{N_0}^1\left(t\right),t_1\le t\le t_2\hfill \\ ⋮\hfill \\ {\mathsf{\Psi }}_{N_0}^k\left(t\right),t_k\le t\le t_{k+1}\hfill \end{array}\right.$$

(41)

The recursive relations are given as follows.

• For $j=0$

  $$\left\{\begin{array}{cc}\hfill x_{10}\left(t\right)& =0,N_{10}(t,x)={\left(x_{20}\right)}^2=1,\hfill \\ \hfill x_{20}\left(t\right)& =1,R_{20}(t,x)=x_{10}\left(t\right)=0,\hfill \\ \hfill x_{30}\left(t\right)& =1,N_{30}(t,x)=x_{20}\left(t\right)x_3\left(t\right)=1.\hfill \end{array}\right.$$

  (42)

For $j=1$

$$\left\{\begin{array}{cc}\hfill x_{11}\left(t\right)& =2{\mathcal{E}}^{-1}\left[s^{{\rho }_1}\mathcal{E}\left[{\mathcal{N}}_{10}(t,x)\right]\right],\hfill \\ \hfill x_{21}\left(t\right)& ={\mathcal{E}}^{-1}\left[s^{{\rho }_2}\mathcal{E}\left[t·{\mathcal{R}}_{20}(t,x)\right]\right],\hfill \\ \hfill x_{31}\left(t\right)& ={\mathcal{E}}^{-1}\left[s^{{\rho }_3}\mathcal{E}\left[{\mathcal{N}}_{30}(t,x)\right]\right].\hfill \end{array}\right.$$

(43)

and for $j=r+1$

$$\left\{\begin{array}{cc}\hfill x_{1(r+1)}\left(t\right)& =2{\mathcal{E}}^{-1}\left[s^{{\rho }_1}\mathcal{E}\left[{\mathcal{N}}_{1r}(t,x)\right]\right],\hfill \\ \hfill x_{2(r+1)}\left(t\right)& ={\mathcal{E}}^{-1}\left[s^{{\rho }_2}\mathcal{E}\left[t·{\mathcal{R}}_{2r}(t,x)\right]\right],\hfill \\ \hfill x_{3(r+1)}\left(t\right)& ={\mathcal{E}}^{-1}\left[s^{{\rho }_3}\mathcal{E}\left[{\mathcal{N}}_{3r}(t,x)\right]\right].\hfill \end{array}\right.$$

(44)

The first few components of $x_{1(r+1)}\left(t\right),x_{2(r+1)}\left(t\right)$, and $x_{3(r+1)}\left(t\right)$, are as follows:

$$\left\{\begin{array}{c}{x}_{11}\left(t\right)=2{\displaystyle \frac{t^{{\rho }_1}}{\mathsf{\Gamma }({\rho }_1+1)}},{\mathcal{N}}_{11}(t,x)=0,\hfill \\ x_{21}\left(t\right)=0,{\mathcal{R}}_{21}(t,x)=x_{11}\left(t\right)=2{\displaystyle \frac{t^{{\rho }_1}}{\mathsf{\Gamma }({\rho }_1+1)}},\hfill \\ x_{31}\left(t\right)={\displaystyle \frac{t^{{\rho }_3}}{\mathsf{\Gamma }({\rho }_3+1)}},{\mathcal{N}}_{31}(t,x)=-1.\hfill \end{array}\right.$$

(45)

and

$$\left\{\begin{array}{cc}\hfill x_{12}\left(\tau \right)& =2{\mathcal{E}}^{-1}\left[s^{{\rho }_1}\mathcal{E}\left[{\mathcal{N}}_{11}(t,x_1,x_2,x_3\right)\right]]=0,\hfill \\ \hfill x_{22}\left(t\right)& ={\mathcal{E}}^{-1}\left[s^{{\rho }_2}\mathcal{E}\left[t·{\mathcal{R}}_{21}(t,x_1,x_2,x_3)\right]\right]={\displaystyle \frac{\mathsf{\Gamma }({\rho }_1+2)}{\mathsf{\Gamma }({\rho }_1+1)\mathsf{\Gamma }({\rho }_1+{\rho }_2+2)}}{t}^{{\rho }_1+{\rho }_2+1}\hfill \\ \hfill x_{32}\left(t\right)& ={\mathcal{E}}^{-1}\left[s^{{\rho }_3}\mathcal{E}\left[{\mathcal{N}}_{31}(t,x_1,x_2,x_3)\right]\right]=-{\displaystyle \frac{t^{{\rho }_3}}{\mathsf{\Gamma }({\rho }_3+1)}}\hfill \end{array}\right.$$

