Analytical Insight into Some Fractional Nonlinear Dynamical Systems Involving the Caputo Fractional Derivative Operator

Abstract

This work explores modern mathematical avenues as part of fractional calculus research. We apply fractional dispersion relations to the fractional wave equation to numerically examine various formulations of the generalized fractional wave equation. The research explores Drinfeld–Sokolov–Wilson and shallow water equations as fundamental differential equations forming the basis of wave theory studies. This work presents effective methods to obtain the numerical solution of the fractional-order FDSW and FSW coupled system equations. The analysis employs Caputo fractional derivatives during studies of fractional orders. This study develops the new iterative transform technique (NITM) and homotopy perturbation transform method (HPTM) using Elzaki transform (ET) with a new iteration method and a homotopy perturbation method. The proposed techniques generate approximation solutions that adopt an infinite fractional series with fractional order solutions converging towards analytic integer solutions. The proposed method demonstrates its precision through tabular simulations of computed approximations and their absolute error values while representing results with 2D and 3D graphics. The paper presents the physical analysis of solution dynamics across diverse $ϵ$ ranges during a suitable time frame. The developed computational techniques yield numerical and graphical output, which are compared to analytic results to verify the solution convergence. The computational algorithms have proven their high accuracy, flexibility, effectiveness, and simplicity in evaluating fractional-order mathematical models.

1. Introduction

Researchers have pursued fractional calculus as a desirable field of research for many years, with ongoing growth in recent development [1,2]. In 1695, two mathematicians, Leibniz and L’Hospital, initially developed the concept [3]. Fractional calculus has managed the infinitesimal calculus as a discipline that explores the capability to differentiate some arbitrary functions with fractional orders, such as order ${\displaystyle \frac{1}{2}}$ differentiation [4]. Fractional calculus and its applications have appeared across multiple scientific fields, including physics, mathematical biology, viscoelasticity, signal processing, electrochemistry, finance, social science, and many more [5,6]. This specific field has emphasized the importance of fractional-order derivatives alongside their integrals [7]. Several researchers have demonstrated that fractional models derived from integer models serve as powerful tools for describing natural phenomena effectively [8,9,10,11,12]. The non-integer order differential-integral operators demonstrate greater flexibility than conventional operators because of their increased freedom. Two distinct approaches exist to define fractional derivatives, with Riemann–Liouville and Caputo establishing the first definitions. The analysis of Reimann–Liouville and Caputo models reveals significant gaps requiring deeper investigation in this field. Classical derivatives exhibit local behavior, whereas Caputo fractional derivatives have non-local characteristics that enable us to inspect interval-based changes instead of point-based changes. The property of the Caputo fractional derivative allows researchers to simulate physical phenomena, including ocean climate, atmospheric physics, dynamical systems, earthquakes, vibrations, and polymers, etc. (see the research papers [13,14,15,16,17,18]).

The recent advancement of real-world developments depends heavily on fractional differential equations (FDEs). Many analytical systems benefit from using fractional differential equations when establishing models for better management. These equations emerge in various fields, including electronic circuits and physics, as well as engineering, along with bioscience [19,20] and other scientific domains [21,22]. Mathematical handling of clients through fractional-order differential equations provides the finance industry with affordable solutions for managing financial crises [23,24]. DE applications can also be found in image and signal processing domains [25,26]. The mathematical models take two possible forms: linear and nonlinear, based on problem geometry requirements. Simple problems require ordinary differential equations (ODEs) for demonstration, whereas complex problems can be shown with partial differential equations (PDEs). The NPFDEs have established advanced modeling techniques to depict some real-life processes through their ability to incorporate multiple non-local effects, memory dynamics, and complex interactive systems [27]. The mathematical framework of the NPFDEs has united two concepts: the partial differential equations, which handle problems with multiple independent variables, and fractional calculus, which deals with non-integer order derivatives [28]. The NPFDEs have developed some practical applications across different domains because they have allowed us to model the complex systems that display the combined features of non-local connectivity, differential memory effects, and nonlinearity [29,30]. The NPFDEs contain fractional derivatives and nonlinear terms, leading to complex challenges during some analysis and solution processes. Approximate solution techniques have become essential for addressing the NPFDEs because most equations do not possess analytic solutions [31,32,33]. Numerous methods exist for obtaining approximate solutions, such as the optimal homotopy asymptotic method [34], the Exp-function method [35], the q-homotopy analysis method [36], the variational iteration method [37], the Yang transform decomposition method [38], the Homotopy analysis transform method [39], the reduced differential transform method [40,41], and the fractional sub-equation method [42], etc.

This research aims to derive solutions for the Fractional Drinfeld–Sokolov–Wilson (FDSW) coupled system and the fractional shallow water (FSW) coupled system. This fractional-order DSW equation incorporates memory effects and genetic consequences that help us understand the complex physical properties of systems. This system appears in the literature as [43].

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^{ϵ}\mathbb{S}(\vartheta ,\varrho )}{\partial {\varrho }^{ϵ}}}+\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}=0,\hfill \\ \hfill & {\displaystyle \frac{{\partial }^{ϵ}\mathbb{T}(\vartheta ,\varrho )}{\partial {\varrho }^{ϵ}}}+2{\displaystyle \frac{{\partial }^3\mathbb{T}(\vartheta ,\varrho )}{\partial {\vartheta }^3}}+2\mathbb{S}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}+\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}=0,\hfill \end{array}$$

(1)

where $0<ϵ\le 1$ indicates the fractional derivative operator in the Caputo manner.

Shallow-water equations, which are crucial to applied mathematics and physics, first appeared in the latter half of the 18th century. When examined physically, shallow-water waves are defined as the noticeable displacements of bodies of water, such as the sea or ocean [44,45,46]. At the same time, many physical phenomena that resemble the motions of shallow-water waves are seen in a variety of scientific domains, such as fluid dynamics, nuclear physics, plasma physics, and other related subjects. This paper’s second coupled system is the fractional-order shallow-water (FSW) equation, which characterizes a thin fluid layer with a constant density in hydrostatic equilibrium. The coupled SW equation represents the associated wave motion. This system appears in the literature as [47]

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^{ϵ}\mathbb{S}(\vartheta ,\varrho )}{\partial {\varrho }^{ϵ}}}+\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}+\mathbb{S}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}=0,\hfill \\ \hfill & {\displaystyle \frac{{\partial }^{ϵ}\mathbb{T}(\vartheta ,\varrho )}{\partial {\varrho }^{ϵ}}}+\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}+{\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}=0,\hfill \end{array}$$

(2)

where $\mathbb{S}(\vartheta ,\varrho )$ and $\mathbb{T}(\vartheta ,\varrho )$ denote the free surface and the horizontal velocity component.

In this paper, we combine the Elzaki transform with a new iterative method (NIM) and the Homotopy perturbation method (HPM) to evaluate the approximate results of nonlinear fractional DSW and SW equations. The main benefits of these methods are their simplicity and the minimal effort required for calculations. Additionally, the analytic solutions found in the literature and the solutions obtained through these methods agree perfectly. Numerical outcomes from these methods are more precise and accurate than those from other methods. To determine the accuracy of each approach, the numerical results are compared to the other methods. We will show that the results of the proposed approaches are predominantly consistent, proving their effectiveness and dependability. Nonetheless, the novel methods surpass the other literature approaches, increasing their value and effectiveness. To obtain precise and approximate results in a few simple steps, researchers may use this investigation as a core reference to explore these techniques and apply them in numerous applications. The presentation of two novel strategies for fractional nonlinear dynamical systems with minimal and progressive phases makes this work interesting. The structure of this article is divided into seven sections: Section 2 covers the fundamental concepts of fractional calculus needed in our research. Section 3 introduces the basic concepts of the NITM. Section 4 introduces the basic concepts of the ETDM. Section 5 of this work demonstrates the convergence results of the proposed approaches. In Section 6, the proposed approaches serve to solve both fractional DSW and fractional SW coupled systems. The conclusion segment appears as the last section.

2. Basic Definitions

In this portion, we expressed the basic results of FC associated with the current work.

Definition 1.

The non-integer derivative in Abel-Riemann manner is as [48,49,50]

$$D^{ϵ}\nu \left(\psi \right)=\left\{\begin{array}{c}{\displaystyle \frac{d^{\delta }}{d{\psi }^{\delta }}}\nu \left(\psi \right),ϵ=\delta ,\hfill \\ {\displaystyle \frac{1}{\Gamma (\delta -ϵ)}}{\displaystyle \frac{d}{d{\psi }^{\delta }}}{\int }_0^{\psi }{\displaystyle \frac{\nu \left(\psi \right)}{{(\psi -\vartheta )}^{ϵ-\delta +1}}}d\vartheta ,\delta -1<ϵ<\delta ,\hfill \end{array}\right.$$

(3)

with $\delta \in Z^{+}$, $ϵ\in R^{+}$ and

$$D^{-ϵ}\nu \left(\psi \right)={\displaystyle \frac{1}{\Gamma \left(ϵ\right)}}{\int }_0^{\psi }{(\psi -\vartheta )}^{ϵ-1}\nu \left(\vartheta \right)d\vartheta ,0<ϵ\le 1.$$

(4)

Definition 2.

The non-integer Abel-Riemann integration operator is as [48,49,50]

$$J^{ϵ}\nu \left(\psi \right)={\displaystyle \frac{1}{\Gamma \left(ϵ\right)}}{\int }_0^{\psi }{(\psi -\vartheta )}^{ϵ-1}\nu \left(\psi \right)d\psi ,\psi >0,ϵ>0,$$

(5)

with below properties:

$$\begin{array}{c}\hfill J^{ϵ}{\psi }^{\delta }={\displaystyle \frac{\Gamma (\delta +1)}{\Gamma (\delta +ϵ+1)}}{\psi }^{\delta +\vartheta },\\ \hfill D^{ϵ}{\psi }^{\delta }={\displaystyle \frac{\Gamma (\delta +1)}{\Gamma (\delta -ϵ+1)}}{\psi }^{\delta -\psi }.\end{array}$$

Definition 3.

