Insights into the Time-Fractional Nonlinear KdV-Type Equations Under Non-Singular Kernel Operators

Abstract

In this study, nonlinear fractional Korteweg–de Vries (KdV) type equations with nonlocal operators are studied using Mittag–Leffler kernels and exponential decay. The KdV equations are well known for its use in modeling ion-acoustic waves in plasma, oceanic dynamics, and shallow-water waves. As a result, mathematicians are working to examine modified and generalized versions of the basic KdV equation. In order to find the solutions of nonlinear fractional KdV equations, an extension of this concept is described in the current paper. The solution of fractional KdV equations is carried out using the well-known natural transform decomposition method (NTDM). To evaluate the problem, we employ the fractional operator in the Caputo–Fabrizio (CF) and the Atangana–Baleanu–Caputo sense (ABC) manner. Nonlinear terms can be handled with Adomian polynomials. The main advantage of this novel approach is that it might offer an approximate solution in the form of convergent series using easy calculations. The dynamical behavior of the resulting solutions have been demonstrated using graphs. Numerical data is represented visually in the tables. The solutions at various fractional orders are found and it is proved that they all tend to an integer-order solution. Additionally, we examine our findings with those of the iterative transform method (ITM) and the residual power series transform method (RPSTM). It is evident from the comparison that our approach offers better outcomes compared to other approaches. The results of the suggested method are very accurate and give helpful details on the real dynamics of each issue. The present technique can be expanded to address other significant fractional order problems due to its straightforward implementation.

1. Introduction

As old as classical calculus, fractional calculus literature dates back to 1695. The idea of the time-fractional derivative was initially brought up in a letter from L’Hospital to Leibnitz, and since then, development has been slow. Due to their extensive applications in the practical sciences, the integral and derivative of non-integer order are regarded as common topics of fractional calculus (FC). Due to its significance for modelling diverse processes in nature, this topic has become very popular among researchers. Systems controlled by memory, hereditary effects, and nonlocal interactions naturally give rise to fractional differential equations (FDEs), where the physical behavior is typically not adequately represented by classical integer-order models. The Riemann–Liouville (RL) fractional derivative, the Caputo fractional derivative [1], the Atangana–Baleanu–Caputo (ABC) fractional derivative [2], and the Caputo–Fabrizio fractional derivative [3] are among the several definitions of fractional calculus. Singular kernel and non-singular kernel types are the two broad categories into which fractional derivatives fall. Power-law memory is incorporated into singular kernel operators (like the RL and Caputo derivatives), but non-singular kernel operators (like the ABC and Caputo–Fabrizio derivatives) employ smooth kernels to prevent algebraic singularities [4]. An additional method for analyzing FC is provided by the CF and ABC derivative, which has uses in many different domains. Systems with memory and hereditary features can be modeled using it, and it offers the possibility of examining and characterizing intricate behaviors that integer-order derivatives are unable to adequately express. Their extensive use in modern research covers a variety of fields, including mathematical epidemiology [5,6], chaotic systems with singular and non-singular kernels [7], catalyzed hydrogenolysis of glycerol [8], oscillatory systems and electric networks [9], propagation of waves in complex media [10], nanofluid flow [11], bifurcation of neural network systems with delays [12], and nonlinear optical fibers [13].

The early theory and advancement of fractional derivatives and fractional differential equations (FDEs) have been rapidly noted throughout the past few decades. Authors like Kilbas et al. [14], Abbas et al. [15], Legnani et al. [16] and Hilfer [17] go into additional detail and grow on the area. Their study is primarily concerned with the theoretical issues related to FDEs and the systematic understanding of FC such as uniqueness and existence. Further concepts of FC can be found in the works by Hern’andez et al. [18], which cover the most recent advances in the theory of FDEs. FC’s applications in interdisciplinary disciplines like image processing and control theory were explored by Magin et al. [19] and Mainardi [20]. The modelling of numerous complicated natural and nonlinear processes benefits greatly from the use of fractional-order differential equations (FPDEs). A wide variety of processes and systems, memory, and other areas of mathematics have all benefited greatly from the work of FPDEs. Many natural occurrences can be accurately described by FPDEs because of their more convergent representation, whether it be in terms of time or space. FPDEs are of interest to scientists studying dielectric polarization, control theory, anomalous diffusion, and other issues relating to physical processes [21,22,23,24]. The largest contribution to explaining these kinds of systems is made by FPDEs because the non-integer order of the derivative describes hereditary features and situations where the future state is influenced by the previous state more precisely than integer order. As the integer order derivative is local and non-integer derivative is global, FPDEs have become more and more widespread. Some of the scientific issues and phenomena that FPDEs are used to represent include continuum mechanics, coloured noises, psychology, biology, acoustics, chemistry, and biology. Many areas of physics and hydrology, including [25,26,27,28,29] contain unique applications of FPDEs.

The construction of numerical and analytical approaches for the solution of FPDEs has become a major topic among scholars in recent years. Mathematicians never have it easy when trying to find numerical or analytical solutions to FPDEs. Fractional calculus has become a vibrant and influential field of study due to the necessity of resolving and exploring these complicated issues. Researchers have shown a great deal of interest in fractional PDEs in particular, which has led to the creation of numerous solutions. A number of efficient methods for examining both accurate and computational solutions to fractional differential equations have been developed in recent years. The following are some essential techniques such as the residual power series method [30], natural homotopy perturbation method [31], new iterative method [32], homotopy analysis method [33] the Legendre base method [34], the differential transform method [35], the Bernstein polynomial [36] and many more.

In particular, the KdV problem is well known as the ideal example of a perfectly solvable model, that is, a nonlinear partial differential equation whose solutions can be precisely and exactly described. Prototypical solitons are included in the solutions as well. The inverse scattering transform can be used to solve the KdV equation. The KdV equation’s basic mathematical theory is extensive, fascinating, and the focus of current mathematical research in general. The Korteweg–de Vries–Burgers equation is stated as [37]

$$D_{\zeta }^{\mu }\omega (\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta }(\vartheta ,\zeta )+{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )=0,0<\mu \le 1,$$

(1)

The modified KdV equation is [37]

$${\omega }_{\zeta }+{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )=0$$

(2)

and differs from the original KdV equation in the last nonlinear term only. The surface similarity is less important because of this, which results in some significant variances. On the other hand, the so-called “Miura transformation” connects these two equations at a deeper level. Understanding the function of nonlinear dispersion and the creation of structures like liquid drops leads to the modification of the KdV equation, which displays compactons, or solitons with compact support.

The modified Korteweg–de Vries (mKdV) equation is regarded as [38]

$$D_{\zeta }^{\mu }\omega (\vartheta ,\zeta )+6{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )=0,0<\mu \le 1,$$

(3)

The mKdV equation occurs in a wide range of disciplines, including biophysics, plasma science, shallow water models, and others. The mKdV equation’s dark multi-soliton solutions were obtained using a Darboux transformation [39]. Wazwaz [40] for obtaining the numerous solitary wave solutions of the mKdV equation using the extended tanh approach. Parkes [41] developed model solutions to the mKdV equation based on his observations on the tanh-coth expansion method. The authors of Ref. [42] developed an effective numerical method to investigate the asymptotic solution of Equation (3). With the use of phase portrait theory [43], the authors investigated compact solitary waves of the mKdV equation. The nonlocal symmetry, ideal systems, and explicit solutions of the mKdV equation were researched using the known Lax pair in [44].

We study the fractional mKdV equation utilising the CF and ABC operators as motivated by the previous literature. For the purpose of determining an approximation of the solution, we employ a hybrid strategy that elegantly combines the natural transform [45] and Adomian decomposition method [46]. By eliminating the necessity to calculate the fractional integrals or the fractional derivative in the recursive mechanism, the suggested approach streamlines the estimate of the series terms compared to the standard Adomian technique. NTDM avoids all round-off errors and doesn’t need discretization, linearization, perturbation, or specified assumptions. The suggested method offers reliable results that give precise solutions to the relevant problems. Our techniques generated infinite series in the numerical examples. After a specific number of iterations, it is found that the computational series approaches the precise outcome extremely closely, and the resulting series provides to us the results quite fast. In order to achieve accurate and approximate results quickly, this study offers a fundamental framework for researchers to analyze this method and use them in a range of applications. The fractional analysis carried out utilizing the suggested technique to look at the problems from a fractional perspective. To demonstrate a novel form of soliton solution that hasn’t been previously researched in the literature, we simulate the obtained solution for a few fractional orders. Recently, NTDM employed to solve time-fractional Fornberg–Whitham equation [47] and time-fractional coupled KdV equations [48]. The following is the paper’s outline: The key relationships are recalled in Section 2. We offer a general examination of the suggested technique in Section 3. The convergence analysis of the proposed method is covered in Section 4. The consideration equation’s solution is the focus of Section 5. We give the conclusions in Section 6.

2. Preliminaries

Here we we will discuss some basic definitions.

Definition 1. 

The fractional R-L integral is presented as [49]

$$\begin{array}{cc}\hfill I^{\mu }j\left(\phi \right)=& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mu \right)}}{\int }_0^{\phi }{(\phi -\varpi )}^{\mu -1}j\left(\varpi \right)d\varpi ,\mu >0,\phi >0,\hfill \\ \hfill and{I}^{0}j\left(\phi \right)=& j\left(\phi \right).\hfill \end{array}$$

(4)

Definition 2. 

The Caputo fractional derivative is presented as [49]

$$\begin{array}{c}\hfill D_{\phi }^{\mu }j\left(\phi \right)=I^{m-\mu }{D}^{m}j\left(\phi \right)={\displaystyle \frac{1}{m-\mu }}{\int }_{\phi }^0{(\phi -\varpi )}^{m-\mu -1}{j}^m\left(\varpi \right)d\varpi ,\end{array}$$

(5)

for $m-1<\mu \le m,m\in N,\phi >0,j\in C_v^m,v\ge -1$.

Definition 3. 

The CF fractional derivative is presented as [49]

$$\begin{array}{c}\hfill D_{\phi }^{\mu }j\left(\phi \right)={\displaystyle \frac{1}{1-\mu }}{\int }_0^{\phi }exp\left({\displaystyle \frac{-\mu (\phi -\varpi )}{1-\mu }}\right)j\left(\varpi \right)d\varpi ,\end{array}$$

(6)

having $0<\mu <1$.