(46)

for the third iteration

$$\left\{\begin{array}{cc}\hfill x_{13}\left(t\right)& =2{\mathcal{E}}^{-1}\left[s^{{\rho }_1}\mathcal{E}\left[N_{12}(t,x_1,x_2,x_3\right)\right]],\hfill \\ \hfill x_{23}\left(t\right)& ={\mathcal{E}}^{-1}\left[s^{{\rho }_2}\mathcal{E}\left[t·R_{22}(t,x_1,x_2,x_3)\right]\right],\hfill \\ \hfill x_{33}\left(t\right)& ={\mathcal{E}}^{-1}\left[s^{{\rho }_3}\mathcal{E}\left[N_{32}(t,x_1,x_2,x_3)\right]\right].\hfill \end{array}\right.$$

(47)

By continuing in the same way, the other components can be found. At last, the series solutions $x_1\left(t\right),x_2\left(t\right)$, and $x_3\left(t\right)$ of Equation (36) are given by

$${\mathsf{\Psi }}_3^0\left(t\right)=\left\{\begin{array}{c}{\mathsf{\Psi }}_{1,3}^0\left(t\right)=x_1\left(t\right)=0+{\displaystyle \frac{t^{{\rho }_1}}{\mathsf{\Gamma }({\rho }_1+1)}}+\cdots ,\hfill \\ {\mathsf{\Psi }}_{2,3}^0\left(t\right)=x_2\left(t\right)=1+0+{\displaystyle \frac{\mathsf{\Gamma }({\rho }_1+2)}{\mathsf{\Gamma }({\rho }_1+1)\mathsf{\Gamma }({\rho }_1+{\rho }_2+2)}}{t}^{{\rho }_1+{\rho }_2+1}+\cdots \hfill \\ {\mathsf{\Psi }}_{3,3}^0\left(t\right)=x_3\left(t\right)=1+{\displaystyle \frac{t^{{\rho }_3}}{\mathsf{\Gamma }({\rho }_3+1)}}-{\displaystyle \frac{t^{{\rho }_3}}{\mathsf{\Gamma }({\rho }_3+1)}}+\cdots .\hfill \end{array}\right.$$

(48)

4. Numerical Experiments and Discussion

All the computations presented in this section have been obtained using Maple software (Maple 2024, Waterloo Maple Inc., Waterloo, Ontario, Canada). The solution of this problem is obtained by applying the technique described in Equation (36).

Example 2.

In this part, an example is presented to demonstrate the method with exact solutions and other methods. This will help readers understand how to compare exact, numerical solutions. Let the system of FDEs be

$$\left\{\begin{array}{cc}{}^c\mathcal{D}^{{\rho }_1}{x}_1\left(t\right)=-x_1\left(t\right),\hfill & x_1\left(0\right)=1\hfill \\ {}^c\mathcal{D}^{{\rho }_2}{x}_2\left(t\right)=-x_1\left(t\right)-x_2^2\left(t\right),\hfill & x_2\left(0\right)=0\hfill \\ {}^c\mathcal{D}^{{\rho }_3}{x}_3\left(t\right)=x_2^2\left(t\right),\hfill & x_3\left(0\right)=0\hfill \end{array}\right.$$

(49)

By applying the ET for the SFDEs in Equation (49), it follows that

$$\left\{\begin{array}{c}{x}_1\left(t\right)={\mathcal{E}}^{-1}\left[s^2\right]+\left[s^{{\rho }_1}\mathcal{E}\left[x_1\left(t\right)\right]\right]\hfill \\ x_2\left(t\right)={\mathcal{E}}^{-1}[s^{{\rho }_2}\mathcal{E}[-x_1\left(t\right)-x_2^2\left(t\right)]\hfill \\ x_3\left(t\right)={\mathcal{E}}^{-1}[s^{{\rho }_3}\mathcal{E}\left[x_2^2\left(t\right)\right].\hfill \end{array}\right.$$

(50)

the initial conditions are

$$\left\{\begin{array}{c}{x}_1\left(0\right)=C_{1,k}=1\hfill \\ x_2\left(0\right)=C_{2,k}=0\hfill \\ x_3\left(0\right)=C_{3,k}=0.\hfill \end{array}\right.$$

(51)

Since, $0<{\rho }_i\le 1$, then $k=0$.