The non-integer derivative in Caputo manner is as [51]

$${}^{C}D^{ϵ}\nu \left(\psi \right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\Gamma (\delta -ϵ)}}{\int }_0^{\psi }{\displaystyle \frac{{\nu }^{\delta }\left(\vartheta \right)}{{(\psi -\vartheta )}^{ϵ-\delta +1}}}d\vartheta ,\delta -1<ϵ<\delta ,\hfill \\ {\displaystyle \frac{d^{\delta }}{d{\psi }^{\delta }}}\nu \left(\psi \right),\delta =ϵ,\hfill \end{array}\right.$$

(6)

with below properties

${\delta }_{\psi }^{ϵ}{D}_{\psi }^{ϵ}g\left(\psi \right)=g\left(\psi \right)-{\sum }_{k=0}^m{g}^k\left(0^{+}\right){\displaystyle \frac{{\psi }^k}{k!}},for\psi >0,$ and $\delta -1<ϵ\le \delta ,\delta \in N.$

$D_{\psi }^{ϵ}{\delta }_{\psi }^{ϵ}g\left(\psi \right)=g\left(\psi \right).$

Definition 4.

The ET of a function is as [52]

$$E\left[g\left(\psi \right)\right]=G\left(r\right)=r{\int }_0^{\infty }h\left(\psi \right)e^{{\displaystyle \frac{-\psi }{r}}}d\psi ,r>0.$$

(7)

Definition 5.

The ET of Caputo operator is given as [52]

$$\mathtt{E}\left[D_{\psi }^{ϵ}g\left(\psi \right)\right]=s^{-ϵ}\mathtt{E}\left[g\left(\psi \right)\right]-\sum _{k=0}^{\delta -1}{s}^{2-ϵ+k}{g}^{\left(k\right)}\left(0\right),where\delta -1<ϵ<\delta .$$

3. Analysis of NITM

In this part, we construct the general solution of the FDE as below.

$$D_{\varrho }^{ϵ}\mathbb{S}(\vartheta ,\varrho )+\mathcal{U}\mathbb{S}(\vartheta ,\varrho )+\mathcal{V}\mathbb{S}(\vartheta ,\varrho )=h(\vartheta ,\varrho ),\varrho >0,1<ϵ\le 0,$$

(8)

with

$${\mathbb{S}}^k(\vartheta ,0)=f\left(\vartheta \right),$$

(9)

with $\mathcal{U},\mathcal{V}$ indicates the linear and nonlinear terms. By implementing the ET to Equation (8), we get

$$E\left[D_{\varrho }^{ϵ}\mathbb{S}(\vartheta ,\varrho )\right]+E[\mathcal{U}\mathbb{S}(\vartheta ,\varrho )+\mathcal{V}\mathbb{S}(\vartheta ,\varrho )]=E\left[h(\vartheta ,\varrho )\right].$$

(10)

After differentiation property

$$E\left[\mathbb{S}(\vartheta ,\varrho )\right]=\sum _{k=0}^m{s}^{2-ϵ+k}{\mathbb{S}}^{\left(k\right)}(\vartheta ,0)+s^{ϵ}E\left[h(\vartheta ,\varrho )\right]-s^{ϵ}E[\mathcal{U}\mathbb{S}(\vartheta ,\varrho )+\mathcal{V}\mathbb{S}(\vartheta ,\varrho )].$$

(11)

On implementing the inverse ET to Equation (11),

$$\mathbb{S}(\vartheta ,\varrho )=E^{-1}\left[\left\{\sum _{k=0}^m{s}^{2-ϵ+k}{\mathbb{S}}^k(\vartheta ,0)+s^{ϵ}E\left[h(\vartheta ,\varrho )\right]\right\}\right]-E^{-1}\left[s^{ϵ}E\left[\mathcal{U}\mathbb{S}(\vartheta ,\varrho )+\mathcal{V}\mathbb{S}(\vartheta ,\varrho )\right]\right].$$

(12)

By iterative approach, we obtain

$$\mathbb{S}(\vartheta ,\varrho )=\sum _{m=0}^{\infty }{\mathbb{S}}_m(\vartheta ,\varrho ),$$

(13)

$$\mathcal{U}\left(\sum _{m=0}^{\infty }{\mathbb{S}}_m(\vartheta ,\varrho )\right)=\sum _{m=0}^{\infty }\mathcal{U}\left[{\mathbb{S}}_m(\vartheta ,\varrho )\right],$$

(14)

the nonlinear term $\mathcal{V}$ is decomposed as

$$\mathcal{V}\left(\sum _{m=0}^{\infty }{\mathbb{S}}_m(\vartheta ,\varrho )\right)={\mathbb{S}}_0(\vartheta ,\varrho )+\mathcal{V}\left(\sum _{k=0}^m{\mathbb{S}}_k(\vartheta ,\varrho )\right)-\mathcal{V}\left(\sum _{k=0}^m{\mathbb{S}}_k(\vartheta ,\varrho )\right).$$

(15)

By using Equations (13)–(15) into Equation (12), we obtain

$$\begin{array}{cc}\hfill & \sum _{m=0}^{\infty }{\mathbb{S}}_m(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}\left(\sum _{k=0}^m{s}^{2-\vartheta +k}{\mathbb{S}}^k(\vartheta ,0)+E\left[h(\vartheta ,\varrho )\right]\right)\right]\hfill \\ \hfill & -E^{-1}\left[s^{ϵ}E\left[N\left(\sum _{k=0}^m{\mathbb{S}}_k(\vartheta ,\varrho )\right)-M\left(\sum _{k=0}^m{\mathbb{S}}_k(\vartheta ,\varrho )\right)\right]\right].\hfill \end{array}$$

(16)

In terms of an iterative formula, we obtain

$${\mathbb{S}}_0(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}\left(\sum _{k=0}^m{s}^{2-\vartheta +k}{\mathbb{S}}^k(\vartheta ,0)+s^{ϵ}E\left(g(\vartheta ,\varrho )\right)\right)\right],$$

(17)

$${\mathbb{S}}_1(\vartheta ,\varrho )=-E^{-1}\left[s^{ϵ}E[\mathcal{U}\left[{\mathbb{S}}_0(\vartheta ,\varrho )\right]+\mathcal{V}\left[{\mathbb{S}}_0(\vartheta ,\varrho )\right]\right],$$

(18)

$$\begin{array}{cc}\hfill & {\mathbb{S}}_{m+1}(\vartheta ,\varrho )=-E^{-1}\left[s^{ϵ}E\left[-\mathcal{U}\left(\sum _{k=0}^m{\mathbb{S}}_k(\vartheta ,\varrho )\right)-\mathcal{V}\left(\sum _{k=0}^m{\mathbb{S}}_k(\vartheta ,\varrho )\right)\right]\right],m\ge 1.\hfill \end{array}$$

(19)

Thus, the solution in series form to Equation (8) illustrates

$$\mathbb{S}(\vartheta ,\varrho )\cong {\mathbb{S}}_0(\vartheta ,\varrho )+{\mathbb{S}}_1(\vartheta ,\varrho )+{\mathbb{S}}_2(\vartheta ,\varrho )+\dots ,m=1,2,\dots .$$

(20)

4. Analysis of ETDM

In this portion, we build the general solution of the FDE as below.

$$\begin{array}{cc}\hfill & D_{\varrho }^{ϵ}\mathbb{S}(\vartheta ,\varrho )+\mathcal{U}\mathbb{S}(\vartheta ,\varrho )+\mathcal{V}\mathbb{S}(\vartheta ,\varrho )=h(\vartheta ,\varrho ),\varrho >0,0<ϵ\le 1,\hfill \\ \hfill & \mathbb{S}(\vartheta ,0)=f\left(\vartheta \right).\hfill \end{array}$$

(21)

By implementing the ET to Equation (21), we get

$$\begin{array}{cc}\hfill & E[D_{\varrho }^{ϵ}\mathbb{S}(\vartheta ,\varrho )+\mathcal{U}\mathbb{S}(\vartheta ,\varrho )+\mathcal{V}\mathbb{S}(\vartheta ,\varrho )]=E\left[h(\vartheta ,\varrho )\right],\varrho >0,0<ϵ\le 1,\hfill \\ \hfill & \mathbb{S}(\vartheta ,\varrho )=s^{2}g\left(\vartheta \right)+s^{ϵ}E\left[h(\vartheta ,\varrho )\right]-s^{ϵ}E[\mathcal{U}\mathbb{S}(\vartheta ,\varrho )+\mathcal{V}\mathbb{S}(\vartheta ,\varrho )].\hfill \end{array}$$

(22)

On implementing the inverse ET, we get

$$\mathbb{S}(\vartheta ,\varrho )=F(x,\varrho )-E^{-1}\left[s^{ϵ}E\{\mathcal{U}\mathbb{S}(\vartheta ,\varrho )+\mathcal{V}\mathbb{S}(\vartheta ,\varrho )\}\right],$$

(23)

where

$$F(\vartheta ,\varrho )=E^{-1}\left[s^{2}g\left(\vartheta \right)+s^{ϵ}E\left[h(\vartheta ,\varrho )\right]\right]=g\left(\nu \right)+E^{-1}\left[s^{ϵ}E\left[h(\vartheta ,\varrho )\right]\right].$$

(24)

Assume the series form as

$$\mathbb{S}(\vartheta ,\varrho )=\sum _{m=0}^{\infty }{\mathbb{S}}_m(\vartheta ,\varrho ).$$

(25)

The decomposition of nonlinear terms is as

$$\mathcal{V}\mathbb{S}(\vartheta ,\varrho )=\sum _{m=0}^{\infty }{\mathcal{A}}_m.$$

(26)

with

$${\mathcal{A}}_m={\displaystyle \frac{1}{m!}}{\left[{\displaystyle \frac{{\partial }^m}{\partial {\varsigma }^m}}\left\{\mathcal{V}\left(\sum _{k=0}^{\infty }{\varsigma }^k{\vartheta }_k,\sum _{k=0}^{\infty }{\varsigma }^k{\varrho }_k\right)\right\}\right]}_{\varsigma =0},$$

(27)

By using Equations (25) and (26) into Equation (23), we get

$$\sum _{k=0}^{\infty }{\mathbb{S}}_k(\vartheta ,\varrho )=F(\vartheta ,\varrho )-\left[E^{-1}\left\{s^{ϵ}E\{\mathcal{U}\left(\sum _{m=0}^{\infty }{\vartheta }_m,\sum _{m=0}^{\infty }{\varrho }_m\right)+\sum _{m=0}^{\infty }{\mathcal{A}}_m\}\right\}\right].$$

(28)

On comparison of both sides, we may have

$$\begin{array}{cc}\hfill & {\mathbb{S}}_0(\vartheta ,\varrho )=F(\vartheta ,\varrho ),\hfill \\ \hfill & {\mathbb{S}}_1(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}E(\mathcal{U}{\mathbb{S}}_0(\vartheta ,\varrho )+{\mathcal{A}}_0)\right],\hfill \\ \hfill & {\mathbb{S}}_2(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}E(\mathcal{U}{\mathbb{S}}_1(\vartheta ,\varrho )+{\mathcal{A}}_1)\right],\hfill \\ \hfill & ⋮\hfill \\ \hfill & {\mathbb{S}}_k(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}E(\mathcal{U}{\mathbb{S}}_{k-1}(\vartheta ,\varrho )+{\mathcal{A}}_{k-1})\right],k>0,k\in N.\hfill \end{array}$$

(29)

5. Convergence Analysis

The convergence analysis of the implemented approaches are discussed below.