Definition 4. 

The fractional ABC derivative is presented as [49]

$$\begin{array}{c}\hfill D_{\phi }^{\mu }j\left(\phi \right)={\displaystyle \frac{B\left(\mu \right)}{1-\mu }}{\int }_0^{\phi }{E}_{\mu }\left({\displaystyle \frac{-\mu {(\phi -\varpi )}^{\mu }}{1-\mu }}\right)j\left(\varpi \right)d\varpi ,\end{array}$$

(7)

with $0<\mu <1$, and $B\left(\mu \right)$ represented the normalization function having $B\left(0\right)=B\left(1\right)=1$ and $E_{\mu }\left(z\right)={\sum }_{m=0}^{\infty }{\displaystyle \frac{z^m}{\mathrm{\Gamma }({\mu }^m+1)}}$ is the Mittag–Leffler function.

Definition 5. 

The NT of $\omega \left(\zeta \right)$ is presented as

$$\begin{array}{c}\hfill \mathit{N}\left(\omega \left(\zeta \right)\right)=\mathcal{P}(\lambda ,\eta )={\int }_{-\infty }^{\infty }{e}^{-\lambda \zeta }\omega \left(\zeta \right)d\zeta ,\lambda \in (-\infty ,\infty ).\end{array}$$

(8)

For $\zeta \in (0,\infty )$, NT of $\omega \left(\zeta \right)$ is presented as

$$\begin{array}{c}\hfill \mathit{N}\left(\omega \left(\zeta \right)H\left(\zeta \right)\right)=\mathit{N}\left(\omega \left(\zeta \right)\right)=\mathcal{P}(\lambda ,\eta )={\int }_{-\infty }^{\infty }{e}^{-\lambda \zeta }\omega \left(\zeta \right)d\zeta ,\lambda \in (0,\infty ),\end{array}$$

(9)

with $H\left(\zeta \right)$ illustrating the Heaviside function.

Definition 6. 

The function $\mathcal{P}(\lambda ,\eta )$ inverse NT is presented as

$$\begin{array}{c}\hfill {\mathit{N}}^{-1}\left[\mathcal{P}(\lambda ,\eta )\right]=\omega \left(\zeta \right),\forall \zeta \ge 0.\end{array}$$

(10)

Definition 7. 

The NT of $D_{\zeta }^{\mu }\omega \left(\zeta \right)$ in Caputo sense is presented as [49]

$$\begin{array}{c}\hfill \mathit{N}\left[D_{\zeta }^{\mu }\omega \left(\zeta \right)\right]={\left({\displaystyle \frac{\lambda }{\eta }}\right)}^{\mu }\left(\mathit{N}\left[\omega \left(\zeta \right)\right]-\left({\displaystyle \frac{1}{\lambda }}\right)\omega \left(0\right)\right).\end{array}$$

(11)

Definition 8. 

The NT of $D_{\zeta }^{\mu }\omega \left(\zeta \right)$ in CF sense is presented as [49]

$$\begin{array}{c}\hfill \mathit{N}\left[D_{\zeta }^{\mu }\omega \left(\zeta \right)\right]={\displaystyle \frac{1}{1-\mu +\mu \left(\frac{\eta }{\lambda }\right)}}\left(\mathit{N}\left[\omega \left(\zeta \right)\right]-\left({\displaystyle \frac{1}{\lambda }}\right)\omega \left(0\right)\right).\end{array}$$

(12)

Definition 9. 

The NT of $D_{\zeta }^{\mu }\omega \left(\zeta \right)$ in ABC sense is presented as [49]

$$\begin{array}{c}\hfill \mathit{N}\left[D_{\zeta }^{\mu }\omega \left(\zeta \right)\right]={\displaystyle \frac{B\left[\mu \right]}{1-\mu +\mu {\left(\frac{\eta }{\lambda }\right)}^{\mu }}}\left(\mathit{N}\left[\omega \left(\zeta \right)\right]-\left({\displaystyle \frac{1}{\lambda }}\right)\omega \left(0\right)\right).\end{array}$$

(13)

3. Methodology

The procedure of NTDM for the below equation is discussed in this section.

$$\begin{array}{c}\hfill D_{\zeta }^{\mu }\omega (\vartheta ,\zeta )=\mathcal{L}\left(\omega (\vartheta ,\zeta )\right)+\mathbb{N}\left(\omega (\vartheta ,\zeta )\right)+h(\vartheta ,\zeta ),\end{array}$$

(14)

with initial source

$$\begin{array}{c}\hfill \omega (\vartheta ,0)=\varphi \left(\vartheta \right),\end{array}$$

(15)

with $\mathcal{L}$, $\mathbb{N}$, $h(\vartheta ,\zeta )$ indicating the linear, nonlinear and the source functions.

3.1. Case I $\left(NTD{M}_{CF}\right)$

On inserting NT to Equation (14) by means of Caputo–Fabrizio, we get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{p(\mu ,\eta ,\lambda )}}\left(\mathit{N}\left[\omega (\vartheta ,\zeta )\right]-{\displaystyle \frac{\varphi \left(\vartheta \right)}{\lambda }}\right)=\mathit{N}\left[M(\vartheta ,\zeta )\right],\hfill \\ \hfill & \mathit{N}\left[\omega (\vartheta ,\zeta )\right]={\displaystyle \frac{\varphi \left(\vartheta \right)}{\lambda }}+p(\mu ,\eta ,\lambda )\mathit{N}\left[M(\vartheta ,\zeta )\right],\hfill \end{array}$$

(16)

with

$$p(\mu ,\eta ,\lambda )=1-\mu +\mu \left({\displaystyle \frac{\eta }{\lambda }}\right),$$

(17)

and

$$M(\vartheta ,\zeta )=\mathcal{L}\left(\omega \right(\vartheta ,\zeta \left)\right)+\mathbb{N}\left(\omega \right(\vartheta ,\zeta \left)\right)+h(\vartheta ,\zeta ).$$

(18)

On inserting inverse NT to Equation (16), we get

$$\begin{array}{c}\hfill \omega (\vartheta ,\zeta )={\mathit{N}}^{-1}\left({\displaystyle \frac{\varphi \left(\vartheta \right)}{\lambda }}+p(\mu ,\eta ,\lambda )\mathit{N}\left[M(\vartheta ,\zeta )\right]\right).\end{array}$$

(19)

The nonlinear term $\mathbb{N}\left(\omega \right(\vartheta ,\zeta \left)\right)$ is represented as

$$\begin{array}{c}\hfill \mathbb{N}\left(\omega (\vartheta ,\zeta )\right)=\sum _{i=0}^{\infty }{A}_i,\end{array}$$

(20)

Here $A_i$ represents the Adomian polynomials. The function $\omega (\vartheta ,\zeta )$ series solution is presented as

$$\begin{array}{c}\hfill \omega (\vartheta ,\zeta )=\sum _{i=0}^{\infty }{\omega }_i(\vartheta ,\zeta ).\end{array}$$

(21)

By using Equations (20) and (21) into (19), we get

$$\begin{array}{cc}\hfill \sum _{i=0}^{\infty }{\omega }_i(\vartheta ,\zeta )=& {\mathit{N}}^{-1}\left({\displaystyle \frac{\varphi \left(\vartheta \right)}{\lambda }}+p(\mu ,\eta ,\lambda )\mathit{N}\left[h(\vartheta ,\zeta )\right]\right)\hfill \\ \hfill & +{\mathit{N}}^{-1}\left(p(\mu ,\eta ,\lambda )\mathit{N}\left[\sum _{i=0}^{\infty }\mathcal{L}\left({\omega }_i(\vartheta ,\zeta )\right)+A_i\right]\right).\hfill \end{array}$$

(22)

By comparison of both sides, we have

$$\begin{array}{cc}\hfill {\omega }_0^{CF}(\vartheta ,\zeta )=& {\mathit{N}}^{-1}\left({\displaystyle \frac{\varphi \left(\vartheta \right)}{\lambda }}+p(\mu ,\eta ,\lambda )\mathit{N}\left[h(\vartheta ,\zeta )\right]\right),\hfill \\ \hfill {\omega }_1^{CF}(\vartheta ,\zeta )=& {\mathit{N}}^{-1}\left(p(\mu ,\eta ,\lambda )\mathit{N}\left[\mathcal{L}\left({\omega }_0(\vartheta ,\zeta )\right)+A_0\right]\right),\hfill \\ \hfill & ⋮\hfill \\ \hfill {\omega }_{l+1}^{CF}(\vartheta ,\zeta )=& {\mathit{N}}^{-1}\left(p(\mu ,\eta ,\lambda )\mathit{N}\left[\mathcal{L}\left({\omega }_l(\vartheta ,\zeta )\right)+A_l\right]\right),l=1,2,3,\dots \hfill \end{array}$$

(23)

Hence, the result of (14) in $NTD{M}_{CF}$ manner by using (23) into (21) as

$$\begin{array}{c}\hfill {\omega }^{CF}(\vartheta ,\zeta )={\omega }_0^{CF}(\vartheta ,\zeta )+{\omega }_1^{CF}(\vartheta ,\zeta )+{\omega }_2^{CF}(\vartheta ,\zeta )+\cdots \end{array}$$

(24)

3.2. Case II $\left(NTD{M}_{ABC}\right)$

On inserting NT to Equation (14) by means of Atangana–Baleanu–Caputo, we get

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{q(\mu ,\eta ,\lambda )}}\left(\mathit{N}\left[\omega (\vartheta ,\zeta )\right]-{\displaystyle \frac{\varphi \left(\vartheta \right)}{\lambda }}\right)=\mathit{N}\left[M(\vartheta ,\zeta )\right],\hfill \\ \hfill & \mathit{N}\left[\omega (\vartheta ,\zeta )\right]={\displaystyle \frac{\varphi \left(\vartheta \right)}{\lambda }}+q(\mu ,\eta ,\lambda )\mathit{N}\left[M(\vartheta ,\zeta )\right],\hfill \end{array}$$

