Dynamical and Computational Analysis of Fractional Korteweg–de Vries-Burgers and Sawada–Kotera Equations in Terms of Caputo Fractional Derivative

Abstract

This work examines the fractional Sawada–Kotera and Korteweg–de Vries (KdV)–Burgers equations, which are essential models of nonlinear wave phenomena in many scientific domains. The homotopy perturbation transform method (HPTM) and the Yang transform decomposition method (YTDM) are two sophisticated techniques employed to derive analytical solutions. The proposed methods are novel and remarkable hybrid integral transform schemes that effectively incorporate the Adomian decomposition method, homotopy perturbation method, and Yang transform method. They efficiently yield rapidly convergent series-type solutions through an iterative process that requires fewer computations. The Caputo operator, used to express the fractional derivatives in the equations, provides a robust framework for analyzing the behavior of non-integer-order systems. To validate the accuracy and reliability of the obtained solutions, numerical simulations and graphical representations are presented. Furthermore, the results are compared with exact solutions using various tables and graphs, illustrating the effectiveness and ease of implementation of the proposed approaches for various fractional partial differential equations. The influence of the non-integer parameter on the solutions behavior is specifically examined, highlighting its function in regulating wave propagation and diffusion. In addition, a comparison with the natural transform iterative method and optimal auxiliary function method demonstrates that the proposed methods are more accurate than these alternative approaches. The results highlight the potential of YTDM and HPTM as reliable tools for solving nonlinear fractional differential equations and affirm their relevance in wave mechanics, fluid dynamics, and other fields where fractional-order models are applied.

1. Introduction

In a letter dating back to 1695, mathematicians Leibniz and L’Hospital discussed the concept of fractional calculus (FC), which extends classical calculus from integer-order to arbitrary-order derivatives. In recent years, FC has attracted growing interest among researchers due to its ability to model a wide range of nonlinear phenomena. FC has become a foundational platform for major advances in science and engineering, owing to its valuable properties such as analyticity, heredity, non-locality, and memory effect. These features have made it one of the most effective tools for analyzing and characterizing complex nonlinear behaviors. The complexity of systems with heterogeneities led to the development of the concept of derivatives of arbitrary order. Differential operators of non-integer order can effectively capture the fundamental behavior of complex media with diffusion mechanisms. The rapid progress in mathematical techniques, computational methods, and software has further enabled researchers to explore both the foundations and diverse applications of fractional-order calculus [1,2,3,4,5,6]. Various innovative definitions of FC have been introduced, laying a solid theoretical foundation for its application in nanotechnology [7], optics [8], human diseases [9], chaos theory [10], and other areas [1,11,12,13,14].

Fractional differential equations (FDEs) are an extension of standard differential equations that exhibit both non-local and genetic effects on material properties. Several renowned researchers have investigated and discussed the concept of FC, proposing novel definitions that serve as the foundation of the field [1,2,15]. By taking into consideration their complex characteristics, such as memory and non-locality, nonlinear FDEs provide a more realistic depiction of biological and physical systems [16]. FDEs are becoming increasingly important for both constructing nonlinear models and for examining dynamical systems. The concept of FC has been used to analyze and evaluate a wide range of phenomena. Real-world applications of the FC theory include financial models [17], noisy environments [18], optics [19], and others [20,21]. Numerous solutions to FDEs illustrate the features of nonlinear problems that occur in nature. Researchers have examined general classes of FDEs, such as mixed, sequential, and hybrid, that remain largely unexplored in this area. In the domain of nonlinear analysis, experiential methods are crucial for understanding the dynamics of systems governed by various schemes. Due to the challenges of solving FDEs describing nonlinear systems with accuracy, several analytical and numerical techniques have been employed. Computational methods that yield analytical solutions offer certain advantages over traditional numerical approaches. The lack of discretization eliminates rounding errors and reduces memory and processing requirements. The Adomian decomposition method (ADM) has recently proven effective in solving fractional ordinary differential equations [22]. A general set of FDEs containing a Caputo fractional derivative with a linear functional argument was solved using the spectral Tau approach [23]. The Caputo-Fabrizio and the Atangana-Baleanu derivatives, two distinct forms of non-integer derivatives, were applied using the natural decomposition approach in [24] to address the time-fractional nonlinear Korteweg–de Vries (KdV)-Burgers equation. In addition, ref. [25] proposed a hybrid technique for solving fractional local Poisson equations.