Theorem 1.

Assuming ϱ is analytic in a neighborhood of $\mathbb{S}$ and $\left|\right|{\varrho }^m\left({\mathbb{S}}_0\right)\left|\right|=sup\left\{\right||{\varrho }^m\left({\mathbb{S}}_0\right)(b_0,b_0,$$\dots b_n)/||b_k\left|\right|\le 1,1\le k\le m\}\le l,$ for every number m as well as for few real number $l>0$ and ||${\mathbb{S}}_k\left|\right|\le M<{\displaystyle \frac{1}{e}},k=1,2,\dots$ thus the series ${\sum }_{m=0}^{\infty }{\mathcal{F}}_m$ is convergent and also

$$\left|\right|{\mathcal{F}}_m\left|\right|\le l{M}^m{e}^{m-1}(e-1),m=1,2,\dots .$$

Also to describe boundedness of ||${\mathbb{S}}_k$||, for every k the conditions on ${\varrho }^j\left({\mathbb{S}}_0\right)$ are given and is sufficient to guarantee series convergence.

Theorem 2.

If ϱ is $C^{\infty }$ and ||${\varrho }^m\left({\mathbb{S}}_0\right)\left|\right|\le M\le e^{-1}$ for all m thus the series ${\sum }_{m=0}^{\infty }{\mathcal{F}}_m$ is convergent.

These are the conditions for series ${\sum }_{j=0}^{\infty }{\mathbb{S}}_j$ to be convergent.

The proofs can be checked in [53].

Theorem 3.

The outcome of (21) is unique at $0<({\vartheta }_1+{\vartheta }_2)\left({\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}\right)<1.$

Proof. 

Suppose $H=\left(C\right[J],||.|\left|\right)$ with the norm $\left|\right|\varphi \left(\varrho \right)\left|\right|={max}_{\varrho \in J}\left|\varphi \left(\varrho \right)\right|$ is Banach space, ∀ continuous function on $J.$ Let $I:H\to H$ is a non-linear mapping, thus

$${\mathbb{S}}_{l+1}^C={\mathbb{S}}_0^C+{\mathtt{E}}^{-1}\left[s^{ϵ}\mathtt{E}[\mathcal{U}\left({\mathbb{S}}_l(\vartheta ,\varrho )\right)+\mathcal{V}\left({\mathbb{S}}_l(\vartheta ,\varrho )\right)]\right],l\ge 0.$$

Suppose that $|\mathcal{U}\left(\mathbb{S}\right)-\mathcal{U}\left({\mathbb{S}}^{*}\right)|<{\vartheta }_1|\mathbb{S}-{\mathbb{S}}^{*}|$ and $|\mathcal{V}\left(\mathbb{S}\right)-\mathcal{V}\left({\mathbb{S}}^{*}\right)|<{\vartheta }_2|\mathbb{S}-{\mathbb{S}}^{*}|$, where $\mathbb{S}:=\mathbb{S}(\vartheta ,\varrho )$ and ${\mathbb{S}}^{*}:={\mathbb{S}}^{*}(\vartheta ,\varrho )$ are are two separate function values and ${\vartheta }_1$,${\vartheta }_2$ are Lipschitz constants.

$$\begin{array}{cc}\hfill \left|\right|I\mathbb{S}-I{\mathbb{S}}^{*}\left|\right|& \le ma{x}_{t\in J}|{\mathtt{E}}^{-1}[s^{ϵ}\mathtt{E}[\mathcal{U}\left(\mathbb{S}\right)-\mathcal{U}\left({\mathbb{S}}^{*}\right)]\hfill \\ \hfill & +s^{ϵ}Y[\mathcal{V}\left(\mathbb{S}\right)-\mathcal{V}\left({\mathbb{S}}^{*}\right)]\left|\right]\hfill \\ \hfill & \le ma{x}_{\varrho \in J}[{\vartheta }_1{\mathtt{E}}^{-1}[s^{ϵ}\mathtt{E}\left[\right|\mathbb{S}-{\mathbb{S}}^{*}\left|\right]]\hfill \\ \hfill & +{\vartheta }_2{\mathtt{E}}^{-1}[s^{ϵ}\mathtt{E}\left[\right|\mathbb{S}-{\mathbb{S}}^{*}\left|\right]\left]\right]\hfill \\ \hfill & \le ma{x}_{t\in J}({\vartheta }_1+{\vartheta }_2)\left[{\mathtt{E}}^{-1}[s^{ϵ}\mathtt{E}|\mathbb{S}-{\mathbb{S}}^{*}\left|\right]\right]\hfill \\ \hfill & \le ({\vartheta }_1+{\vartheta }_2)\left[{\mathtt{E}}^{-1}[s^{ϵ}\mathtt{E}\left|\right|\mathbb{S}-{\mathbb{S}}^{*}\left|\right|]\right]\hfill \\ \hfill & =({\vartheta }_1+{\vartheta }_2)\left({\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}\right)\left|\right|\mathbb{S}-{\mathbb{S}}^{*}\left|\right|\hfill \end{array}$$

(30)

I is contraction as $0<({\vartheta }_1+{\vartheta }_2)\left({\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}\right)<1$. The outcome of (21) is unique in terms of Banach fixed point theorem. □

Theorem 4.

The outcome of (21) is convergent.

Proof. 

Assume ${\mathbb{S}}_m={\sum }_{r=0}^m{\mathbb{S}}_r(\vartheta ,\varrho )$. To show that ${\mathbb{S}}_m$ is a Cauchy sequence in H. Consider,

$$\begin{array}{cc}\hfill \left|\right|{\mathbb{S}}_m-{\mathbb{S}}_n\left|\right|& =ma{x}_{\varrho \in J}\left|\sum _{r=n+1}^m{\mathbb{S}}_r\right|,n=1,2,3,\dots \hfill \\ \hfill & \le ma{x}_{\varrho \in J}\left|{\mathtt{E}}^{-1}\left[s^{ϵ}\mathtt{E}\left[\sum _{r=n+1}^m(\mathcal{U}\left({\mathbb{S}}_{r-1}\right)+\mathcal{V}\left({\mathbb{S}}_{r-1}\right))\right]\right]\right|\hfill \\ \hfill & =ma{x}_{\varrho \in J}\left|{\mathtt{E}}^{-1}\left[s^{ϵ}\mathtt{E}\left[\sum _{r=n+1}^{m-1}(\mathcal{U}\left({\mathbb{S}}_r\right)+\mathcal{V}\left({\mathbb{S}}_r\right))\right]\right]\right|\hfill \\ \hfill & \le ma{x}_{\varrho \in J}\left|{\mathtt{E}}^{-1}\left[s^{ϵ}\mathtt{E}\left[(\mathcal{U}\left({\mathbb{S}}_{m-1}\right)-\mathcal{U}\left({\mathbb{S}}_{n-1}\right)+\mathcal{V}\left({\mathbb{S}}_{m-1}\right)-\mathcal{V}\left({\mathbb{S}}_{n-1}\right))\right]\right]\right|\hfill \\ \hfill & \le {\vartheta }_{1}ma{x}_{\varrho \in J}\left|{\mathtt{E}}^{-1}\left[s^{ϵ}\mathtt{E}\left[(\mathcal{U}\left({\mathbb{S}}_{m-1}\right)-\mathcal{U}\left({\mathbb{S}}_{n-1}\right))\right]\right]\right|\hfill \\ \hfill & +{\vartheta }_{2}ma{x}_{\varrho \in J}\left|{\mathtt{E}}^{-1}\left[s^{ϵ}\mathtt{E}\left[(\mathcal{V}\left({\mathbb{S}}_{m-1}\right)-\mathcal{V}\left({\mathbb{S}}_{n-1}\right))\right]\right]\right|\hfill \\ \hfill & =({\vartheta }_1+{\vartheta }_2)\left({\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}\right)\left|\right|{\mathbb{S}}_{m-1}-{\mathbb{S}}_{n-1}\left|\right|\hfill \end{array}$$

(31)

Let $m=n+1$, then

$$\begin{array}{c}\hfill \left|\right|{\mathbb{S}}_{n+1}-{\mathbb{S}}_n\left|\right|\le \vartheta \left|\right|{\mathbb{S}}_n-{\mathbb{S}}_{n-1}\left|\right|\le {\vartheta }^2\left|\right|{\mathbb{S}}_{n-1}{\mathbb{S}}_{n-2}\left|\right|\le \dots \le {\vartheta }^n\left|\right|{\mathbb{S}}_1-{\mathbb{S}}_0\left|\right|,\end{array}$$

(32)

where $\vartheta =({\vartheta }_1+{\vartheta }_2)\left({\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}\right)$. Similarly, we have

$$\begin{array}{cc}\hfill \left|\right|{\mathbb{S}}_m-{\mathbb{S}}_n\left|\right|& \le \left|\right|{\mathbb{S}}_{n+1}-{\mathbb{S}}_n\left|\right|+\left|\right|{\mathbb{S}}_{n+2}{\mathbb{S}}_{n+1}\left|\right|+\dots +\left|\right|{\mathbb{S}}_m-{\mathbb{S}}_{m-1}\left|\right|,\hfill \\ \hfill & ({\vartheta }^n+{\vartheta }^{n+1}+\dots +{\vartheta }^{m-1})\left|\right|{\mathbb{S}}_1-{\mathbb{S}}_0\left|\right|\hfill \\ \hfill & \le {\vartheta }^n\left({\displaystyle \frac{1-{\vartheta }^{m-n}}{1-\vartheta }}\right)\left|\right|{\mathbb{S}}_1\left|\right|,\hfill \end{array}$$