(25)

with

$$\begin{array}{c}\hfill q(\mu ,\eta ,\lambda )={\displaystyle \frac{1-\mu +\mu {\left(\frac{\eta }{\lambda }\right)}^{\mu }}{B\left(\mu \right)}}.\end{array}$$

(26)

On inserting inverse NT to Equation (25), we get

$$\begin{array}{c}\hfill \omega (\vartheta ,\zeta )={\mathit{N}}^{-1}\left({\displaystyle \frac{\varphi \left(\vartheta \right)}{\lambda }}+q(\mu ,\eta ,\lambda )\mathit{N}\left[M(\vartheta ,\zeta )\right]\right).\end{array}$$

(27)

The nonlinear term $\mathbb{N}\left(\omega \right(\vartheta ,\zeta \left)\right)$ is represented as

$$\begin{array}{c}\hfill \mathbb{N}\left(\omega (\vartheta ,\zeta )\right)=\sum _{i=0}^{\infty }{A}_i.\end{array}$$

(28)

The function $\omega (\vartheta ,\zeta )$ series solution is presented as

$$\begin{array}{c}\hfill \omega (\vartheta ,\zeta )=\sum _{i=0}^{\infty }{\omega }_i(\vartheta ,\zeta ).\end{array}$$

(29)

By using Equations (28) and (29) into (27), we get

$$\begin{array}{cc}\hfill \sum _{i=0}^{\infty }{\omega }_i(\vartheta ,\zeta )=& {\mathit{N}}^{-1}\left({\displaystyle \frac{\varphi \left(\vartheta \right)}{\lambda }}+q(\mu ,\eta ,\lambda )\mathit{N}\left[h(\vartheta ,\zeta )\right]\right)\hfill \\ \hfill & +{\mathit{N}}^{-1}\left(q(\mu ,\eta ,\lambda )\mathit{N}\left[\sum _{i=0}^{\infty }\mathcal{L}\left({\omega }_i(\vartheta ,\zeta )\right)+A_i\right]\right).\hfill \end{array}$$

(30)

By comparing both sides, we have

$$\begin{array}{cc}\hfill {\omega }_0^{ABC}(\vartheta ,\zeta )=& {\mathit{N}}^{-1}\left({\displaystyle \frac{\varphi \left(\vartheta \right)}{\lambda }}+q(\mu ,\eta ,\lambda )\mathit{N}\left[h(\vartheta ,\zeta )\right]\right),\hfill \\ \hfill {\omega }_1^{ABC}(\vartheta ,\zeta )=& {\mathit{N}}^{-1}\left(q(\mu ,\eta ,\lambda )\mathit{N}\left[\mathcal{L}\left({\omega }_0(\vartheta ,\zeta )\right)+A_0\right]\right),\hfill \\ \hfill & ⋮\hfill \\ \hfill {\omega }_{l+1}^{ABC}(\vartheta ,\zeta )=& {\mathit{N}}^{-1}\left(q(\mu ,\eta ,\lambda )\mathit{N}\left[\mathcal{L}\left({\omega }_l(\vartheta ,\zeta )\right)+A_l\right]\right),l=1,2,3,\dots \hfill \end{array}$$

(31)

Hence, the result of (14) in $NTD{M}_{ABC}$ manner by using (31) into (29) as

$$\begin{array}{c}\hfill {\omega }^{ABC}(\vartheta ,\zeta )={\omega }_0^{ABC}(\vartheta ,\zeta )+{\omega }_1^{ABC}(\vartheta ,\zeta )+{\omega }_2^{ABC}(\vartheta ,\zeta )+\cdots \end{array}$$

(32)

4. Convergence Analysis

The convergence for $NTD{M}_{CF}$ and $NTD{M}_{ABC}$ is stated as given.

Theorem 1. 

The $NTD{M}_{CF}$ result of (14) is unique if $0<({\vartheta }_1+{\vartheta }_2)(1-\mu +\mu \zeta )<1.$

Proof. 

If $\mathcal{G}=\left(C\right[J],||.|\left|\right)$ is Banach space with J represents compact interval having norm $\left|\right|\varphi \left(\zeta \right)\left|\right|={max}_{\zeta \in J}\left|\varphi \left(\zeta \right)\right|$,∀ continuous function on $J.$ Assume $I:\mathcal{G}\to \mathcal{G}$ is a nonlinear mapping; thus,

$${\omega }_{l+1}^C={\omega }_0^C+{\mathit{N}}^{-1}\left[p(\mu ,\eta ,\lambda )\mathit{N}[\mathcal{L}\left({\omega }_l(\vartheta ,\zeta )\right)+\mathbb{N}\left({\omega }_l(\vartheta ,\zeta )\right)]\right],l\ge 0.$$

Let $|\mathcal{L}\left(\omega \right)-\mathcal{L}\left({\omega }^{*}\right)|<{\vartheta }_1|\omega -{\omega }^{*}|$ and $|\mathbb{N}\left(\omega \right)-\mathbb{N}\left({\omega }^{*}\right)| <{\vartheta }_2|\omega -{\omega }^{*}|$, where $\omega :=\omega (\vartheta ,\zeta )$ and ${\omega }^{*}:={\omega }^{*}(\vartheta ,\zeta )$ are values of two separate functions and ${\vartheta }_1$,${\vartheta }_2$ are Lipschitz constants.

$$\begin{array}{cc}\hfill \left|\right|I\omega -I{\omega }^{*}\left|\right|& \le ma{x}_{t\in J}|{\mathit{N}}^{-1}[p(\mu ,\eta ,\lambda )\mathit{N}[\mathcal{L}\left(\omega \right)-\mathcal{L}\left({\omega }^{*}\right)]\hfill \\ \hfill & +p(\mu ,\eta ,\lambda )\mathit{N}[\mathbb{N}\left(\omega \right)-\mathbb{N}\left({\omega }^{*}\right)]\left|\right]\hfill \\ \hfill & \le ma{x}_{\zeta \in J}[{\vartheta }_1{\mathit{N}}^{-1}\left[p(\mu ,\eta ,\lambda )\mathit{N}\right[|\omega -{\omega }^{*}\left|\right]]\hfill \\ \hfill & +{\vartheta }_2{\mathit{N}}^{-1}\left[p(\mu ,\eta ,\lambda )\mathit{N}\right[|\omega -{\omega }^{*}\left|\right]\left]\right]\hfill \\ \hfill & \le ma{x}_{\zeta \in J}({\vartheta }_1+{\vartheta }_2)\left[{\mathit{N}}^{-1}\left[p(\mu ,\eta ,\lambda )\mathit{N}\right|\omega -{\omega }^{*}\left|\right]\right]\hfill \\ \hfill & \le ({\vartheta }_1+{\vartheta }_2)\left[{\mathit{N}}^{-1}\left[p(\mu ,\eta ,\lambda )\mathit{N}\right||\omega -{\omega }^{*}\left|\right|]\right]\hfill \\ \hfill & =({\vartheta }_1+{\vartheta }_2)(1-\mu +\mu \zeta )\left|\right|\omega -{\omega }^{*}\left|\right|\hfill \end{array}$$

(33)

I is contraction as $0<({\vartheta }_1+{\vartheta }_2)(1-\mu +\mu \zeta )<1$. Thus the outcome of (14) is unique. □

Theorem 2. 

The $NTD{M}_{ABC}$ outcome of (14) is unique when $0<({\vartheta }_1+{\vartheta }_2)\left(1-\mu +{\displaystyle \frac{\mu {\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\right)<1.$

Proof. 

The proof of this theorem is same as of Theorem 1. □

Theorem 3. 

The $NTD{M}_{CF}$ result of (14) is convergent.

Proof. 

Let ${\omega }_m={\sum }_{r=0}^m{\omega }_r(\phi ,\zeta )$. To show that ${\omega }_m$ is a Cauchy sequence in $\mathcal{G}$, assume

$$\begin{array}{cc}\hfill \left|\right|{\omega }_m-{\omega }_n\left|\right|& =ma{x}_{\zeta \in J}|\sum _{r=n+1}^m{\omega }_r|,n=1,2,3,\dots \hfill \\ \hfill & \le ma{x}_{\zeta \in J}\left|{\mathit{N}}^{-1}\left[p(\mu ,\eta ,\lambda )\mathit{N}\left[\sum _{r=n+1}^m(\mathcal{L}\left({\omega }_{r-1}\right)+\mathbb{N}\left({\omega }_{r-1}\right))\right]\right]\right|\hfill \\ \hfill & =ma{x}_{\zeta \in J}\left|{\mathit{N}}^{-1}\left[p(\mu ,\eta ,\lambda )\mathit{N}\left[\sum _{r=n+1}^{m-1}(\mathcal{L}\left({\omega }_r\right)+\mathbb{N}\left({\omega }_r\right))\right]\right]\right|\hfill \\ \hfill & \le ma{x}_{\zeta \in J}\left|{\mathit{N}}^{-1}\left[p(\mu ,\eta ,\lambda )\mathit{N}\left[(\mathcal{L}\left({\omega }_{m-1}\right)-\mathcal{L}\left({\omega }_{n-1}\right)+\mathbb{N}\left({\omega }_{m-1}\right)-\mathbb{N}\left({\omega }_{n-1}\right))\right]\right]\right|\hfill \\ \hfill & \le {\vartheta }_{1}ma{x}_{\zeta \in J}\left|{\mathit{N}}^{-1}\left[p(\mu ,\eta ,\lambda )\mathit{N}\left[(\mathcal{L}\left({\omega }_{m-1}\right)-\mathcal{L}\left({\omega }_{n-1}\right))\right]\right]\right|\hfill \\ \hfill & +{\vartheta }_{2}ma{x}_{\zeta \in J}\left|{\mathit{N}}^{-1}\left[p(\mu ,\eta ,\lambda )\mathit{N}\left[(\mathbb{N}\left({\omega }_{m-1}\right)-\mathbb{N}\left({\omega }_{n-1}\right))\right]\right]\right|\hfill \\ \hfill & =({\vartheta }_1+{\vartheta }_2)(1-\mu +\mu \zeta )\left|\right|{\omega }_{m-1}-{\omega }_{n-1}\left|\right|\hfill \end{array}$$

(34)

Let $m=n+1$, then

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

(35)

where $\vartheta =({\vartheta }_1+{\vartheta }_2)(1-\mu +\mu \zeta )$. So

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

(36)

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

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

(37)

Since $\left|\right|{\omega }_1\left|\right|<\infty ,\left|\right|{\omega }_m-{\omega }_n\left|\right|\to 0$ when $n\to \infty$. Thus, the series ${\omega }_m$ is convergent as ${\omega }_m$ is a Cauchy sequence in $\mathcal{G}$. □

Theorem 4. 