• So,

  $$\left\{\begin{array}{c}{x}_1\left(0\right)=C_{1,0}=1\hfill \\ x_2\left(0\right)=C_{2,0}=0\hfill \\ x_3\left(0\right)=C_{3,0}=0.\hfill \end{array}\right.$$

  (52)

Example 3.

$$\left\{\begin{array}{cc}{}^c\mathcal{D}^{{\rho }_1}{x}_1\left(t\right)=-x_1\left(t\right)+x_2\left(t\right)x_3\left(t\right),\hfill & x_1\left(0\right)=1\hfill \\ {}^c\mathcal{D}^{{\rho }_2}{x}_2\left(t\right)=-x_2\left(t\right)x_3\left(t\right)-2x_2^2\left(t\right),\hfill & x_2\left(0\right)=2\hfill \\ {}^{c}D^{{\rho }_3}{x}_3\left(t\right)=x_2^2\left(t\right),\hfill & x_3\left(0\right)=0\hfill \end{array}\right.$$

(53)

Continuing the same steps explained in example 2, it follows that

$$\left\{\begin{array}{c}{x}_1\left(t\right)={\mathcal{E}}^{-1}\left[s^2\right]+\left[s^{{\rho }_1}\mathcal{E}\left[x_1\left(t\right)\right]\right]\hfill \\ x_2\left(t\right)=2{\mathcal{E}}^{-1}\left[s^2\right]+{\mathcal{E}}^{-1}[s^{{\rho }_2}\mathcal{E}[-x_2\left(t\right)x_3\left(t\right)-2x_2^2\left(t\right)]\hfill \\ x_3\left(t\right)={\mathcal{E}}^{-1}[s^{{\rho }_3}\mathcal{E}\left[x_2^2\left(t\right)\right].\hfill \end{array}\right.$$

(54)

4.1. TabularResults

Table 1. Comparison between the results from the exact solution, FTDM and MTDEM ($N_0=5$), for Equation (49).

${\mathit{t}}_{\mathit{i}}$	Numerical Results of ${\mathit{x}}_1\left(\mathit{t}\right)$
	Exact Solution	FTDM	MTDEM
0	1	1	1
0.5	0.6065306555	0.606532118	0.6065321181
1	0.3678794322	0.368055555	0.3678812102
1.5	0.2231301511	0.225976562	0.2231317696
2	0.1353352650	0.155555555	0.1353365849
2.5	0.0820849592	0.173719618	0.08227101598
3	0.0497870322	0.36250000	0.04990001359
3.5	0.0301973534	0.90809461	0.03026596092
4	0.0183156116	2.15555555	0.01835727739
4.5	0.0111089688	4.6791015	0.01113427834
5	0.00673791685	9.3680555	0.006753297424
${\mathit{t}}_{\mathit{i}}$	Numerical Results of  ${\mathit{x}}_{\mathbf{2}}\left(\mathit{t}\right)$
	Exact Solution	FTDM	MTDEM
0	0	0	0
0.5	0.3666759377	0.366672398	0.3666723985
1	0.503346689	0.502558020	0.5033397218
1.5	0.5151197386	0.498914047	0.5151123990
2	0.4784218368	0.3709221000	0.4784160375
2.5	0.428589959	−0.278 × ${10}^{-1}$	0.4285857901
3	0.3793248078	−1.711547 × ${10}^6$	0.3793810916
3.5	0.335194617	−1.1185865 × ${10}^{11}$	0.3352707834
4	0.297187977	−9.3550388 × ${10}^{14}$	0.2972663259
4.5	0.264975565	−2.0961374 × ${10}^{18}$	0.2650488343
5	0.2378134754	−1.8273720 × ${10}^{21}$	0.2378791642
${\mathit{t}}_{\mathit{i}}$	Numerical Results of  ${\mathit{x}}_{\mathbf{3}}\left(\mathit{t}\right)$
	Exact Solution	FTDM	MTDEM
0	0	0	0
0.5	0.0267934066	0.0267954831	0.02679548328
1	0.1287738779	0.129386423	0.1287790677
1.5	0.2617501102	0.275109392	0.2617558307
2	0.3862428981	0.473522300	0.3862473769
2.5	0.4893250815	0.85370000000	0.4893282235
3	0.5708881599	1.711548 × ${10}^6$	0.570939244
3.5	0.6346080284	1.1185864 × ${10}^{11}$	0.634682854
4	0.6844964111	9.3550387 × ${10}^{14}$	0.6845614266
4.5	0.7239154655	2.09613743 × ${10}^{18}$	0.7240019172
5	0.7554486076	1.8273720 × ${10}^{21}$	0.7555525682

and

Table 2. Comparison between the results from the exact solution, FTDM and MTDEM ($N_0=4$), for Equation (53).