(33)

As $0<\vartheta <1$, we get $1-{\vartheta }^{m-n}<1$. Hence,

$$\begin{array}{c}\hfill \left|\right|{\mathbb{S}}_m-{\mathbb{S}}_n\left|\right|\le {\displaystyle \frac{{\vartheta }^n}{1-\vartheta }}ma{x}_{\varrho \in J}\left|\right|{\mathbb{S}}_1\left|\right|.\end{array}$$

(34)

Since $\left|\right|{\mathbb{S}}_1\left|\right|<\infty ,\left|\right|{\mathbb{S}}_m-{\mathbb{S}}_n\left|\right|\to 0$ when $n\to \infty$. Hence, ${\mathbb{S}}_m$ is a Cauchy sequence in H, demonstrating the series ${\mathbb{S}}_m$ is convergent. □

6. Applications

6.1. Example

Consider the nonlinear fractional DSW equation

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^{ϵ}\mathbb{S}(\vartheta ,\varrho )}{\partial {\varrho }^{ϵ}}}+\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}=0,\hfill \\ \hfill & {\displaystyle \frac{{\partial }^{ϵ}\mathbb{T}(\vartheta ,\varrho )}{\partial {\varrho }^{ϵ}}}+2{\displaystyle \frac{{\partial }^3\mathbb{T}(\vartheta ,\varrho )}{\partial {\vartheta }^3}}+2\mathbb{S}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}+\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}=0,\hfill \end{array}$$

(35)

with

$$\begin{array}{cc}\hfill & \mathbb{S}(\vartheta ,0)={\displaystyle \frac{3\kappa }{2}}{sech}^2\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right),\hfill \\ \hfill & \mathbb{T}(\vartheta ,0)=\kappa sech\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right).\hfill \end{array}$$

(36)

• CaseI: Application of NITM

By implementing the ET to Equation (35), we obtain

$$\begin{array}{cc}\hfill & E\left[\mathbb{S}(\vartheta ,\varrho )\right]=s^2\left({\displaystyle \frac{3\kappa }{2}}{sech}^2\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)\right)-s^{ϵ}E\left[\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}\right],\hfill \\ \hfill & E\left[\mathbb{T}(\vartheta ,\varrho )\right]=s^2\left(\kappa sech\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)\right)-s^{ϵ}E\left[2{\displaystyle \frac{{\partial }^3\mathbb{T}(\vartheta ,\varrho )}{\partial {\vartheta }^3}}+2\mathbb{S}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}+\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}\right].\hfill \end{array}$$

(37)

On implementing the inverse ET, we obtain

$$\begin{array}{cc}\hfill & \mathbb{S}(\vartheta ,\varrho )={\displaystyle \frac{3\kappa }{2}}{sech}^2\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)-E^{-1}\left[s^{ϵ}E\left[\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}\right]\right],\hfill \\ \hfill & \mathbb{T}(\vartheta ,\varrho )=\kappa sech\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)-E^{-1}\left[s^{ϵ}E\left[2{\displaystyle \frac{{\partial }^3\mathbb{T}(\vartheta ,\varrho )}{\partial {\vartheta }^3}}+2\mathbb{S}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}+\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}\right]\right].\hfill \end{array}$$

(38)

By NITM, we have

$$\begin{array}{cc}& {\mathbb{S}}_0(\vartheta ,\varrho )={\displaystyle \frac{3\kappa }{2}}{sech}^2\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right),\hfill \\ & {\mathbb{T}}_0(\vartheta ,\varrho )=\kappa sech\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right),\hfill \\ & {\mathbb{S}}_1(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}E\left\{-{\mathbb{T}}_0(\vartheta ,\varrho ){\displaystyle \frac{\partial {\mathbb{T}}_0(\vartheta ,\varrho )}{\partial \vartheta }}\right\}\right]={\displaystyle \frac{3}{2}}{\displaystyle \frac{{\kappa }^{\frac{5}{2}}\sqrt{2}sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}{{cosh}^3\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}},\hfill \\ & {\mathbb{T}}_1(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}E\left\{-2{\displaystyle \frac{{\partial }^3{\mathbb{T}}_0(\vartheta ,\varrho )}{\partial {\vartheta }^3}}-2{\mathbb{S}}_0(\vartheta ,\varrho ){\displaystyle \frac{\partial {\mathbb{T}}_0(\vartheta ,\varrho )}{\partial \vartheta }}-{\mathbb{T}}_0(\vartheta ,\varrho ){\displaystyle \frac{\partial {\mathbb{S}}_0(\vartheta ,\varrho )}{\partial \vartheta }}\right\}\right]=\hfill \\ & {\displaystyle \frac{1}{2}}{\displaystyle \frac{{\kappa }^{\frac{5}{2}}\sqrt{2}sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}{{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}},\hfill \end{array}$$

$$\begin{array}{cc}& {\mathbb{S}}_2(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}E\left\{\left(-({\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho )){\displaystyle \frac{\partial ({\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho ))}{\partial \vartheta }}\right)-\left(-\left({\mathbb{T}}_0(\vartheta ,\varrho )\right){\displaystyle \frac{\partial \left({\mathbb{T}}_0(\vartheta ,\varrho )\right)}{\partial \vartheta }}\right)\right\}\right]=\hfill \\ & {\displaystyle \frac{3}{2}}{\displaystyle \frac{{\kappa }^4\left(2{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-3\right)}{{cosh}^4\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}}+\hfill \\ & {\displaystyle \frac{3}{4}}{\displaystyle \frac{sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\left({cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-2\right)\sqrt{2}{\kappa }^{\frac{11}{2}}\Gamma (2ϵ+1)}{{cosh}^5\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\Gamma {(ϵ+1)}^2}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}},\hfill \\ & {\mathbb{T}}_2(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}E\right\{(-2{\displaystyle \frac{{\partial }^3({\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho ))}{\partial {\vartheta }^3}}-2({\mathbb{S}}_0(\vartheta ,\varrho )+{\mathbb{S}}_1(\vartheta ,\varrho )){\displaystyle \frac{\partial ({\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho ))}{\partial \vartheta }}-\hfill \\ & ({\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho )){\displaystyle \frac{\partial ({\mathbb{S}}_0(\vartheta ,\varrho )+{\mathbb{S}}_1(\vartheta ,\varrho ))}{\partial \vartheta }})-(-2{\displaystyle \frac{{\partial }^2{\mathbb{T}}_0(\vartheta ,\varrho )}{\partial {\vartheta }^2}}-2{\mathbb{S}}_0(\vartheta ,\varrho ){\displaystyle \frac{\partial {\mathbb{T}}_0(\vartheta ,\varrho )}{\partial \vartheta }}-{\mathbb{T}}_0(\vartheta ,\varrho )\hfill \\ & {\displaystyle \frac{\partial {\mathbb{S}}_0(\vartheta ,\varrho )}{\partial \vartheta }}\left)\right\}]={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\kappa }^4\left(2{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-2\right)}{{cosh}^3\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}}+\hfill \\ & {\displaystyle \frac{3}{4}}{\displaystyle \frac{sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\left(4{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-7\right)\sqrt{2}{\kappa }^{\frac{11}{2}}\Gamma (2ϵ+1)}{{cosh}^6\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\Gamma {(ϵ+1)}^2}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}},\hfill \\ & \hfill ⋮\hfill \end{array}$$

At the end, the solution in series form is taken as

$$\begin{array}{cc}& \mathbb{S}(\vartheta ,\varrho )={\mathbb{S}}_0(\vartheta ,\varrho )+{\mathbb{S}}_1(\vartheta ,\varrho )+{\mathbb{S}}_2(\vartheta ,\varrho )+\dots ,\hfill \\ & \mathbb{S}(\vartheta ,\varrho )={\displaystyle \frac{3\kappa }{2}}{sech}^2\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)+{\displaystyle \frac{3}{2}}{\displaystyle \frac{{\kappa }^{\frac{5}{2}}\sqrt{2}sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}{{cosh}^3\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}+{\displaystyle \frac{3}{2}}{\displaystyle \frac{{\kappa }^4\left(2{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-3\right)}{{cosh}^4\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}}+\hfill \\ & {\displaystyle \frac{3}{4}}{\displaystyle \frac{sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\left({cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-2\right)\sqrt{2}{\kappa }^{\frac{11}{2}}\Gamma (2ϵ+1)}{{cosh}^5\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\Gamma {(ϵ+1)}^2}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}}+\dots ,\hfill \\ & \mathbb{T}(\vartheta ,\varrho )={\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho )+{\mathbb{T}}_2(\vartheta ,\varrho )+\dots ,\hfill \\ & \mathbb{T}(\vartheta ,\varrho )=\kappa sech\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\kappa }^{\frac{5}{2}}\sqrt{2}sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}{{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\kappa }^4\left(2{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-2\right)}{{cosh}^3\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}}+\hfill \\ & {\displaystyle \frac{3}{4}}{\displaystyle \frac{sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\left(4{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-7\right)\sqrt{2}{\kappa }^{\frac{11}{2}}\Gamma (2ϵ+1)}{{cosh}^6\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\Gamma {(ϵ+1)}^2}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}}+\dots ,\hfill \end{array}$$

(39)

• Case II: Application of ETDM

By implementing the ET to Equation (35), we obtain

$$\begin{array}{cc}\hfill & E\left[\mathbb{S}(\vartheta ,\varrho )\right]=s^2\left({\displaystyle \frac{3\kappa }{2}}{sech}^2\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)\right)-s^{ϵ}E\left[\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}\right],\hfill \\ \hfill & E\left[\mathbb{T}(\vartheta ,\varrho )\right]=s^2\left(\kappa sech\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)\right)-s^{ϵ}E\left[2{\displaystyle \frac{{\partial }^3\mathbb{T}(\vartheta ,\varrho )}{\partial {\vartheta }^3}}+2\mathbb{S}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}+\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}\right].\hfill \end{array}$$

(40)