$NTD{M}_{ABC}$ solution of (14) is convergent.

Proof. 

The proof of this theorem is similar to that Theorem 3. □

5. Applications

We implemented NTDM to solve the nonlinear fractional KDV equations in this section.

Example 1. 

Assume fractional mKdV equation of the form [37]:

$$\begin{array}{c}\hfill D_{\zeta }^{\mu }\omega (\vartheta ,\zeta )+6{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )=0,0<\mu \le 1,\end{array}$$

(38)

with

$$\omega (\vartheta ,0)=\chi -{\displaystyle \frac{4\chi }{4{\chi }^2{\vartheta }^2+1}}.$$

(39)

On inserting NT to Equation (38), we get

$$\begin{array}{cc}\hfill & \mathit{N}\left[D_{\zeta }^{\mu }\omega (\vartheta ,\zeta )\right]=-\mathit{N}\left\{6{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )\right\}-\mathit{N}\left\{{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )\right\}.\hfill \end{array}$$

(40)

Now by using transform property

$$\begin{array}{cc}\hfill & \mathit{N}\left[\omega (\vartheta ,\zeta )\right]={\displaystyle \frac{\omega (\vartheta ,0)}{\lambda }}+\mathit{N}\left[-6{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )-{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )\right].\hfill \end{array}$$

(41)

On inserting inverse inverse NT, we have

$$\begin{array}{cc}\hfill & \omega (\vartheta ,\zeta )=\left[\chi -{\displaystyle \frac{4{\chi }^2}{4{\chi }^2{\vartheta }^2+1}}\right]-{\mathit{N}}^{-1}\left[{\left({\displaystyle \frac{\eta }{\lambda }}\right)}^{\mu }\mathit{N}\left\{6{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )\right\}\right].\hfill \end{array}$$

(42)

• Implementation of $NTD{M}_{CF}$

The series solution of $\omega (\vartheta ,\zeta )$ is represented as

$$\omega (\vartheta ,\zeta )=\sum _{l=0}^{\infty }{\omega }_l(\vartheta ,\zeta ).$$

(43)

The nonlinear term is indicated as ${\omega }^2{\omega }_{\vartheta }={\sum }_{l=0}^{\infty }{\mathcal{A}}_l,$ So Equation (42) is taken as

$$\begin{array}{cc}\hfill & \sum _{l=0}^{\infty }{\omega }_{l+1}(\vartheta ,\zeta )=\left[\chi -{\displaystyle \frac{4\chi }{4{\chi }^2{\vartheta }^2+1}}\right]\hfill \\ \hfill & -{\mathit{N}}^{-1}\left[{\displaystyle \frac{\eta (\lambda -\mu (\lambda -\eta \left)\right)}{{\lambda }^2}}\mathit{N}\left\{6\sum _{l=0}^{\infty }{\mathcal{A}}_l+{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )\right\}\right].\hfill \end{array}$$

(44)

Some Adomian polynomials are calculated as below:

$$\begin{array}{cc}\hfill & {\mathcal{A}}_0={\omega }_0^2{\left({\omega }_0\right)}_{\zeta },\hfill \\ \hfill & {\mathcal{A}}_1={\omega }_0^2{\left({\omega }_1\right)}_{\zeta }+2{\omega }_0{\omega }_1{\left({\omega }_0\right)}_{\zeta },\hfill \\ \hfill & ⋮\hfill \end{array}$$

(45)

By comparison of both sides of Equation (44), we get

$${\omega }_0(\vartheta ,\zeta )=\chi -{\displaystyle \frac{4\chi }{4{\chi }^2{\vartheta }^2+1}},$$

$$\begin{array}{cc}\hfill & {\omega }_1(\vartheta ,\zeta )=-{\displaystyle \frac{192{\chi }^6\vartheta (16{\vartheta }^4{\chi }^4-32{\vartheta }^2{\chi }^3+40{\vartheta }^2{\chi }^2+16{\chi }^2-8\chi -7)}{{(4{\chi }^2{\vartheta }^2+1)}^4}}\left(\mu (\zeta -1)+1\right),\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & {\omega }_2(\vartheta ,\zeta )=-{\displaystyle \frac{1}{{(4{\chi }^2{\vartheta }^2+1)}^7}}\left(1152{\chi }^8\right(3072{\chi }^{10}{\vartheta }^{10}-20480{\chi }^9{\vartheta }^8+23296{\chi }^8{\vartheta }^8+43008{\chi }^8{\vartheta }^6-114688{\chi }^7{\vartheta }^6+\hfill \\ \hfill & 72576{\chi }^6{\vartheta }^6-36864{\chi }^7{\vartheta }^4+124416{\chi }^6{\vartheta }^4+5632{\chi }^5{\vartheta }^4+11264{\chi }^6{\vartheta }^2-93088{\chi }^4{\vartheta }^4-8192{\chi }^5{\vartheta }^2-23680{\chi }^4{\vartheta }^2\hfill \\ \hfill & +7680{\chi }^3{\vartheta }^2-256{\chi }^4+12924{\chi }^2{\vartheta }^2+256{\chi }^3+288{\chi }^2-144\chi -145\left)\right)\left({(1-\mu )}^2+2\mu (1-\mu )\zeta +{\displaystyle \frac{{\mu }^2{\zeta }^2}{2}}\right).\hfill \end{array}$$

(47)

Lastly, the solution in $NTD{M}_{CF}$ manner is as follows:

$$\begin{array}{cc}\hfill \omega (\vartheta ,\zeta )& =\sum _{l=0}^{\infty }{\omega }_l(\vartheta ,\zeta )={\omega }_0(\vartheta ,\zeta )+{\omega }_1(\vartheta ,\zeta )+{\omega }_2(\vartheta ,\zeta )+\dots ,\hfill \\ \hfill \omega (\vartheta ,\zeta )& =\chi -{\displaystyle \frac{4\chi }{4{\chi }^2{\vartheta }^2+1}}-{\displaystyle \frac{192{\chi }^6\vartheta (16{\vartheta }^4{\chi }^4-32{\vartheta }^2{\chi }^3+40{\vartheta }^2{\chi }^2+16{\chi }^2-8\chi -7)}{{(4{\chi }^2{\vartheta }^2+1)}^4}}\left(\mu (\zeta -1)+1\right)-\hfill \\ \hfill & {\displaystyle \frac{1}{{(4{\chi }^2{\vartheta }^2+1)}^7}}\left(1152{\chi }^8\right(3072{\chi }^{10}{\vartheta }^{10}-20480{\chi }^9{\vartheta }^8+23296{\chi }^8{\vartheta }^8+43008{\chi }^8{\vartheta }^6-114688{\chi }^7{\vartheta }^6+\hfill \\ \hfill & 72576{\chi }^6{\vartheta }^6-36864{\chi }^7{\vartheta }^4+124416{\chi }^6{\vartheta }^4+5632{\chi }^5{\vartheta }^4+11264{\chi }^6{\vartheta }^2-93088{\chi }^4{\vartheta }^4-8192{\chi }^5{\vartheta }^2-\hfill \\ \hfill & 23680{\chi }^4{\vartheta }^2+7680{\chi }^3{\vartheta }^2-256{\chi }^4+12924{\chi }^2{\vartheta }^2+256{\chi }^3+288{\chi }^2-144\chi -145\left)\right)({(1-\mu )}^2+\hfill \\ \hfill & 2\mu (1-\mu )\zeta +{\displaystyle \frac{{\mu }^2{\zeta }^2}{2}})+\cdots .\hfill \end{array}$$

(48)

• Implementation of $NTD{M}_{ABC}$

The series solution of $\omega (\vartheta ,\zeta )$ is represented as

$$\omega (\vartheta ,\zeta )=\sum _{l=0}^{\infty }{\omega }_l(\vartheta ,\zeta ).$$

(49)

The nonlinear term is indicated as ${\omega }^2{\omega }_{\vartheta }={\sum }_{l=0}^{\infty }{\mathcal{A}}_l,$ So Equation (42) is taken as

$$\begin{array}{cc}\hfill \sum _{l=0}^{\infty }{\omega }_{l+1}(\vartheta ,\zeta )& =\left[\chi -{\displaystyle \frac{4\chi }{4{\chi }^2{\vartheta }^2+1}}\right]\hfill \\ \hfill & -{\mathit{N}}^{-1}\left[{\displaystyle \frac{{\eta }^{\mu }({\lambda }^{\mu }+\mu ({\eta }^{\mu }-{\lambda }^{\mu }))}{{\lambda }^{2\mu }}}\mathit{N}\left\{6\sum _{l=0}^{\infty }{\mathcal{A}}_l+{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )\right\}\right].\hfill \end{array}$$

(50)