This paper focuses on the fractional Sawada–Kotera and KdV–Burgers equations. The KdV–Burgers model has attracted considerable attention in recent decades. It is used in several real-world scenarios, including wave propagation through an elastic pipe filled with a viscous liquid [26], the transport of liquids carrying gas bubbles [27], weakly nonlinear plasma waves with specific dissipative properties [28], and the turbulence of undular bores in shallow water [29]. In addition, it serves as a nonlinear model in circuit theory, turbulence, ferroelectricity theory, and other domains [30,31]. Assume the fractional KdV-Burgers equation as follows

$${\displaystyle \frac{{\partial }^{\epsilon }\mathit{w}}{\partial {\tau }^{\epsilon }}}=6\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau ),$$

(1)

where $0<\epsilon \le 1$ indicates the non-integer derivative, taken in the Caputo sense. Here, $\mathit{w}$ is a function of $\varkappa$ and $\tau$, where $\varkappa$ denotes the spatial variable and $\tau$ the time variable. The classical Sawada–Kotera equation is another important model that describes the dynamics of long waves in shallow water under gravity, as well as in a one-dimensional nonlinear lattice. It also has numerous applications in nonlinear optics and quantum physics. The use of kink-shaped tanh solutions or bell-shaped sech solutions to represent wave phenomena and fluid dynamics in plasma media is well known. This equation is also used to simulate waves propagating in opposite directions. Assume the fractional Sawada–Kotera equation as follows [32]:

$${\displaystyle \frac{{\partial }^{\epsilon }\mathit{w}}{\partial {\tau }^{\epsilon }}}+45{\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )+15{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )+15\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )+{\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )=0.$$

(2)

This is the fifth-order variation of the Sawada–Kotera equation [33,34,35]. The condition $0<\epsilon \le 1$ indicates the non-integer derivative. The fractional derivative is considered in the Caputo sense and is denoted by $D_{\tau }^{\epsilon }$, ${\mathit{w}}_{\varkappa }$ indicates the ordinary integer-order derivative of $\mathit{w}(\varkappa ,\tau )$, and $\tau$ signifies time.

The ADM was proposed by Grorge Adomian in 1980 [36] and has since attracted significant attention in applied mathematics. Cherruault [37] conducted an important investigation into the convergence of the method. ADM enables the efficient and accurate solution of a wide class of linear and nonlinear partial and ordinary differential equations (PDEs and ODEs). Notably, it can be applied to most physical models of PDEs without relying on linearization or other restrictive assumptions that might distort the actual physical behavior of the systems under study [36,38]. In parallel, Ji-Huan He introduced the homotopy perturbation method (HPM) [39,40,41], which combines the homotopy method with traditional perturbation techniques [40]. The primary advantage of the HPM lies in its simplicity of handling nonlinear terms. Burgers-type equations [39,40,41,42] were among the linear or nonlinear ODEs and PDEs successfully investigated using this method. The HPM and its convergence have been examined and refined by numerous researchers. In this study, the Yang transform decomposition method (YTDM) and homotopy perturbation transform method (HPTM) are employed to obtain analytical and approximate solutions for the nonlinear fractional Sawada–Kotera and KdV–Burgers equations. Interestingly, YTDM combines Hes polynomials, the Yang transformation, and the homotopy perturbation method, whereas HPTM combines the Yang transformation, the ADM, and Adomian polynomials. The primary aim of this paper is to address certain limitations by proposing a novel modification of the ADM and HPM. These requirements are met through the application of new techniques that merge the Yang transform with ADM and HPM. The main advantages of these approaches are their ease of use and low computational effort. Moreover, the analytical solutions derived using these methods show complete agreement with those found in existing literature. These techniques offer more accurate and exact numerical results compared to alternative methods. The numerical results are contrasted with those obtained through other approaches to assess both methods’ accuracy. The findings demonstrate that the outcomes of the proposed techniques are largely consistent, confirming their efficacy and reliability. The novel methods also outperform existing approaches, making them more valuable and effective. In addition, just one or two iteration steps yield highly accurate results over a broad range. In the numerical scenarios considered, our methods produced infinite series solutions. Once expressed in closed form, these series yield accurate solutions to the corresponding equations. The remainder of this paper is structured as follows. In Section 2, we outline several key concepts and notations related to FC and the Yang transform used in the current context. Section 3 presents the Laplace–Yang duality property. Section 4 and Section 5 provide a concise overview of the general principles underlying the proposed methods. In Section 6, we present the main results of the recommended strategies. Section 7 introduces the proposed techniques along with the validity of the error-bound theorem. Section 8 includes the implementation of the proposed techniques on the fractional Sawada–Kotera and KdV–Burgers equations, supported by numerical results and observations. Finally, the conclusion is presented in Section 9.