On implementing the inverse ET, we obtain

$$\begin{array}{cc}\hfill & \mathbb{S}(\vartheta ,\varrho )={\displaystyle \frac{3\kappa }{2}}{sech}^2\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)-E^{-1}\left[s^{ϵ}E\left[\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}\right]\right],\hfill \\ \hfill & \mathbb{T}(\vartheta ,\varrho )=\kappa sech\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)-E^{-1}\left[s^{ϵ}E\left[2{\displaystyle \frac{{\partial }^3\mathbb{T}(\vartheta ,\varrho )}{\partial {\vartheta }^3}}+2\mathbb{S}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}+\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}\right]\right].\hfill \end{array}$$

(41)

Assume the series form solution as

$$\mathbb{S}(\vartheta ,\varrho )=\sum _{m=0}^{\infty }{\mathbb{S}}_m(\vartheta ,\varrho ),\mathbb{T}(\vartheta ,\varrho )=\sum _{m=0}^{\infty }{\mathbb{T}}_m(\vartheta ,\varrho ).$$

(42)

The nonlinear terms decomposes as $\mathbb{T}(\vartheta ,\varrho ){\mathbb{T}}_{\vartheta }(\vartheta ,\varrho )={\sum }_{m=0}^{\infty }{\mathcal{A}}_m,\mathbb{S}(\vartheta ,\varrho ){\mathbb{T}}_{\vartheta }(\vartheta ,\varrho )={\sum }_{m=0}^{\infty }{\mathcal{B}}_m,\mathbb{T}(\vartheta ,\varrho ){\mathbb{S}}_{\vartheta }(\vartheta ,\varrho )={\sum }_{m=0}^{\infty }{\mathcal{C}}_m.$ So, we have

$$\begin{array}{cc}\hfill & \sum _{m=0}^{\infty }{\mathbb{S}}_m(\vartheta ,\varrho )={\displaystyle \frac{3\kappa }{2}}{sech}^2\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)-{\mathtt{E}}^{-1}\left[s^{ϵ}\mathtt{E}\left[\sum _{m=0}^{\infty }{\mathcal{A}}_m\right]\right],\hfill \\ \hfill & \sum _{m=0}^{\infty }{\mathbb{T}}_m(\vartheta ,\varrho )=\kappa sech\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)-{\mathtt{E}}^{-1}\left[s^{ϵ}\mathtt{E}\left[2{\mathbb{T}}_{\vartheta \vartheta \vartheta }(\vartheta ,\varrho )+2\sum _{m=0}^{\infty }{\mathcal{B}}_m+\sum _{m=0}^{\infty }{\mathcal{C}}_m\right]\right].\hfill \end{array}$$

(43)

Similarly,

$${\mathbb{S}}_0(\vartheta ,\varrho )={\displaystyle \frac{3\kappa }{2}}{sech}^2\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right),$$

$${\mathbb{T}}_0(\vartheta ,\varrho )=\kappa sech\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right),$$

On $m=0$

$${\mathbb{S}}_1(\vartheta ,\varrho )={\displaystyle \frac{3}{2}}{\displaystyle \frac{{\kappa }^{\frac{5}{2}}\sqrt{2}sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}{{cosh}^3\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}},$$

$${\mathbb{T}}_1(\vartheta ,\varrho )={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\kappa }^{\frac{5}{2}}\sqrt{2}sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}{{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}},$$

On $m=1$

$$\begin{array}{cc}\hfill & {\mathbb{S}}_2(\vartheta ,\varrho )={\displaystyle \frac{3}{2}}{\displaystyle \frac{{\kappa }^4\left(2{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-3\right)}{{cosh}^4\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\mathbb{T}}_2(\vartheta ,\varrho )={\displaystyle \frac{1}{2}}{\displaystyle \frac{{\kappa }^4\left(2{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-2\right)}{{cosh}^3\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}},\hfill \end{array}$$

On $m=2$

$$\begin{array}{cc}\hfill & {\mathbb{S}}_3(\vartheta ,\varrho )={\displaystyle \frac{3}{2}}{\displaystyle \frac{sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\left({cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-4\right)\sqrt{2}{\kappa }^{\frac{11}{2}}}{{cosh}^5\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}}+\hfill \\ \hfill & {\displaystyle \frac{3}{4}}{\displaystyle \frac{sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\left({cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-2\right)\sqrt{2}{\kappa }^{\frac{11}{2}}\Gamma (2ϵ+1)}{{cosh}^5\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\Gamma {(ϵ+1)}^2}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\mathbb{T}}_3(\vartheta ,\varrho )={\displaystyle \frac{1}{4}}{\displaystyle \frac{sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\left({cosh}^4\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-30{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)+42\right)\sqrt{2}{\kappa }^{\frac{11}{2}}}{{cosh}^6\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}}+\hfill \\ \hfill & {\displaystyle \frac{3}{4}}{\displaystyle \frac{sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\left(4{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-7\right)\sqrt{2}{\kappa }^{\frac{11}{2}}\Gamma (2ϵ+1)}{{cosh}^6\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\Gamma {(ϵ+1)}^2}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}},\hfill \end{array}$$

Thus, the solution in series form illustrates as

$$\begin{array}{cc}\hfill & \mathbb{S}(\vartheta ,\varrho )={\mathbb{S}}_0(\vartheta ,\varrho )+{\mathbb{S}}_1(\vartheta ,\varrho )+{\mathbb{S}}_2(\vartheta ,\varrho )+{\mathbb{S}}_3(\vartheta ,\varrho )+\dots \hfill \\ \hfill & \mathbb{S}(\vartheta ,\varrho )={\displaystyle \frac{3\kappa }{2}}{sech}^2\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)+{\displaystyle \frac{3}{2}}{\displaystyle \frac{{\kappa }^{\frac{5}{2}}\sqrt{2}sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}{{cosh}^3\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}+{\displaystyle \frac{3}{2}}{\displaystyle \frac{{\kappa }^4\left(2{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-3\right)}{{cosh}^4\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}\hfill \\ \hfill & {\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}}+{\displaystyle \frac{3}{2}}{\displaystyle \frac{sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\left({cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-4\right)\sqrt{2}{\kappa }^{\frac{11}{2}}}{{cosh}^5\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}}+\hfill \\ \hfill & {\displaystyle \frac{3}{4}}{\displaystyle \frac{sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\left({cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-2\right)\sqrt{2}{\kappa }^{\frac{11}{2}}\Gamma (2ϵ+1)}{{cosh}^5\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\Gamma {(ϵ+1)}^2}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}}+\dots \hfill \\ \hfill & \mathbb{T}(\vartheta ,\varrho )={\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho )+{\mathbb{T}}_2(\vartheta ,\varrho )+{\mathbb{T}}_3(\vartheta ,\varrho )+\dots \hfill \\ \hfill & \mathbb{T}(\vartheta ,\varrho )=\kappa sech\left(\sqrt{{\displaystyle \frac{\kappa }{2}}}\vartheta \right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\kappa }^{\frac{5}{2}}\sqrt{2}sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}{{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\kappa }^4\left(2{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-2\right)}{{cosh}^3\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}\hfill \\ \hfill & {\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}}+{\displaystyle \frac{1}{4}}{\displaystyle \frac{sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\left({cosh}^4\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-30{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)+42\right)\sqrt{2}{\kappa }^{\frac{11}{2}}}{{cosh}^6\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}}+\hfill \\ \hfill & {\displaystyle \frac{3}{4}}{\displaystyle \frac{sinh\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\left(4{cosh}^2\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)-7\right)\sqrt{2}{\kappa }^{\frac{11}{2}}\Gamma (2ϵ+1)}{{cosh}^6\left(\frac{1}{2}\sqrt{2}\sqrt{\kappa }\vartheta \right)\Gamma {(ϵ+1)}^2}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}}+\dots \hfill \end{array}$$

(44)

Inserting $ϵ=1$, we obtain

$$\mathbb{S}(\vartheta ,\varrho )=3{sech}^2(\vartheta -2\varrho ),\mathbb{T}(\vartheta ,\varrho )=2sech(\vartheta -2\varrho ).$$

(45)

Figure 1 depicts the NITM approximate outcome (39) of $\mathbb{S}(\vartheta ,\varrho )$ for different values of $ϵ$. Figure 2 depicts the correspondence between NITM solution (39) and accurate solution (45) for $\mathbb{S}(\vartheta ,\varrho )$ at $ϵ=1$. Figure 3 displays the ETDM approximate solution (44) at different $ϵ$ values while Figure 4 shows the comparison of ETDM solution (44) against accurate solution (45) for $\mathbb{S}(\vartheta ,\varrho )$ at $ϵ=1$. Figure 5 depicts the NITM approximate outcome (39) of $\mathbb{S}(\vartheta ,\varrho )$ for different values of $ϵ$. The Figure 6 depicts the correspondence between NITM solution (39) and accurate solution (45) for $\mathbb{S}(\vartheta ,\varrho )$ at $ϵ=1$. Figure 7 displays the ETDM approximate solution (44) at different $ϵ$ values while Figure 8 shows the comparison of ETDM solution (44) against accurate solution (45) for $\mathbb{S}(\vartheta ,\varrho )$ at $ϵ=1$. All figures show that when fractional order approaches 1, the approximate solution predicts analytic solution behavior while matching the analytic solution identically for the case of $ϵ=1$, thus confirming the reliability of our proposed approaches. The NITM approximation (39) and ETDM approximation (44) get compared to the analytic solution (45) for $\mathbb{S}(\vartheta ,\varrho )$ through

Table 1. Accurate solution along with NITM solution at numerous orders of $ϵ$ for $\mathbb{S}(\vartheta ,\varrho )$.

$\mathit{\vartheta }$	$\mathit{ϵ}=0.85$	$\mathit{ϵ}=0.90$	$\mathit{ϵ}=0.95$	$\mathit{ϵ}=1\left(\mathit{NITM}\right)$	$\mathit{ϵ}=1\left(\mathit{Accurate}\right)$
0.0	3.0000000000	2.9964040800	2.9979184430	2.9988000000	2.9988003200
0.1	2.9701988730	2.9862500920	2.9834093360	2.9808860040	2.9808816240
0.2	2.8831289490	2.9175725200	2.9106003540	2.9048700200	2.9048616140
0.3	2.7454108850	2.7956573370	2.7850839530	2.7765796730	2.7765684510
0.4	2.5669163580	2.6294405240	2.6160196600	2.6053424550	2.6053298790
0.5	2.3593431990	2.4301774040	2.4147785420	2.4026124610	2.4025999210
0.6	2.1347332880	2.2099449710	2.1934453770	2.1804739570	2.1804625460
0.7	1.9042187700	1.9802880960	1.9634847600	1.9503240100	1.9503144140
0.8	1.6771655030	1.7512214490	1.7347736840	1.7219293630	1.7219218680
0.9	1.4607520830	1.5306671140	1.5150710680	1.5029206560	1.5029152310
1.0	1.2599230250	1.3242946060	1.3098840780	1.2986788210	1.2986752290

and

Table 2. Accurate solution along with ETDM solution at numerous orders of $ϵ$ for $\mathbb{S}(\vartheta ,\varrho )$.