The Adomian polynomials are calculated as stated in Equation (45). By comparing both sides of Equation (50), we obtain

$${\omega }_0(\vartheta ,\zeta )=\chi -{\displaystyle \frac{4\chi }{4{\chi }^2{\vartheta }^2+1}},$$

$$\begin{array}{cc}\hfill & {\omega }_1(\vartheta ,\zeta )=-{\displaystyle \frac{192{\chi }^6\vartheta (16{\vartheta }^4{\chi }^4-32{\vartheta }^2{\chi }^3+40{\vartheta }^2{\chi }^2+16{\chi }^2-8\chi -7)}{{(4{\chi }^2{\vartheta }^2+1)}^4}}\left(1-\mu +{\displaystyle \frac{\mu {\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\right),\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill & {\omega }_2(\vartheta ,\zeta )=-{\displaystyle \frac{1}{{(4{\chi }^2{\vartheta }^2+1)}^7}}\left(1152{\chi }^8\right(3072{\chi }^{10}{\vartheta }^{10}-20480{\chi }^9{\vartheta }^8+23296{\chi }^8{\vartheta }^8+43008{\chi }^8{\vartheta }^6-114688{\chi }^7{\vartheta }^6\hfill \\ \hfill & +72576{\chi }^6{\vartheta }^6-36864{\chi }^7{\vartheta }^4+124416{\chi }^6{\vartheta }^4+5632{\chi }^5{\vartheta }^4+11264{\chi }^6{\vartheta }^2-93088{\chi }^4{\vartheta }^4-8192{\chi }^5{\vartheta }^2-23680{\chi }^4{\vartheta }^2\hfill \\ \hfill & +7680{\chi }^3{\vartheta }^2-256{\chi }^4+12924{\chi }^2{\vartheta }^2+256{\chi }^3+288{\chi }^2-144\chi -145\left)\right)[{\displaystyle \frac{{\mu }^2{\zeta }^{2\mu }}{\mathrm{\Gamma }(2\mu +1)}}+2\mu (1-\mu ){\displaystyle \frac{{\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\hfill \\ \hfill & +{(1-\mu )}^2].\hfill \end{array}$$

(52)

Lastly, the solution in $NTD{M}_{ABC}$ manner is as follows:

$$\begin{array}{cc}\hfill \omega (\vartheta ,\zeta )& =\sum _{l=0}^{\infty }{\omega }_l(\vartheta ,\zeta )={\omega }_0(\vartheta ,\zeta )+{\omega }_1(\vartheta ,\zeta )+{\omega }_2(\vartheta ,\zeta )+\dots ,\hfill \\ \hfill \omega (\vartheta ,\zeta )& =\chi -{\displaystyle \frac{4\chi }{4{\chi }^2{\vartheta }^2+1}}-{\displaystyle \frac{192{\chi }^6\vartheta (16{\vartheta }^4{\chi }^4-32{\vartheta }^2{\chi }^3+40{\vartheta }^2{\chi }^2+16{\chi }^2-8\chi -7)}{{(4{\chi }^2{\vartheta }^2+1)}^4}}\left(1-\mu +{\displaystyle \frac{\mu {\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{{(4{\chi }^2{\vartheta }^2+1)}^7}}\left(1152{\chi }^8\right(3072{\chi }^{10}{\vartheta }^{10}-20480{\chi }^9{\vartheta }^8+23296{\chi }^8{\vartheta }^8+43008{\chi }^8{\vartheta }^6-114688{\chi }^7{\vartheta }^6\hfill \\ \hfill & +72576{\chi }^6{\vartheta }^6-36864{\chi }^7{\vartheta }^4+124416{\chi }^6{\vartheta }^4+5632{\chi }^5{\vartheta }^4+11264{\chi }^6{\vartheta }^2-93088{\chi }^4{\vartheta }^4-8192{\chi }^5{\vartheta }^2-\hfill \\ \hfill & 23680{\chi }^4{\vartheta }^2+7680{\chi }^3{\vartheta }^2-256{\chi }^4+12924{\chi }^2{\vartheta }^2+256{\chi }^3+288{\chi }^2-144\chi -145\left)\right)[{\displaystyle \frac{{\mu }^2{\zeta }^{2\mu }}{\mathrm{\Gamma }(2\mu +1)}}+\hfill \\ \hfill & 2\mu (1-\mu ){\displaystyle \frac{{\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}+{(1-\mu )}^2]+\dots .\hfill \end{array}$$

(53)

By taking $\mu =1$, we get the exact solution as

$$\omega (\vartheta ,\zeta )=\chi -{\displaystyle \frac{4\chi }{4{\chi }^2{({\vartheta }^2-6{\chi }^2\zeta )}^2+1}}.$$

(54)

With the use of 2D and 3D simulations, the governing model’s physical interpretation is shown graphically. Using figures and tables, we compared the solutions from the suggested method with the precise solutions. The 3D simulations of the precise and suggested approach solutions for $\mu =1$ are shown in Figure 1. The 3D simulations of the suggested technique solutions for $\mu =0.75$ and $\mu =0.50$ are shown in Figure 2 and Figure 3. Similarly, the 2D simulations of the precise solution and the NTDM solution in terms of both operators are shown in Figure 4 and Figure 5. Figure 1, Figure 2, Figure 3, Figure 4 and Figure 5 demonstrate that fractional values of $\mu$ will result in smooth solution curves, whereas $\mu \to 1$ will result in a solution curve that approaches the integer-order solution. The accurate and derived results for various $\mu$ values are shown in

Table 1. Analysis of the accurate and proposed method solution.

$\mathit{\zeta }$	$\mathit{\vartheta }$	${\mathit{NTDM}}_{\mathit{ABC}}$ at	${\mathit{NTDM}}_{\mathit{CF}}$ at	${\mathit{NTDM}}_{\mathit{ABC}}$ at	${\mathit{NTDM}}_{\mathit{CF}}$ at	$\mathit{Exact}$ at
		$\mathit{\mu }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{99}$	$\mathit{\mu }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{99}$	$\mathit{\mu }\mathbf{=}\mathbf{1}$	$\mathit{\mu }\mathbf{=}\mathbf{1}$	$\mathit{\mu }\mathbf{=}\mathbf{1}$
	0.0	0.2999999582	0.2999999600	0.2999999942	0.2999999942	0.2999999944
	0.2	0.2993687935	0.2993686000	0.2993648444	0.2993648444	0.2999751640
	0.4	0.2974717159	0.2974713307	0.2974638569	0.2974638569	0.2995938792
0.01	0.6	0.2943445780	0.2943440086	0.2943329566	0.2943329566	0.2979439296
	0.8	0.2900453693	0.2900446259	0.2900301985	0.2900301985	0.2935639333
	1	0.2846515829	0.2846506787	0.2846331316	0.2846331316	0.2846331317
	0.0	0.2999998618	0.2999998692	0.2999999482	0.2999999482	0.2999999480
	0.2	0.2993765368	0.2993760890	0.2993724543	0.2993724543	0.2999766534
	0.4	0.2974871516	0.2974862570	0.2974789778	0.2974789778	0.2995999645
0.03	0.6	0.2943674165	0.2943660919	0.2943553049	0.2943553049	0.2979575664
	0.8	0.2900751919	0.2900734617	0.2900593645	0.2900593645	0.2935876813
	1	0.2846878610	0.2846857557	0.2846685984	0.2846685984	0.2846685983
	0.0	0.2999997178	0.2999997333	0.2999998560	0.2999998560	0.2999998560
	0.2	0.2993841762	0.2993835333	0.2993800185	0.2993800185	0.2999780972
	0.4	0.2975024290	0.2975011399	0.2974940544	0.2974940544	0.2996060045
0.05	0.6	0.2943900451	0.2943881339	0.2943776109	0.2943776109	0.2979711590
	0.8	0.2901047569	0.2901022589	0.2900884912	0.2900884912	0.2936113880
	1	0.2847238380	0.2847207973	0.2847040293	0.2847040293	0.2847040293

. The figures and tables further show that both derivative solutions converge to the exact result as the fractional order gets closer to 1. Additionally, the absolute errors of the suggested method solution and the outcomes obtained by ITM and RPSTM are compared in

Table 2. The comparison of our method solution with the ITM [37] and the RPSTM [37] at $\mu =1$.

$\mathit{\zeta }$	$\mathit{ITM}$ Error	$\mathit{RPSTM}$ Error	${\mathit{NTDM}}_{\mathit{ABC}}$ Error	${\mathit{NTDM}}_{\mathit{CF}}$ Error
0.1	2.0016544 × ${10}^{-5}$	1.9744012 × ${10}^{-5}$	2.3990000 × ${10}^{-7}$	2.3990000 × ${10}^{-7}$
0.2	7.7213786 × ${10}^{-5}$	7.6672159 × ${10}^{-5}$	1.7980000 × ${10}^{-7}$	1.7980000 × ${10}^{-7}$
0.3	1.6432297 × ${10}^{-4}$	1.6351377 × ${10}^{-4}$	1.1970000 × ${10}^{-7}$	1.1970000 × ${10}^{-7}$
0.4	2.6923314 × ${10}^{-4}$	2.6815863 × ${10}^{-4}$	5.9800000 × ${10}^{-8}$	5.9800000 × ${10}^{-8}$
0.5	3.7501932 × ${10}^{-4}$	3.7368249 × ${10}^{-4}$	0.0000000000000	0.0000000000000
0.6	4.6000623 × ${10}^{-4}$	4.5841080 × ${10}^{-4}$	5.9500000 × ${10}^{-8}$	5.9500000 × ${10}^{-8}$
0.7	4.9789793 × ${10}^{-4}$	4.9604829 × ${10}^{-4}$	1.1880000 × ${10}^{-7}$	1.1880000 × ${10}^{-7}$
0.8	4.5800758 × ${10}^{-4}$	4.5590880 × ${10}^{-4}$	1.7770000 × ${10}^{-7}$	1.7770000 × ${10}^{-7}$
0.9	3.0562718 × ${10}^{-4}$	3.0328498 × ${10}^{-4}$	2.3620000 × ${10}^{-7}$	2.3620000 × ${10}^{-7}$
1.0	2.5792975 × ${10}^{-6}$	2.9475186 × ${10}^{-7}$	2.9410000 × ${10}^{-7}$	2.9410000 × ${10}^{-7}$

. This implies that the solutions generated by NTDM are better suited than those generated by ITM and RPSTM. Interestingly, the fractional-order situation gives wave profiles a damping-like quality that provides further modeling flexibility for physical systems with memory effects and anomalous diffusion.

Example 2. 