2. Preliminaries

Some well-known definitions from FC, which are related to our analysis, are presented in this section.

Definition 1.

The Riemann–Liouville’s (RL) non-integer derivative is defined as [43,44]

$$D^{\epsilon }\mathit{w}\left(\varkappa \right)=\left\{\begin{array}{c}{\displaystyle \frac{d^{\varpi }}{d{\varkappa }^{\varpi }}}\mathit{w}\left(\varkappa \right),\epsilon =\varpi ,\hfill \\ {\displaystyle \frac{1}{\Gamma (\varpi -\epsilon )}}{\displaystyle \frac{d^{\varpi }}{d{\varkappa }^{\varpi }}}{\int }_0^{\varkappa }{\displaystyle \frac{\mathit{w}\left(\tau \right)}{{(\varkappa -\tau )}^{\epsilon -\varpi +1}}}d\tau ,\varpi -1<\epsilon <\varpi ,\hfill \end{array}\right.$$

(3)

with $\varpi \in Z^{+}$, $\epsilon \in R^{+}$ and

$$D^{-\epsilon }\mathit{w}\left(\varkappa \right)={\displaystyle \frac{1}{\Gamma \left(\epsilon \right)}}{\int }_0^{\varkappa }{(\varkappa -\tau )}^{\epsilon -1}\mathit{w}\left(\tau \right)d\tau ,0<\epsilon \le 1.$$

(4)

Definition 2.

The Caputo non-integer derivative is defined as [43,44]

$$D^{\epsilon }\mathit{w}\left(\varkappa \right)=\left\{\begin{array}{c}{\displaystyle \frac{1}{\Gamma (\varpi -\epsilon )}}{\int }_0^{\varkappa }{\displaystyle \frac{{\mathit{w}}^{\varpi }\left(\tau \right)}{{(\varkappa -\tau )}^{\epsilon -\varpi +1}}}d\tau ,\varpi -1<\epsilon <\varpi ,\hfill \\ {\displaystyle \frac{d^{\varpi }}{d{\varkappa }^{\varpi }}}\mathit{w}\left(\varkappa \right),\varpi =\epsilon ,\hfill \end{array}\right.$$

(5)

with

$$\begin{array}{cc}\hfill J_{\tau }^{\epsilon }{D}_{\tau }^{\epsilon }\mathit{w}\left(\tau \right)& =\mathit{w}\left(\tau \right)-{\displaystyle \sum _{k=0}^{m-1}}{\displaystyle \frac{{\tau }^k}{k!}}{\displaystyle \frac{{\partial }^k\mathit{w}\left(0^{+}\right)}{\partial {\tau }^k}},for\tau >0,and\varpi -1<\epsilon \le \varpi ,\varpi \in N,\hfill \\ \hfill D_{\tau }^{\epsilon }{J}_{\tau }^{\epsilon }\mathit{w}\left(\tau \right)& =\mathit{w}\left(\tau \right).\hfill \end{array}$$

(6)

Definition 3.