$\mathit{\vartheta }$	$\mathit{ϵ}=0.85$	$\mathit{ϵ}=0.90$	$\mathit{ϵ}=0.95$	$\mathit{ϵ}=1\left(\mathit{ETDM}\right)$	$\mathit{ϵ}=1\left(\mathit{Accurate}\right)$
0.0	2.9938145530	2.9964040800	2.9979184430	2.9988000000	2.9988003200
0.1	2.9891613720	2.9862233240	2.9833981130	2.9808813300	2.9808816240
0.2	2.9257584010	2.9175230820	2.9105796270	2.9048613880	2.9048616140
0.3	2.8084976120	2.7955923410	2.7850567040	2.7765683250	2.7765684510
0.4	2.6460176770	2.6293683540	2.6159894030	2.6053298540	2.6053298790
0.5	2.4494086370	2.4301059420	2.4147485810	2.4025999830	2.4025999210
0.6	2.2307189300	2.2098803150	2.1934182700	2.1804626680	2.1804625460
0.7	2.0015800170	1.9802339970	1.9634620790	1.9503145640	1.9503144140
0.8	1.7721713510	1.7511793950	1.7347560530	1.7219220200	1.7219218680
0.9	1.5506177240	1.5306368260	1.5150583700	1.5029153680	1.5029152310
1.0	1.3427948840	1.3242746780	1.3098757230	1.2986753420	1.2986752290

. Additionally, the NITM approximation (39) and ETDM approximation (44) get compared to the analytic solution (45) for $\mathbb{T}(\vartheta ,\varrho )$ through

Table 3. Accurate solution along with NITM solution at numerous orders of $ϵ$ for $\mathbb{T}(\vartheta ,\varrho )$.

$\mathit{\vartheta }$	$\mathit{ϵ}=0.85$	$\mathit{ϵ}=0.90$	$\mathit{ϵ}=0.95$	$\mathit{ϵ}=1\left(\mathit{NITM}\right)$	$\mathit{ϵ}=1\left(\mathit{Accurate}\right)$
0.0	1.9979381840	1.9988013600	1.9993061480	1.9996000670	1.9996000000
0.1	1.9963469540	1.9953856280	1.9944508290	1.9936170220	1.9936136620
0.2	1.9750281010	1.9722846220	1.9699540180	1.9680317120	1.9680260040
0.3	1.9350238390	1.9306221500	1.9270038120	1.9240819290	1.9240754130
0.4	1.8782093890	1.8723431300	1.8675981660	1.8638060980	1.8638003820
0.5	1.8070880220	1.8000018110	1.7943282680	1.7898230530	1.7898193530
0.6	1.7245421570	1.7165090690	1.7101237150	1.7050757740	1.7050746750
0.7	1.6335821890	1.6248810730	1.6180019120	1.6125815390	1.6125829990
0.8	1.5371273600	1.5280232740	1.5208550690	1.5152213780	1.5152249070
0.9	1.4378399520	1.4285698240	1.4212939010	1.4155871490	1.4155920420
1.0	1.3380200640	1.3287842350	1.3215527850	1.3158901820	1.3158957260

and

Table 4. Accurate solution along with ETDM solution at numerous orders of $ϵ$ for $\mathbb{T}(\vartheta ,\varrho )$.

$\mathit{\vartheta }$	$\mathit{ϵ}=0.85$	$\mathit{ϵ}=0.90$	$\mathit{ϵ}=0.95$	$\mathit{ϵ}=1\left(\mathit{ETDM}\right)$	$\mathit{ϵ}=1\left(\mathit{Accurate}\right)$
0.0	1.9979381840	1.9988013600	1.9993061480	1.9996000000	1.9996000000
0.1	1.9963916830	1.9954045120	1.9944587460	1.9936169590	1.9936136620
0.2	1.9751048170	1.9723170110	1.9699675970	1.9680316590	1.9680260040
0.3	1.9351117460	1.9306592630	1.9270193720	1.9240818930	1.9240754130
0.4	1.8782866870	1.8723757640	1.8676118480	1.8638060800	1.8638003820
0.5	1.8071381840	1.8000229890	1.7943371470	1.7898230510	1.7898193530
0.6	1.7245572190	1.7165154280	1.7101263810	1.7050757850	1.7050746750
0.7	1.6335626690	1.6248728320	1.6179984570	1.6125815600	1.6125829990
0.8	1.5370798560	1.5280032180	1.5208466610	1.5152214050	1.5152249070
0.9	1.4377739580	1.4285419620	1.4212822200	1.4155871770	1.4155920420
1.0	1.3379452300	1.3287526410	1.3215395390	1.3158902100	1.3158957260

. The obtained approximations are shown in Table 1, Table 2, Table 3 and Table 4 for a range of $ϵ$ values, which are 1, 0.95, 0.90, and 0.85, respectively, with $\kappa =2$ and temporal variable $\varrho =0.01$.

Table 5. Comparative analysis of our methods solution with analytic and homotopy perturbation transform method (HPTM) solution for $\mathbb{S}(\vartheta ,\varrho )$.

$\mathit{\varrho }$	$\mathit{\vartheta }$	$\mathit{Analytic}$	$\mathit{HPTM}$	$\mathit{ETDM}$	$\mathit{NITM}$
	−4	0.00286522	0.00386526	0.00386525	0.00386523
	−3	0.0284431	0.0284434	0.0284433	0.0284432
0.01	−2	0.203929	0.203931	0.203930	0.203930
	−1	1.22192	1.22191	1.22191	1.22191
	0	2.9988	2.9988	2.9988	2.9988
	−4	0.00371375	0.00371409	0.00371408	0.00371391
	−3	0.0273329	0.0273353	0.0273352	0.0273340
0.02	−2	0.196199	0.196213	0.196212	0.196205
	−1	1.18467	1.18465	1.18465	1.18464
	0	2.99521	2.9952	2.9952	2.9952

and

Table 6. Comparative analysis of our methods solution with analytic and homotopy perturbation transform method (HPTM) solution for $\mathbb{T}(\vartheta ,\varrho )$.

$\mathit{\varrho }$	$\mathit{\vartheta }$	$\mathit{Analytic}$	$\mathit{HPTM}$	$\mathit{ETDM}$	$\mathit{NITM}$
	−4	0.0717887	0.0717888	0.0717888	0.00386523
	−3	0.194741	0.194741	0.194741	0.194741
0.01	−2	0.521446	0.521446	0.521446	0.521446
	−1	1.27641	1.27641	1.27641	1.27641
	0	1.9996	1.9996	1.9996	1.9996
	−4	0.0703681	0.0703689	0.0703688	0.0703688
	−3	0.190903	0.190905	0.190904	0.190904
0.02	−2	0.511467	0.511467	0.511470	0.511462
	−1	1.25681	1.25681	1.25679	1.25676
	0	1.9984	1.9984	1.9984	1.9984

shows a comparison among our methods solution with analytic and HPTM solution. The tables visualize the close match between our computational results and actual solutions at $ϵ=1$. Also, the solutions obtained by NITM are more suitable and efficient than those obtained by HPTM. Overall, the tabular and graphical data support the finding that both approaches work well, while NITM provides better accuracy in particular fractional contexts.

6.2. Example

Consider the nonlinear fractional DSW equation

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\partial }^{ϵ}\mathbb{S}(\vartheta ,\varrho )}{\partial {\varrho }^{ϵ}}}+\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}+\mathbb{S}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}=0,\hfill \\ \hfill & {\displaystyle \frac{{\partial }^{ϵ}\mathbb{T}(\vartheta ,\varrho )}{\partial {\varrho }^{ϵ}}}+\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}+{\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}=0,\hfill \end{array}$$

(46)

with

$$\begin{array}{cc}\hfill & \mathbb{S}(\vartheta ,0)={\displaystyle \frac{1}{9}}({\vartheta }^2-2\vartheta +1),\hfill \\ \hfill & \mathbb{T}(\vartheta ,0)={\displaystyle \frac{2(1-\vartheta )}{3}}.\hfill \end{array}$$

(47)

• Case I: Application of NITM

By implementing the ET to Equation (46), we obtain

$$\begin{array}{cc}\hfill & E\left[\mathbb{S}(\vartheta ,\varrho )\right]=s^2\left({\displaystyle \frac{1}{9}}({\vartheta }^2-2\vartheta +1)\right)-s^{ϵ}E\left[\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}+\mathbb{S}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}\right],\hfill \\ \hfill & E\left[\mathbb{T}(\vartheta ,\varrho )\right]=s^2\left({\displaystyle \frac{2(1-\vartheta )}{3}}\right)-s^{ϵ}E\left[\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}+{\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}\right].\hfill \end{array}$$

(48)

On implementing the inverse ET, we obtain

$$\begin{array}{cc}\hfill & \mathbb{S}(\vartheta ,\varrho )={\displaystyle \frac{1}{9}}({\vartheta }^2-2\vartheta +1)-E^{-1}\left[s^{ϵ}E\left[\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}+\mathbb{S}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}\right]\right],\hfill \\ \hfill & \mathbb{T}(\vartheta ,\varrho )={\displaystyle \frac{2(1-\vartheta )}{3}}-E^{-1}\left[s^{ϵ}E\left[\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}+{\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}\right]\right].\hfill \end{array}$$

(49)