Assume fractional KdV Burger’s equation of the form [37]:

$$\begin{array}{c}\hfill D_{\zeta }^{\mu }\omega (\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta }(\vartheta ,\zeta )+{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )=0,0<\mu \le 1,\end{array}$$

(55)

with

$$\omega (\vartheta ,0)={\displaystyle \frac{\sqrt{6}(exp\left(\vartheta \right)-exp(-\vartheta ))}{(exp\left(\vartheta \right)+exp(-\vartheta ))}}+{\displaystyle \frac{\sqrt{6}}{6}}.$$

(56)

On inserting NT to Equation (55), we get

$$\begin{array}{cc}\hfill & \mathit{N}\left[D_{\zeta }^{\mu }\omega (\vartheta ,\zeta )\right]=-\mathit{N}\left\{{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )\right\}-\mathit{N}\left\{{\omega }_{\vartheta \vartheta }(\vartheta ,\zeta )\right\}-\mathit{N}\left\{{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )\right\}.\hfill \end{array}$$

(57)

Now by using transform property

$$\begin{array}{cc}\hfill & \mathit{N}\left[\omega (\vartheta ,\zeta )\right]={\displaystyle \frac{\omega (\vartheta ,0)}{\lambda }}+\mathit{N}\left[-{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )-{\omega }_{\vartheta \vartheta }(\vartheta ,\zeta )-{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )\right].\hfill \end{array}$$

(58)

On inserting inverse inverse NT, we have

$$\begin{array}{cc}\hfill & \omega (\vartheta ,\zeta )=\left[{\displaystyle \frac{\sqrt{6}(exp\left(\vartheta \right)-exp(-\vartheta ))}{(exp\left(\vartheta \right)+exp(-\vartheta ))}}+{\displaystyle \frac{\sqrt{6}}{6}}\right]-{\mathit{N}}^{-1}\left[{\left({\displaystyle \frac{\eta }{\lambda }}\right)}^{\mu }\mathit{N}\right\{{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta }(\vartheta ,\zeta )+\hfill \\ \hfill & {\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )\left\}\right].\hfill \end{array}$$

(59)

• Implementation of $NTD{M}_{CF}$

The series solution of $\omega (\vartheta ,\zeta )$ is represented as

$$\omega (\vartheta ,\zeta )=\sum _{l=0}^{\infty }{\omega }_l(\vartheta ,\zeta ).$$

(60)

The nonlinear term is indicated as ${\omega }^2{\omega }_{\vartheta }={\sum }_{l=0}^{\infty }{\mathcal{A}}_l,$ So Equation (59) is taken as

$$\begin{array}{cc}\hfill & \sum _{l=0}^{\infty }{\omega }_{l+1}(\vartheta ,\zeta )=\left[{\displaystyle \frac{\sqrt{6}(exp\left(\vartheta \right)-exp(-\vartheta ))}{(exp\left(\vartheta \right)+exp(-\vartheta ))}}+{\displaystyle \frac{\sqrt{6}}{6}}\right]\hfill \\ \hfill & -{\mathit{N}}^{-1}\left[{\displaystyle \frac{\eta (\lambda -\mu (\lambda -\eta \left)\right)}{{\lambda }^2}}\mathit{N}\left\{{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta }(\vartheta ,\zeta )+\sum _{l=0}^{\infty }{\mathcal{A}}_l\right\}\right].\hfill \end{array}$$

(61)

The Adomian polynomials are calculated as stated in Equation (45). By comparison of both sides of Equation (61), we get

$${\omega }_0(\vartheta ,\zeta )={\displaystyle \frac{\sqrt{6}(exp\left(\vartheta \right)-exp(-\vartheta ))}{(exp\left(\vartheta \right)+exp(-\vartheta ))}}+{\displaystyle \frac{\sqrt{6}}{6}},$$

$$\begin{array}{cc}\hfill & {\omega }_1(\vartheta ,\zeta )=-{\displaystyle \frac{2}{3}}{\displaystyle \frac{\sqrt{6}(61exp\left(4\vartheta \right)-166exp\left(2\vartheta \right)+61)exp\left(2\vartheta \right)}{{(exp\left(2\vartheta \right)+1)}^4}}\left(\mu (\zeta -1)+1\right),\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill & {\omega }_2(\vartheta ,\zeta )=-{\displaystyle \frac{2}{9{(exp\left(2\vartheta \right)+1)}^7}}\left(\right(3721exp\left(10\vartheta \right)-93669exp\left(8\vartheta \right)+421010exp\left(6\vartheta \right)-448658exp\left(4\vartheta \right)\hfill \\ \hfill & +107493exp\left(2\vartheta \right)-3721)exp\left(2\vartheta \right)\sqrt{6})\left({(1-\mu )}^2+2\mu (1-\mu )\zeta +{\displaystyle \frac{{\mu }^2{\zeta }^2}{2}}\right).\hfill \end{array}$$

(63)

Lastly, the solution in $NTD{M}_{CF}$ manner is as follows:

$$\begin{array}{cc}\hfill \omega (\vartheta ,\zeta )& =\sum _{l=0}^{\infty }{\omega }_l(\vartheta ,\zeta )={\omega }_0(\vartheta ,\zeta )+{\omega }_1(\vartheta ,\zeta )+{\omega }_2(\vartheta ,\zeta )+\dots ,\hfill \\ \hfill \omega (\vartheta ,\zeta )& ={\displaystyle \frac{\sqrt{6}(exp\left(\vartheta \right)-exp(-\vartheta ))}{(exp\left(\vartheta \right)+exp(-\vartheta ))}}+{\displaystyle \frac{\sqrt{6}}{6}}-{\displaystyle \frac{2}{3}}{\displaystyle \frac{\sqrt{6}(61exp\left(4\vartheta \right)-166exp\left(2\vartheta \right)+61)exp\left(2\vartheta \right)}{{(exp\left(2\vartheta \right)+1)}^4}}\hfill \\ \hfill & \left(\mu (\zeta -1)+1\right)-{\displaystyle \frac{2}{9{(exp\left(2\vartheta \right)+1)}^7}}\left(\right(3721exp\left(10\vartheta \right)-93669exp\left(8\vartheta \right)+421010exp\left(6\vartheta \right)-\hfill \\ \hfill & 448658exp\left(4\vartheta \right)+107493exp\left(2\vartheta \right)-3721)exp\left(2\vartheta \right)\sqrt{6})\left({(1-\mu )}^2+2\mu (1-\mu )\zeta +{\displaystyle \frac{{\mu }^2{\zeta }^2}{2}}\right)\hfill \\ \hfill & +\cdots .\hfill \end{array}$$

(64)

• Implementation of $NTD{M}_{ABC}$

The series solution of $\omega (\vartheta ,\zeta )$ is represented as

$$\omega (\vartheta ,\zeta )=\sum _{l=0}^{\infty }{\omega }_l(\vartheta ,\zeta ).$$

(65)

The nonlinear term is indicated as ${\omega }^2{\omega }_{\vartheta }={\sum }_{l=0}^{\infty }{\mathcal{A}}_l,$ So Equation (59) is taken as

$$\begin{array}{cc}\hfill \sum _{l=0}^{\infty }{\omega }_{l+1}(\vartheta ,\zeta )& =\left[{\displaystyle \frac{\sqrt{6}(exp\left(\vartheta \right)-exp(-\vartheta ))}{(exp\left(\vartheta \right)+exp(-\vartheta ))}}+{\displaystyle \frac{\sqrt{6}}{6}}\right]\hfill \\ \hfill & -{\mathit{N}}^{-1}\left[{\displaystyle \frac{{\eta }^{\mu }({\lambda }^{\mu }+\mu ({\eta }^{\mu }-{\lambda }^{\mu }))}{{\lambda }^{2\mu }}}\mathit{N}\left\{{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta }(\vartheta ,\zeta )+\sum _{l=0}^{\infty }{\mathcal{A}}_l\right\}\right].\hfill \end{array}$$

(66)

The Adomian polynomials are calculated as stated in Equation (45). By comparison of both sides of Equation (66) as

$${\omega }_0(\vartheta ,\zeta )={\displaystyle \frac{\sqrt{6}(exp\left(\vartheta \right)-exp(-\vartheta ))}{(exp\left(\vartheta \right)+exp(-\vartheta ))}}+{\displaystyle \frac{\sqrt{6}}{6}},$$

$$\begin{array}{cc}\hfill & {\omega }_1(\vartheta ,\zeta )=-{\displaystyle \frac{2}{3}}{\displaystyle \frac{\sqrt{6}(61exp\left(4\vartheta \right)-166exp\left(2\vartheta \right)+61)exp\left(2\vartheta \right)}{{(exp\left(2\vartheta \right)+1)}^4}}\left(1-\mu +{\displaystyle \frac{\mu {\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\right),\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill & {\omega }_2(\vartheta ,\zeta )=-{\displaystyle \frac{2}{9{(exp\left(2\vartheta \right)+1)}^7}}\left(\right(3721exp\left(10\vartheta \right)-93669exp\left(8\vartheta \right)+421010exp\left(6\vartheta \right)-448658exp\left(4\vartheta \right)\hfill \\ \hfill & +107493exp\left(2\vartheta \right)-3721)exp\left(2\vartheta \right)\sqrt{6})\left[{\displaystyle \frac{{\mu }^2{\zeta }^{2\mu }}{\mathrm{\Gamma }(2\mu +1)}}+2\mu (1-\mu ){\displaystyle \frac{{\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}+{(1-\mu )}^2\right].\hfill \end{array}$$

(68)

Lastly, the solution in $NTD{M}_{ABC}$ manner is as follows:

$$\begin{array}{cc}\hfill \omega (\vartheta ,\zeta )& =\sum _{l=0}^{\infty }{\omega }_l(\vartheta ,\zeta )={\omega }_0(\vartheta ,\zeta )+{\omega }_1(\vartheta ,\zeta )+{\omega }_2(\vartheta ,\zeta )+\dots ,\hfill \\ \hfill \omega (\vartheta ,\zeta )& ={\displaystyle \frac{\sqrt{6}(exp\left(\vartheta \right)-exp(-\vartheta ))}{(exp\left(\vartheta \right)+exp(-\vartheta ))}}+{\displaystyle \frac{\sqrt{6}}{6}}-{\displaystyle \frac{2}{3}}{\displaystyle \frac{\sqrt{6}(61exp\left(4\vartheta \right)-166exp\left(2\vartheta \right)+61)exp\left(2\vartheta \right)}{{(exp\left(2\vartheta \right)+1)}^4}}\hfill \\ \hfill & \left(1-\mu +{\displaystyle \frac{\mu {\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\right)-{\displaystyle \frac{2}{9{(exp\left(2\vartheta \right)+1)}^7}}\left(\right(3721exp\left(10\vartheta \right)-93669exp\left(8\vartheta \right)+421010exp\left(6\vartheta \right)\hfill \\ \hfill & -448658exp\left(4\vartheta \right)+107493exp\left(2\vartheta \right)-3721)exp\left(2\vartheta \right)\sqrt{6})[{\displaystyle \frac{{\mu }^2{\zeta }^{2\mu }}{\mathrm{\Gamma }(2\mu +1)}}+2\mu (1-\mu ){\displaystyle \frac{{\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\hfill \\ \hfill & +{(1-\mu )}^2]+\cdots .\hfill \end{array}$$