The Laplace transform (LT) of function $\mathit{w}\left(\tau \right)$ for $\tau \ge 0$ is defined as [45]

$$L\left(\mathit{w}\left(\tau \right)\right)={\int }_0^{\infty }{e}^{-u\tau }\mathit{w}\left(\tau \right)d\tau .$$

(7)

Definition 4.

The Yang transform (YT) of the $\mathit{w}\left(\varkappa \right)$ is defined as [46]:

$$Y\left\{\mathit{w}\left(\varkappa \right)\right\}=M\left(u\right)={\int }_0^{\infty }{e}^{{\displaystyle \frac{-\tau }{u}}}\mathit{w}\left(\tau \right)d\tau ,\tau >0,$$

(8)

with u indicating the transform variable.

Few properties of YT are defined as

$$\begin{array}{cc}\hfill Y\left[1\right]=& u,\hfill \\ \hfill Y\left[\varkappa \right]=& u^2,\hfill \\ \hfill Y\left[{\varkappa }^m\right]=& \Gamma (m+1)u^{m+1},\hfill \end{array}$$

(9)

and the inverse is given by

$$Y^{-1}\left\{M\left(u\right)\right\}=\mathit{w}\left(\tau \right).$$

(10)

Definition 5.

The YT of Caputo derivative is defined as [46]

$$Y\left\{{\mathit{w}}^{\epsilon }\left(\varkappa \right)\right\}={\displaystyle \frac{M\left(u\right)}{u^{\epsilon }}}-\sum _{k=0}^{n-1}{\displaystyle \frac{{\mathit{w}}^k\left(0\right)}{u^{\epsilon -(k+1)}}},n-1<\epsilon \le n.$$

(11)

3. Laplace-Yang Duality

Lemma 1.

Suppose the LT of $\mathbb{J}\left(\tau \right)$ is $F\left(u\right)$, then $M\left(u\right)=F\left({\displaystyle \frac{1}{u}}\right)$.

Proof. 

By switching ${\displaystyle \frac{\tau }{u}}=\varkappa$ in Equation (8), we obtain an alternative form of the YT as follows:

$$Y\left[\mathbb{J}\left(\tau \right)\right]=M\left(u\right)=u{\int }_0^{\infty }\mathbb{J}\left(u\varkappa \right)e^{\varkappa }d\varkappa .\varkappa >0.$$

(12)

Given $L\left[\mathbb{J}\left(\tau \right)\right]=F\left(u\right)$, so

$$F\left(u\right)=L\left[\mathbb{J}\left(\tau \right)\right]={\int }_0^{\infty }\mathbb{J}\left(\tau \right)e^{-u\tau }d\tau .$$

(13)

Substitute $\tau =\varkappa /u$ in (13), we have:

$$F\left(u\right)={\displaystyle \frac{1}{u}}{\int }_0^{\infty }\mathbb{J}\left({\displaystyle \frac{\varkappa }{u}}\right)e^{\varkappa }d\varkappa .$$

(14)

Thus, from Equation (12), we obtain

$$F\left(u\right)=\zeta \left({\displaystyle \frac{1}{u}}\right).$$

(15)

Therefore, from Equations (8) and (13), we have:

$$F\left({\displaystyle \frac{1}{u}}\right)=\zeta \left(u\right).$$

(16)

Equations (15) and (16) demonstrate the duality relation between the YT and LT. □

4. Methodology of HPTM

Consider the FDE of the form:

$$D_{\tau }^{\epsilon }\mathit{w}(\varkappa ,\tau )=\mathcal{G}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau )+\mathcal{H}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau ),0<\epsilon \le 1,$$

(17)

with

$$\mathit{w}(\varkappa ,0)=\theta \left(\varkappa \right).$$

Here, $D_{\tau }^{\epsilon }={\displaystyle \frac{{\partial }^{\epsilon }}{\partial {\tau }^{\epsilon }}}$ represents the Caputo operator, $\mathcal{G}\left[\varkappa \right]$ is linear, and $\mathcal{H}\left[\varkappa \right]$ is the nonlinear operator.