By NITM, we have

$$\begin{array}{cc}\hfill & {\mathbb{S}}_0(\vartheta ,\varrho )={\displaystyle \frac{1}{9}}({\vartheta }^2-2\vartheta +1),\hfill \\ \hfill & {\mathbb{T}}_0(\vartheta ,\varrho )={\displaystyle \frac{2(1-\vartheta )}{3}},\hfill \\ \hfill & {\mathbb{S}}_1(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}E\left\{-{\mathbb{T}}_0(\vartheta ,\varrho ){\displaystyle \frac{\partial {\mathbb{S}}_0(\vartheta ,\varrho )}{\partial \vartheta }}-{\mathbb{S}}_0(\vartheta ,\varrho ){\displaystyle \frac{\partial {\mathbb{T}}_0(\vartheta ,\varrho )}{\partial \vartheta }}\right\}\right]={\displaystyle \frac{2{(\vartheta -1)}^2}{9}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}},\hfill \\ \hfill & {\mathbb{T}}_1(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}E\left\{-{\mathbb{T}}_0(\vartheta ,\varrho ){\displaystyle \frac{\partial {\mathbb{T}}_0(\vartheta ,\varrho )}{\partial \vartheta }}-{\displaystyle \frac{\partial {\mathbb{S}}_0(\vartheta ,\varrho )}{\partial \vartheta }}\right\}\right]=-{\displaystyle \frac{2(\vartheta -1)}{3}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\mathbb{S}}_2(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}E\right\{(-({\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho )){\displaystyle \frac{\partial ({\mathbb{S}}_0(\vartheta ,\varrho )+{\mathbb{S}}_1(\vartheta ,\varrho ))}{\partial \vartheta }}-({\mathbb{S}}_0(\vartheta ,\varrho )+{\mathbb{S}}_1(\vartheta ,\varrho ))\hfill \\ \hfill & {\displaystyle \frac{\partial ({\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho ))}{\partial \vartheta }})-\left(-\left({\mathbb{T}}_0(\vartheta ,\varrho )\right){\displaystyle \frac{\partial \left({\mathbb{S}}_0(\vartheta ,\varrho )\right)}{\partial \vartheta }}-\left({\mathbb{S}}_0(\vartheta ,\varrho )\right){\displaystyle \frac{\partial \left({\mathbb{T}}_0(\vartheta ,\varrho )\right)}{\partial \vartheta }}\right)\}]=\hfill \\ \hfill & {\displaystyle \frac{2}{3}}{(\vartheta -1)}^2{\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}}+{\displaystyle \frac{4}{9}}{\displaystyle \frac{{(\vartheta -1)}^2\Gamma (2ϵ+1)}{\Gamma {(ϵ+1)}^2}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}},\hfill \\ \hfill & {\mathbb{T}}_2(\vartheta ,\varrho )=E^{-1}\left[s^{ϵ}E\right\{\left(-({\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho )){\displaystyle \frac{\partial ({\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho ))}{\partial \vartheta }}-{\displaystyle \frac{\partial ({\mathbb{S}}_0(\vartheta ,\varrho )+{\mathbb{S}}_1(\vartheta ,\varrho ))}{\partial {\vartheta }^3}}\right)-\hfill \\ \hfill & \left({\mathbb{T}}_0(\vartheta ,\varrho ){\displaystyle \frac{\partial {\mathbb{T}}_0(\vartheta ,\varrho )}{\partial \vartheta }}-{\displaystyle \frac{\partial {\mathbb{S}}_0(\vartheta ,\varrho )}{\partial \vartheta }}\right)\left\}\right]=-{\displaystyle \frac{4}{3}}(\vartheta -1){\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}}-{\displaystyle \frac{4}{9}}{\displaystyle \frac{(\vartheta -1)\Gamma (2ϵ+1)}{\Gamma {(ϵ+1)}^2}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}},\hfill \\ & \hfill ⋮\hfill \end{array}$$

At the end, the solution in series form is taken as

$$\begin{array}{cc}\hfill & \mathbb{S}(\vartheta ,\varrho )={\mathbb{S}}_0(\vartheta ,\varrho )+{\mathbb{S}}_1(\vartheta ,\varrho )+{\mathbb{S}}_2(\vartheta ,\varrho )+\dots ,\hfill \\ \hfill & \mathbb{S}(\vartheta ,\varrho )={\displaystyle \frac{1}{9}}({\vartheta }^2-2\vartheta +1)+{\displaystyle \frac{2{(\vartheta -1)}^2}{9}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}+{\displaystyle \frac{2}{3}}{(\vartheta -1)}^2{\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}}+{\displaystyle \frac{4}{9}}{\displaystyle \frac{{(\vartheta -1)}^2\Gamma (2ϵ+1)}{\Gamma {(ϵ+1)}^2}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}}+\dots ,\hfill \\ \hfill & \mathbb{T}(\vartheta ,\varrho )={\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho )+{\mathbb{T}}_2(\vartheta ,\varrho )+\dots ,\hfill \\ \hfill & \mathbb{T}(\vartheta ,\varrho )={\displaystyle \frac{2(1-\vartheta )}{3}}-{\displaystyle \frac{2(\vartheta -1)}{3}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}-{\displaystyle \frac{4}{3}}(\vartheta -1){\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}}-{\displaystyle \frac{4}{9}}{\displaystyle \frac{(\vartheta -1)\Gamma (2ϵ+1)}{\Gamma {(ϵ+1)}^2}}{\displaystyle \frac{{\varrho }^{3ϵ}}{\Gamma (3ϵ+1)}}+\dots ,\hfill \end{array}$$

(50)

• Case II: Application of ETDM

By implementing the ET to Equation (46), we obtain

$$\begin{array}{cc}\hfill & E\left[\mathbb{S}(\vartheta ,\varrho )\right]=s^2\left({\displaystyle \frac{1}{9}}({\vartheta }^2-2\vartheta +1)\right)-s^{ϵ}E\left[\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}+\mathbb{S}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}\right],\hfill \\ \hfill & E\left[\mathbb{T}(\vartheta ,\varrho )\right]=s^2\left({\displaystyle \frac{2(1-\vartheta )}{3}}\right)-s^{ϵ}E\left[\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}+{\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}\right].\hfill \end{array}$$

(51)

On implementing the inverse ET, we obtain

$$\begin{array}{cc}\hfill & \mathbb{S}(\vartheta ,\varrho )={\displaystyle \frac{1}{9}}({\vartheta }^2-2\vartheta +1)-E^{-1}\left[s^{ϵ}E\left[\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}+\mathbb{S}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}\right]\right],\hfill \\ \hfill & \mathbb{T}(\vartheta ,\varrho )={\displaystyle \frac{2(1-\vartheta )}{3}}-E^{-1}\left[s^{ϵ}E\left[\mathbb{T}(\vartheta ,\varrho ){\displaystyle \frac{\partial \mathbb{T}(\vartheta ,\varrho )}{\partial \vartheta }}+{\displaystyle \frac{\partial \mathbb{S}(\vartheta ,\varrho )}{\partial \vartheta }}\right]\right].\hfill \end{array}$$

(52)

Assume the series form solution as

$$\mathbb{S}(\vartheta ,\varrho )=\sum _{m=0}^{\infty }{\mathbb{S}}_m(\vartheta ,\varrho ),\mathbb{T}(\vartheta ,\varrho )=\sum _{m=0}^{\infty }{\mathbb{T}}_m(\vartheta ,\varrho ).$$

(53)

The nonlinear terms decomposes as $\mathbb{T}(\vartheta ,\varrho ){\mathbb{S}}_{\vartheta }(\vartheta ,\varrho )={\sum }_{m=0}^{\infty }{\mathcal{A}}_m,\mathbb{S}(\vartheta ,\varrho ){\mathbb{T}}_{\vartheta }(\vartheta ,\varrho )={\sum }_{m=0}^{\infty }{\mathcal{B}}_m,\mathbb{T}(\vartheta ,\varrho ){\mathbb{T}}_{\vartheta }(\vartheta ,\varrho )={\sum }_{m=0}^{\infty }{\mathcal{C}}_m.$ So, we have

$$\begin{array}{cc}\hfill & \sum _{m=0}^{\infty }{\mathbb{S}}_m(\vartheta ,\varrho )={\displaystyle \frac{1}{9}}({\vartheta }^2-2\vartheta +1)-{\mathtt{E}}^{-1}\left[s^{ϵ}\mathtt{E}\left[\sum _{m=0}^{\infty }{\mathcal{A}}_m+\sum _{m=0}^{\infty }{\mathcal{B}}_m\right]\right],\hfill \\ \hfill & \sum _{m=0}^{\infty }{\mathbb{T}}_m(\vartheta ,\varrho )={\displaystyle \frac{2(1-\vartheta )}{3}}-{\mathtt{E}}^{-1}\left[s^{ϵ}\mathtt{E}\left[\sum _{m=0}^{\infty }{\mathcal{C}}_m+{\mathbb{S}}_{\vartheta }(\vartheta ,\varrho )\right]\right].\hfill \end{array}$$

(54)

Similarly,

$${\mathbb{S}}_0(\vartheta ,\varrho )={\displaystyle \frac{1}{9}}({\vartheta }^2-2\vartheta +1),$$

$${\mathbb{T}}_0(\vartheta ,\varrho )={\displaystyle \frac{2(1-\vartheta )}{3}},$$

On $m=0$

$${\mathbb{S}}_1(\vartheta ,\varrho )={\displaystyle \frac{2{(\vartheta -1)}^2}{9}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}},$$

$${\mathbb{T}}_1(\vartheta ,\varrho )=-{\displaystyle \frac{2(\vartheta -1)}{3}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}},$$

On $m=1$

$$\begin{array}{cc}\hfill & {\mathbb{S}}_2(\vartheta ,\varrho )={\displaystyle \frac{2}{3}}{(\vartheta -1)}^2{\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\mathbb{T}}_2(\vartheta ,\varrho )=-{\displaystyle \frac{4}{3}}(\vartheta -1){\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}},\hfill \end{array}$$