(69)

By taking $\mu =1$, we get the exact solution as

$$\omega (\vartheta ,\zeta )={\displaystyle \frac{\sqrt{6}(exp(\vartheta -\frac{13\zeta }{6})-exp(\frac{13\zeta }{6}-\vartheta ))}{(exp(\vartheta +\frac{13\zeta }{6})+exp(\frac{13\zeta }{6}-\vartheta ))}}+{\displaystyle \frac{\sqrt{6}}{6}}.$$

(70)

The 3D simulations of the precise and suggested approach solutions for $\mu =1$ are shown in Figure 6. The 3D simulations of the suggested technique solutions for $\mu =0.75$ and $\mu =0.50$ are shown in Figure 7 and Figure 8. Similarly, the 2D simulations of the precise solution and the NTDM solution in terms of both operators are shown in Figure 9 and Figure 10. The accurate and derived results for various $\mu$ values are shown in

Table 3. Analysis of the accurate and proposed method solution.

$\mathit{\zeta }$	$\mathit{\vartheta }$	${\mathit{NTDM}}_{\mathit{ABC}}$ at	${\mathit{NTDM}}_{\mathit{CF}}$ at	${\mathit{NTDM}}_{\mathit{ABC}}$ at	${\mathit{NTDM}}_{\mathit{CF}}$ at	$\mathit{Exact}$ at
		$\mathit{\mu }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{99}$	$\mathit{\mu }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{99}$	$\mathit{\mu }\mathbf{=}\mathbf{1}$	$\mathit{\mu }\mathbf{=}\mathbf{1}$	$\mathit{\mu }\mathbf{=}\mathbf{1}$
	1.0	2.2239746050	2.2242598810	2.2685262870	2.2685262870	2.2715327840
	1.5	2.5861161990	2.5863840770	2.6218588960	2.6218588960	2.6244387180
	2.0	2.7494847070	2.7496359850	2.7680018420	2.7680018420	2.7692482160
0.001	2.5	2.8165262730	2.8165918310	2.8243030110	2.8243030110	2.8248084610
	3.0	2.8423733970	2.8423990280	2.8453795530	2.8453795530	2.8455722300
	3.5	2.8520572810	2.8520669240	2.8531836370	2.8531836370	2.8532554480
	1.0	2.2161834750	2.2168770940	2.2582590810	2.2582590810	2.2670454940
	1.5	2.5786556070	2.5793353450	2.6147747000	2.6147747000	2.6225055120
	2.0	2.7452319590	2.7456236030	2.7647011300	2.7647011300	2.7684920020
0.003	2.5	2.8146774520	2.8148483290	2.8229766080	2.8229766080	2.8245237740
	3.0	2.8416497540	2.8417167200	2.8448753670	2.8448753670	2.8454665980
	3.5	2.8517849100	2.8518101270	2.8529959030	2.8529959030	2.8532164650
	1.0	2.2087573820	2.2097699750	2.2482731410	2.2482731410	2.2625285960
	1.5	2.5712452170	2.5722831070	2.6076869250	2.6076869250	2.6205571020
	2.0	2.7409274730	2.7415374230	2.7613251260	2.7613251260	2.7677294490
0.005	2.5	2.8127943140	2.8130621780	2.8216066890	2.8216066890	2.8242366420
	3.0	2.8409110490	2.8410162640	2.8443526580	2.8443526580	2.8453600550
	3.5	2.8515066450	2.8515462970	2.8528009970	2.8528009970	2.8531771460

. A smoother profile that tends to achieve the classical solution at $\mu =1$ is generated by the fractional orders, which modify the solution amplitude and wave shape with lower values of $\mu$. The figures and tables further show that both derivative solutions converge to the exact result as the fractional order gets closer to 1. Additionally, the absolute errors of the suggested method solution and the outcomes obtained by RPSTM are compared in

Table 4. The comparison of our method solution with the RPSTM at $\mu =1$.

$\mathit{\zeta }$	$\mathit{RPSTM}$ Error	${\mathit{NTDM}}_{\mathit{ABC}}$ Error	${\mathit{NTDM}}_{\mathit{CF}}$ Error
1.0	2.6567180 $\times {10}^{-2}$	2.6567179$\times {10}^{-2}$	2.6567179$\times {10}^{-2}$
1.6	2.3155500$\times {10}^{-2}$	2.3155499$\times {10}^{-2}$	2.3155499$\times {10}^{-2}$
2.2	9.4697320$\times {10}^{-3}$	9.4697322$\times {10}^{-3}$	9.4697322$\times {10}^{-3}$
2.8	3.1283675$\times {10}^{-3}$	3.1283658$\times {10}^{-3}$	3.1283658$\times {10}^{-3}$
3.4	9.6889137$\times {10}^{-4}$	9.6889321$\times {10}^{-4}$	9.6889321$\times {10}^{-4}$
4.0	2.9428743$\times {10}^{-4}$	2.9428659$\times {10}^{-4}$	2.9428659$\times {10}^{-4}$
4.6	8.8862376$\times {10}^{-5}$	8.8862588$\times {10}^{-5}$	8.8862588$\times {10}^{-5}$
5.2	2.6785252$\times {10}^{-5}$	2.6785170$\times {10}^{-5}$	2.6785170$\times {10}^{-5}$
5.8	8.0694165$\times {10}^{-6}$	8.0710687$\times {10}^{-6}$	8.0710687$\times {10}^{-6}$

. The fractional-order analysis generally shows that the dynamical representation of nonlinear PDEs can greatly benefit from the inclusion of the fractional order parameter $\mu$.

Example 3. 

Assume fractional mKdV equation of the form [38]:

$$\begin{array}{c}\hfill D_{\zeta }^{\mu }\omega (\vartheta ,\zeta )+6{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )=0,0<\mu \le 1,\end{array}$$

(71)

with

$$\omega (\vartheta ,0)=-{\displaystyle \frac{2\psi exp\left(\psi \vartheta \right)}{exp\left(2\psi \vartheta \right)+1}}.$$

(72)

On inserting NT to Equation (71), we get

$$\begin{array}{cc}\hfill & \mathit{N}\left[D_{\zeta }^{\mu }\omega (\vartheta ,\zeta )\right]=-\mathit{N}\left\{6{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )\right\}-\mathit{N}\left\{{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )\right\}.\hfill \end{array}$$

(73)

Now by using transform property

$$\begin{array}{cc}\hfill & \mathit{N}\left[\omega (\vartheta ,\zeta )\right]={\displaystyle \frac{\omega (\vartheta ,0)}{\lambda }}+\mathit{N}\left[-6{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )-{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )\right].\hfill \end{array}$$

(74)

On inserting inverse inverse NT, we have

$$\begin{array}{cc}\hfill & \omega (\vartheta ,\zeta )=\left[-{\displaystyle \frac{2\psi exp\left(\psi \vartheta \right)}{exp\left(2\psi \vartheta \right)+1}}\right]-{\mathit{N}}^{-1}\left[{\left({\displaystyle \frac{\eta }{\lambda }}\right)}^{\mu }\mathit{N}\left\{6{\omega }^2(\vartheta ,\zeta ){\omega }_{\vartheta }(\vartheta ,\zeta )+{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )\right\}\right].\hfill \end{array}$$

(75)

• Implementation of $NTD{M}_{CF}$

The series solution of $\omega (\vartheta ,\zeta )$ is represented as

$$\omega (\vartheta ,\zeta )=\sum _{l=0}^{\infty }{\omega }_l(\vartheta ,\zeta ).$$

(76)

The nonlinear term is indicated as ${\omega }^2{\omega }_{\vartheta }={\sum }_{l=0}^{\infty }{\mathcal{A}}_l,$ So Equation (75) is taken as

$$\begin{array}{cc}\hfill & \sum _{l=0}^{\infty }{\omega }_{l+1}(\vartheta ,\zeta )=\left[-{\displaystyle \frac{2\psi exp\left(\psi \vartheta \right)}{exp\left(2\psi \vartheta \right)+1}}\right]\hfill \\ \hfill & -{\mathit{N}}^{-1}\left[{\displaystyle \frac{\eta (\lambda -\mu (\lambda -\eta \left)\right)}{{\lambda }^2}}\mathit{N}\left\{6\sum _{l=0}^{\infty }{\mathcal{A}}_l+{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )\right\}\right].\hfill \end{array}$$

(77)

The Adomian polynomials are calculated as stated in Equation (45). By comparing both sides of Equation (77), we get

$${\omega }_0(\vartheta ,\zeta )=-{\displaystyle \frac{2\psi exp\left(\psi \vartheta \right)}{exp\left(2\psi \vartheta \right)+1}},$$

$$\begin{array}{cc}\hfill & {\omega }_1(\vartheta ,\zeta )=-{\displaystyle \frac{2{\psi }^{4}exp\left(\psi \vartheta \right)(exp\left(2\psi \vartheta \right)-1)}{{(exp\left(2\psi \vartheta \right)+1)}^2}}\left(\mu (\zeta -1)+1\right),\hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill & {\omega }_2(\vartheta ,\zeta )=-{\displaystyle \frac{2{\psi }^{7}exp\left(\psi \vartheta \right)(exp\left(4\psi \vartheta \right)-6exp\left(2\psi \vartheta \right)-1)}{{(exp\left(2\psi \vartheta \right)+1)}^3}}\left({(1-\mu )}^2+2\mu (1-\mu )\zeta +{\displaystyle \frac{{\mu }^2{\zeta }^2}{2}}\right).\hfill \end{array}$$

(79)