${\mathit{t}}_{\mathit{i}}$	Numerical Results of ${\mathit{x}}_1\left(\mathit{t}\right)$
	Exact Solution	FTDM	MTDEM
0	1	1	1
0.5	0.7607628130	0.72313384	0.7574710997
1	0.5620657341	−2.366865	0.5630034014
1.5	0.3977588680	−233387.5294	0.3999142051
2	0.2756515018	−5.075903071 × ${10}^6$	0.2778574593
2.5	0.1889960647	−2.537093652 × ${10}^8$	0.1908877708
3	0.1288690558	−5.567442828 × ${10}^9$	0.1303783537
3.5	0.0876386351	−7.171360175 × ${10}^{10}$	0.08879997703
4	0.0595422272	−6.351686445 × ${10}^{11}$	0.06041790370
4.5	0.0111089688	−4.259538599 × ${10}^{12}$	0.01113427834
5	0.00673791685	−2.303969156 × ${10}^{13}$	0.006753297424
${\mathit{t}}_{\mathit{i}}$	Numerical Results of  ${\mathit{x}}_{\mathbf{2}}\left(\mathit{t}\right)$
	Exact Solution	FTDM	MTDEM
0	2	2	2
0.5	0.56737111575	0.79840051	0.5963406996
1	0.2735835848	−0.097095	0.2839292143
1.5	0.1555627924	−32212.1151	0.1607265228
2	0.0956977104	−7.627578871 × ${10}^6$	0.09863836696
2.5	0.0614374820	−3.937826779 × ${10}^8$	0.06328816258
3	0.0404791844	−8.787034972 × ${10}^9$	0.04170595050
3.5	0.02711140440	−1.143495031 × ${10}^{11}$	0.02795104991
4	0.0183532975	−1.019837235 × ${10}^{12}$	0.01893960688
4.5	0.264975565	−6.873555629 × ${10}^{12}$	0.2650488343
5	0.2378134754	−3.732045475 × ${10}^{13}$	0.2378791642
${\mathit{t}}_{\mathit{i}}$	Numerical Results of  ${\mathit{x}}_{\mathbf{3}}\left(\mathit{t}\right)$
	Exact Solution	FTDM	MTDEM
0	0	0	0
0.5	0.6193923396	0.52228093	0.6063342002
1	0.7007036555	3.023075	0.6950151510
1.5	0.7227514273	27829.6859	0.7186485735
2	0.7304239536	6.352213879 × ${10}^6$	0.7268158212
2.5	0.7334444988	3.237496837 × ${10}^8$	0.7300226878
3	0.7347200720	7.177257515 × ${10}^9$	0.7313763410
3.5	0.7352823904	9.303162403 × ${10}^{10}$	0.7319735550
4	0.7355371750	8.275031673 × ${10}^{11}$	0.7322445803
4.5	0.7239154655	5.566547764 × ${10}^{12}$	0.7240019172
5	0.7554486076	3.018007469 × ${10}^{13}$	0.7555525682

illustrate the comparison between the approximate solutions obtained using the proposed method, MFTDM, the exact solutions, and the solution of FTDM for each example, where $\mathsf{\Delta }{t}_i=0.5$. The tables clearly demonstrate a strong agreement between the exact solutions and those produced by the proposed method.

4.2. Graphical Analysis

The graphs in Examles 2 and 3 in Figure 1 and Figure 2, respectively, are given blow. The graph in Figure 1 shows the solutions of Example 2, and the graph in Figure 2 shows the solutions of Example 3. The X-axis is variable t, and the Y-axis is the solution $x_i$.

The proposed method’s approximate solutions in Figure 1 and Figure 2 are compared with the exact solutions. Thus, these comparisons give us valuable information about how the approximate solutions converge in the exact solution. Based on these results, the MTDEM method is more accurate and comes closer to exact solutions along the domain $\left[0.T\right]$.

5. Conclusions

We examine the key characteristics of the MTDEM, focusing on its application in solving various SFDEs. Additionally, graphs and tables demonstrate the accuracy and effectiveness of the proposed method by comparing it with exact solutions.

The findings confirm that the proposed approach is practical, reliable, and well suited for solving SFDEs with initial conditions. Consequently, this method can be effectively applied of SFDEs.

One of the notable advantages of the MTDEM is its ability to transform differential equations into algebraic equations, making them more comprehensible and easier to handle. This transformation significantly reduces the computational effort and time required to obtain exact solutions.

In the future, we hope we can use the MTDEM to solve the SFDEs with other types of derivatives. We could combine it with other numerical techniques to enhance its capabilities, which potentially could lead to more specialized solutions and broader applications across fields.