Thus, the solution in series form illustrates as

$$\begin{array}{cc}\hfill & \mathbb{S}(\vartheta ,\varrho )={\mathbb{S}}_0(\vartheta ,\varrho )+{\mathbb{S}}_1(\vartheta ,\varrho )+{\mathbb{S}}_2(\vartheta ,\varrho )+{\mathbb{S}}_3(\vartheta ,\varrho )+\dots \hfill \\ \hfill & \mathbb{S}(\vartheta ,\varrho )={\displaystyle \frac{1}{9}}({\vartheta }^2-2\vartheta +1)+{\displaystyle \frac{2{(\vartheta -1)}^2}{9}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}+{\displaystyle \frac{2}{3}}{(\vartheta -1)}^2{\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}}+\dots ,\hfill \\ \hfill & \mathbb{T}(\vartheta ,\varrho )={\mathbb{T}}_0(\vartheta ,\varrho )+{\mathbb{T}}_1(\vartheta ,\varrho )+{\mathbb{T}}_2(\vartheta ,\varrho )+{\mathbb{T}}_3(\vartheta ,\varrho )+\dots \hfill \\ \hfill & \mathbb{T}(\vartheta ,\varrho )={\displaystyle \frac{2(1-\vartheta )}{3}}-{\displaystyle \frac{2(\vartheta -1)}{3}}{\displaystyle \frac{{\varrho }^{ϵ}}{\Gamma (ϵ+1)}}-{\displaystyle \frac{4}{3}}(\vartheta -1){\displaystyle \frac{{\varrho }^{2ϵ}}{\Gamma (2ϵ+1)}}+\dots ,\hfill \end{array}$$

(55)

Inserting $ϵ=1$, we obtain

$$\mathbb{S}(\vartheta ,\varrho )={\displaystyle \frac{{(\vartheta -1)}^2}{9{(\varrho -1)}^2}},\mathbb{T}(\vartheta ,\varrho )={\displaystyle \frac{2(\vartheta -1)}{3(\varrho -1)}}.$$

(56)

Figure 9 depicts the NITM approximate outcome (50) of $\mathbb{S}(\vartheta ,\varrho )$ for different values of $ϵ$. The Figure 10 depicts the correspondence between the NITM solution (50) and the accurate solution (56) for $\mathbb{S}(\vartheta ,\varrho )$ at $ϵ=1$. Figure 11 displays the ETDM approximate solution (55) at different $ϵ$ values, while Figure 12 shows the comparison of ETDM solution (55) against accurate solution (45) for $\mathbb{S}(\vartheta ,\varrho )$ at $ϵ=1$. Figure 13 depicts the NITM approximate outcome (50) of $\mathbb{S}(\vartheta ,\varrho )$ for different values of $ϵ$. The Figure 14 depicts the correspondence between NITM solution (50) and accurate solution (56) for $\mathbb{S}(\vartheta ,\varrho )$ at $ϵ=1$. Figure 15 displays the ETDM approximate solution (55) at different $ϵ$ values while Figure 16 shows the comparison of the ETDM solution (55) against the accurate solution (56) for $\mathbb{S}(\vartheta ,\varrho )$ at $ϵ=1$. All figures show that when the fractional order approaches 1, the approximate solution predicts analytic solution behavior while matching the analytic solution identically for the case of $ϵ=1$, thus confirming the reliability of our proposed approaches. The NITM approximation (50) and the ETDM approximation (55) are compared to the analytic solution (56) for $\mathbb{S}(\vartheta ,\varrho )$ through

Table 7. Accurate solution along with NITM solution at numerous orders of $ϵ$ for $\mathbb{S}(\vartheta ,\varrho )$.

$\mathit{\vartheta }$	$\mathit{ϵ}=0.85$	$\mathit{ϵ}=0.90$	$\mathit{ϵ}=0.95$	$\mathit{ϵ}=1\left(\mathit{NITM}\right)$	$\mathit{ϵ}=1\left(\mathit{Accurate}\right)$
0.0	0.1159736073	0.1148737650	0.1140243245	0.1133668148	0.1133671168
0.1	0.0939386218	0.0930477496	0.0923597028	0.0918271199	0.0918273646
0.2	0.0742231086	0.0735192096	0.0729755676	0.0725547614	0.0725549547
0.3	0.0568270675	0.0562881448	0.0558719189	0.0555497392	0.0555498872
0.4	0.0417504986	0.0413545554	0.0410487568	0.0408120533	0.0408121620
0.5	0.0289934018	0.0287184412	0.0285060811	0.0283417037	0.0283417792
0.6	0.0185557771	0.0183798024	0.0182438919	0.0181386903	0.0181387386
0.7	0.0104376245	0.0103386387	0.0102621891	0.0102030132	0.0102030405
0.8	0.0046389442	0.0045949505	0.0045609729	0.0045346725	0.0045346846
0.9	0.0011597360	0.0011487376	0.0011402432	0.0011336681	0.0011336711
1.0	0.0000000000	0.0000000000	0.0000000000	0.0000000000	0.0000000000

and

Table 8. Accurate solution along with ETDM solution at numerous orders of $ϵ$ for $\mathbb{S}(\vartheta ,\varrho )$.

$\mathit{\vartheta }$	$\mathit{ϵ}=0.85$	$\mathit{ϵ}=0.90$	$\mathit{ϵ}=0.95$	$\mathit{ϵ}=1\left(\mathit{ETDM}\right)$	$\mathit{ϵ}=1\left(\mathit{Accurate}\right)$
0.0	0.1159718714	0.1148729961	0.1140239860	0.1133666667	0.1133671168
0.1	0.0939372158	0.0930471268	0.0923594286	0.0918269999	0.0918273646
0.2	0.0742219976	0.0735187175	0.0729753510	0.0725546666	0.0725549547
0.3	0.0568262169	0.0562877680	0.0558717531	0.0555496666	0.0555498872
0.4	0.0417498736	0.0413542785	0.0410486349	0.0408119999	0.0408121620
0.5	0.0289929678	0.0287182490	0.0285059965	0.0283416666	0.0283417792
0.6	0.0185554994	0.0183796794	0.0182438377	0.0181386666	0.0181387386
0.7	0.0104374683	0.0103385695	0.0102621586	0.0102029999	0.0102030405
0.8	0.0046388748	0.0045949198	0.0045609593	0.0045346666	0.0045346846
0.9	0.0011597187	0.0011487299	0.0011402398	0.0011336666	0.0011336711
1.0	0.0000000000	0.0000000000	0.0000000000	0.0000000000	0.0000000000

. Additionally, the NITM approximation (50) and the ETDM approximation (55) are compared to the analytic solution (56) for $\mathbb{T}(\vartheta ,\varrho )$ through

Table 9. Accurate solution along with NITM solution at numerous orders of $ϵ$ for $\mathbb{T}(\vartheta ,\varrho )$.

$\mathit{\vartheta }$	$\mathit{ϵ}=0.85$	$\mathit{ϵ}=0.90$	$\mathit{ϵ}=0.95$	$\mathit{ϵ}=1\left(\mathit{NITM}\right)$	$\mathit{ϵ}=1\left(\mathit{Accurate}\right)$
0.0	0.6810788654	0.6778532040	0.6753478089	0.6734001482	0.6734006734
0.1	0.6129709789	0.6100678836	0.6078130280	0.6060601334	0.6060606061
0.2	0.5448630923	0.5422825632	0.5402782471	0.5387201186	0.5387205387
0.3	0.4767552058	0.4744972428	0.4727434662	0.4713801037	0.4713804714
0.4	0.4086473192	0.4067119224	0.4052086853	0.4040400889	0.4040404040
0.5	0.3405394327	0.3389266020	0.3376739045	0.3367000741	0.3367003367
0.6	0.2724315462	0.2711412816	0.2701391236	0.2693600593	0.2693602694
0.7	0.2043236596	0.2033559612	0.2026043427	0.2020200445	0.2020202020
0.8	0.1362157731	0.1355706408	0.1350695618	0.1346800296	0.1346801347
0.9	0.0681078865	0.0677853204	0.0675347809	0.0673400148	0.0673400673
1.0	0.0000000000	0.0000000000	0.0000000000	0.0000000000	0.0000000000

and

Table 10. Accurate solution along with ETDM solution at numerous orders of $ϵ$ for $\mathbb{T}(\vartheta ,\varrho )$.

$\mathit{\vartheta }$	$\mathit{ϵ}=0.85$	$\mathit{ϵ}=0.90$	$\mathit{ϵ}=0.95$	$\mathit{ϵ}=1\left(\mathit{ETDM}\right)$	$\mathit{ϵ}=1\left(\mathit{Accurate}\right)$
0.0	0.6810771295	0.6778524351	0.6753474704	0.6734000001	0.6734006734
0.1	0.6129694166	0.6100671916	0.6078127234	0.6060600001	0.6060606061
0.2	0.5448617036	0.5422819481	0.5402779763	0.5387200001	0.5387205387
0.3	0.4767539907	0.4744967046	0.4727432293	0.4713800001	0.4713804714
0.4	0.4086462777	0.4067114611	0.4052084822	0.4040400001	0.4040404040
0.5	0.3405385647	0.3389262175	0.3376737352	0.3367000001	0.3367003367
0.6	0.2724308518	0.2711409740	0.2701389882	0.2693600000	0.2693602694
0.7	0.2043231389	0.2033557305	0.2026042411	0.2020200000	0.2020202020
0.8	0.1362154259	0.1355704870	0.1350694941	0.1346800000	0.1346801347
0.9	0.0681077129	0.0677852435	0.0675347470	0.0673400000	0.0673400673
1.0	0.0000000000	0.0000000000	0.0000000000	0.0000000000	0.0000000000

. The obtained approximations are shown in Table 7, Table 8, Table 9 and Table 10 for a range of $ϵ$ values, which are 1, 0.95, 0.90, and 0.85, respectively, with temporal variable $\varrho =0.01$. The tables visualize the close match between our computational results and actual solutions at $ϵ=1$.

7. Conclusions

In this paper, we have successfully developed novel strategies for obtaining the approximate solution of fractional DSW and fractional SW equations with fractional derivatives. The implementation of the proposed methods for recurrence relations is straightforward, making it possible to achieve iterative results. The methods demonstrate excellent capability when handling nonlinear terms while producing series results through successive iterations. The technique requires an initial condition to generate simple iterative methods that produce approximate solutions near the analytic solution. The proposed methods need less computational effort while providing better efficiency in their operations. The displayed graphs and tables validate that the proposed schemes generate highly accurate results. The proposed schemes demonstrate high accuracy, making them suitable for exploring complex nonlinear differential equations of fractional order. The application of fractional derivatives establishes novel possibilities within mathematical modeling fields. Due to the beneficial results of using the suggested techniques, these techniques can be used to analyze different kinds of fractional wave equations that occur in different plasma models to understand the ambiguity surrounding specific behaviors that accompany certain nonlinear phenomena (such as solitary, shock, and cnoidal waves, etc.) that occur in different plasma models.