Lastly, the solution in $NTD{M}_{CF}$ manner is as follows:

$$\begin{array}{cc}\hfill \omega (\vartheta ,\zeta )& =\sum _{l=0}^{\infty }{\omega }_l(\vartheta ,\zeta )={\omega }_0(\vartheta ,\zeta )+{\omega }_1(\vartheta ,\zeta )+{\omega }_2(\vartheta ,\zeta )+\dots ,\hfill \\ \hfill \omega (\vartheta ,\zeta )& =-{\displaystyle \frac{2\psi exp\left(\psi \vartheta \right)}{exp\left(2\psi \vartheta \right)+1}}-{\displaystyle \frac{2{\psi }^{4}exp\left(\psi \vartheta \right)(exp\left(2\psi \vartheta \right)-1)}{{(exp\left(2\psi \vartheta \right)+1)}^2}}\left(\mu (\zeta -1)+1\right)\hfill \\ \hfill & -{\displaystyle \frac{2{\psi }^{7}exp\left(\psi \vartheta \right)(exp\left(4\psi \vartheta \right)-6exp\left(2\psi \vartheta \right)-1)}{{(exp\left(2\psi \vartheta \right)+1)}^3}}\left({(1-\mu )}^2+2\mu (1-\mu )\zeta +{\displaystyle \frac{{\mu }^2{\zeta }^2}{2}}\right)+\cdots .\hfill \end{array}$$

(80)

• Implementation of $NTD{M}_{ABC}$

The series solution of $\omega (\vartheta ,\zeta )$ is represented as

$$\omega (\vartheta ,\zeta )=\sum _{l=0}^{\infty }{\omega }_l(\vartheta ,\zeta ).$$

(81)

The nonlinear term is indicated as ${\omega }^2{\omega }_{\vartheta }={\sum }_{l=0}^{\infty }{\mathcal{A}}_l,$ So Equation (75) is taken as

$$\begin{array}{cc}\hfill \sum _{l=0}^{\infty }{\omega }_{l+1}(\vartheta ,\zeta )& =\left[-{\displaystyle \frac{2\psi exp\left(\psi \vartheta \right)}{exp\left(2\psi \vartheta \right)+1}}\right]\hfill \\ \hfill & -{\mathit{N}}^{-1}\left[{\displaystyle \frac{{\eta }^{\mu }({\lambda }^{\mu }-\mu ({\lambda }^{\mu }-{\eta }^{\mu }))}{{\lambda }^{2\mu }}}\mathit{N}\left\{6\sum _{l=0}^{\infty }{\mathcal{A}}_l+{\omega }_{\vartheta \vartheta \vartheta }(\vartheta ,\zeta )\right\}\right].\hfill \end{array}$$

(82)

The Adomian polynomials are calculated as stated in Equation (45). By comparing both sides of Equation (82), we get

$${\omega }_0(\vartheta ,\zeta )=-{\displaystyle \frac{2\psi exp\left(\psi \vartheta \right)}{exp\left(2\psi \vartheta \right)+1}},$$

$$\begin{array}{cc}\hfill & {\omega }_1(\vartheta ,\zeta )=-{\displaystyle \frac{2{\psi }^{4}exp\left(\psi \vartheta \right)(exp\left(2\psi \vartheta \right)-1)}{{(exp\left(2\psi \vartheta \right)+1)}^2}}\left(1-\mu +{\displaystyle \frac{\mu {\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\right),\hfill \end{array}$$

(83)

$$\begin{array}{cc}\hfill & {\omega }_2(\vartheta ,\zeta )=-{\displaystyle \frac{2{\psi }^{7}exp\left(\psi \vartheta \right)(exp\left(4\psi \vartheta \right)-6exp\left(2\psi \vartheta \right)-1)}{{(exp\left(2\psi \vartheta \right)+1)}^3}}\left[{\displaystyle \frac{{\mu }^2{\zeta }^{2\mu }}{\mathrm{\Gamma }(2\mu +1)}}+2\mu (1-\mu ){\displaystyle \frac{{\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}+{(1-\mu )}^2\right].\hfill \end{array}$$

(84)

Lastly, the solution in $NTD{M}_{ABC}$ manner is as follows:

$$\begin{array}{cc}\hfill \omega (\vartheta ,\zeta )& =\sum _{l=0}^{\infty }{\omega }_l(\vartheta ,\zeta )={\omega }_0(\vartheta ,\zeta )+{\omega }_1(\vartheta ,\zeta )+{\omega }_2(\vartheta ,\zeta )+\dots ,\hfill \\ \hfill \omega (\vartheta ,\zeta )& =-{\displaystyle \frac{2\psi exp\left(\psi \vartheta \right)}{exp\left(2\psi \vartheta \right)+1}}-{\displaystyle \frac{2{\psi }^{4}exp\left(\psi \vartheta \right)(exp\left(2\psi \vartheta \right)-1)}{{(exp\left(2\psi \vartheta \right)+1)}^2}}\left(1-\mu +{\displaystyle \frac{\mu {\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}\right)\hfill \\ \hfill & -{\displaystyle \frac{2{\psi }^{7}exp\left(\psi \vartheta \right)(exp\left(4\psi \vartheta \right)-6exp\left(2\psi \vartheta \right)-1)}{{(exp\left(2\psi \vartheta \right)+1)}^3}}\left[{\displaystyle \frac{{\mu }^2{\zeta }^{2\mu }}{\mathrm{\Gamma }(2\mu +1)}}+2\mu (1-\mu ){\displaystyle \frac{{\zeta }^{\mu }}{\mathrm{\Gamma }(\mu +1)}}+{(1-\mu )}^2\right]+\cdots .\hfill \end{array}$$

(85)

By taking $\mu =1$, we get the exact solution as

$$\omega (\vartheta ,\zeta )=-{\displaystyle \frac{2\psi exp\left(\psi (\vartheta -{\psi }^2\zeta )\right)}{exp\left(2\psi (\vartheta -{\psi }^2\zeta )\right)+1}}.$$

(86)

The 3D simulations of the precise and suggested approach solutions for $\mu =1$ are shown in Figure 11. The 3D simulations of the suggested technique solutions for $\mu =0.75$ and $\mu =0.50$ are shown in Figure 12 and Figure 13. Similarly, the 2D simulations of the precise solution and the suggested technique solution in terms of both operators are shown in Figure 14 and Figure 15. The accurate and derived results for various $\mu$ values are shown in

Table 5. Analysis of the accurate and proposed method solution.

$\mathit{\zeta }$	$\mathit{\vartheta }$	${\mathit{NTDM}}_{\mathit{ABC}}$ at	${\mathit{NTDM}}_{\mathit{CF}}$ at	${\mathit{NTDM}}_{\mathit{ABC}}$ at	${\mathit{NTDM}}_{\mathit{CF}}$ at	$\mathit{Exact}$ at
		$\mathit{\mu }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{99}$	$\mathit{\mu }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{99}$	$\mathit{\mu }\mathbf{=}\mathbf{1}$	$\mathit{\mu }\mathbf{=}\mathbf{1}$	$\mathit{\mu }\mathbf{=}\mathbf{1}$
0.01	0.0	0.9996370674	0.9996529950	0.9999500000	0.9999500000	0.9999500020
	0.2	0.9839491393	0.9838648254	0.9822177258	0.9822177258	0.9822175738
	0.4	0.9319419297	0.9317731115	0.9284891113	0.9284891113	0.9284888698
	0.6	0.8526675213	0.8524420909	0.8480631252	0.8480631252	0.8480628778
	0.8	0.7578015152	0.7575496286	0.7526605050	0.7526605050	0.7526603096
	1	0.6581654397	0.6579119973	0.6529950034	0.6529950034	0.6529948774
0.03	0.0	0.9988003546	0.9988649550	0.9995500000	0.9995500000	0.9995501686
	0.2	0.9871553470	0.9869836317	0.9857259982	0.9857259982	0.9857219806
	0.4	0.9385889168	0.9382134520	0.9352550489	0.9352550489	0.9352485588
	0.6	0.8616464663	0.8611307624	0.8569809147	0.8569809147	0.8569742204
	0.8	0.7678954975	0.7673107406	0.7625551854	0.7625551854	0.7625498584
	1	0.6683597693	0.6677661105	0.6629075791	0.6629075791	0.6629041338
0.05	0.0	0.9975507086	0.9976848750	0.9987500000	0.9987500000	0.9987513004
	0.2	0.9899595376	0.9897480549	0.9888726918	0.9888726918	0.9888545122
	0.4	0.9449120035	0.9443958550	0.9417578118	0.9417578118	0.9417279294
	0.6	0.8704107063	0.8696794941	0.8657559228	0.8657559228	0.8657248512
	0.8	0.7778792473	0.7770372312	0.7724145412	0.7724145412	0.7723896738
	1	0.6785236010	0.6776608868	0.6728616436	0.6728616436	0.6728454780

. The tabular and graphical representations show how easy it is to apply NTDM and how it offers extremely valid and nearly accurate solutions. It is noteworthy that the formation of condensed soliton gas dynamics is mostly mediated by pairwise collisions of particle-like phenomena, such as breathers and solitary waves. These waves manifest as deep ocean waves, shallow groundwater waves, internal waves in a segmented sea, and fiber optics.

6. Conclusions

This work employs a novel methodology to offer the solutions to several nonlinear fractional KdV equations. The simplest and most effective method that may be used to solve FPDEs and associated systems is called NTDM. Each targeted problem’s fractional derivatives are represented by the Caputo–Fabrizio (CF) and Atangana–Baleanu derivative (ABC). After a discussion of the suggested technique for general nonlinear issues, a few nonlinear problems involving nonlinear FPDEs are resolved. Through the use of graphs and tables, the suggested technique results are presented. The comparison of the solutions revealed a very close correspondence between the precise and suggested approach solutions. More emphasis and important information about the dynamics of the targeted problems are offered by fractional order solutions. The fractional solutions are discovered to be convergent with the targeted problems actual solution. The current work supports the actual dynamics of physical processes in complete detail and can be expanded to include the solutions to other complex and nonlinear FPDEs and associated systems. To get better results, readers could combine our proposed approach with unique transforms and operators as a future study topic. We anticipate that this method will be used in the future to swiftly and efficiently tackle other fractional differential problems in scientific domains.