Taking the YT of Equation (17), we obtain

$$Y\left[D_{\tau }^{\epsilon }\mathit{w}(\varkappa ,\tau )\right]=Y[\mathcal{G}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau )+\mathcal{H}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau )],$$

(18)

$${\displaystyle \frac{1}{u^{\epsilon }}}\{M\left(u\right)-u\mathit{w}\left(0\right)\}=Y[\mathcal{G}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau )+\mathcal{H}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau )].$$

(19)

After some simplification,

$$M\left(\mathit{w}\right)=u\mathit{w}\left(0\right)+u^{\epsilon }Y[\mathcal{G}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau )+\mathcal{H}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau )].$$

(20)

Apply inverse YT to Equation (20)

$$\mathit{w}(\varkappa ,\tau )=\mathit{w}\left(0\right)+Y^{-1}\left[u^{\epsilon }Y[\mathcal{G}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau )+\mathcal{H}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau )]\right].$$

(21)

By implementing the homotopy perturbation method [41], we obtain

$$\mathit{w}(\varkappa ,\tau )=\mathit{w}\left(0\right)+ϵ\left[Y^{-1}\left[u^{\epsilon }Y[\mathcal{G}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau )+\mathcal{H}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau )]\right]\right].$$

(22)

having parameter $ϵ\in [0,1]$.

Assume a series solution of the form:

$$\mathit{w}(\varkappa ,\tau )=\sum _{k=0}^{\infty }{ϵ}^k{\mathit{w}}_k(\varkappa ,\tau ).$$

(23)

with

$$\mathcal{H}\left[\varkappa \right]\mathit{w}(\varkappa ,\tau )=\sum _{k=0}^{\infty }{ϵ}^k{H}_n\left(\mathit{w}\right).$$

(24)

He’s polynomials are defined as [47]:

$$H_n({\mathit{w}}_0,{\mathit{w}}_1,...,{\mathit{w}}_n)={\displaystyle \frac{1}{\Gamma (n+1)}}{D}_{ϵ}^k\left(\mathcal{H}\left(\sum _{k=0}^{\infty }{ϵ}^i{\mathit{w}}_i\right)\right){|}_{ϵ=0}.$$

(25)

where $D_{ϵ}^k={\displaystyle \frac{{\partial }^k}{\partial {ϵ}^k}}.$

By utilizing (23) and (24) in (22), we obtain

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathit{w}}_k(\varkappa ,\tau )=\mathit{w}\left(0\right)+ϵ\times \left(Y^{-1}\left[u^{\epsilon }Y\{\mathcal{G}\sum _{k=0}^{\infty }{ϵ}^k{\mathit{w}}_k(\varkappa ,\tau )+\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left(\mathit{w}\right)\}\right]\right).$$

(26)

By applying the recursive relationship:

$$\begin{array}{cc}\hfill & {ϵ}^0:{\mathit{w}}_0(\varkappa ,\tau )=\mathit{w}\left(0\right),\hfill \\ \hfill & {ϵ}^1:{\mathit{w}}_1(\varkappa ,\tau )=Y^{-1}\left[u^{\epsilon }Y(\mathcal{G}\left[\varkappa \right]{\mathit{w}}_0(\varkappa ,\tau )+H_0\left(\mathit{w}\right))\right],\hfill \\ \hfill & {ϵ}^2:{\mathit{w}}_2(\varkappa ,\tau )=Y^{-1}\left[u^{\epsilon }Y(\mathcal{G}\left[\varkappa \right]{\mathit{w}}_1(\varkappa ,\tau )+H_1\left(\mathit{w}\right))\right],\hfill \\ \hfill & .\hfill \\ \hfill & .\hfill \\ \hfill & .\hfill \\ \hfill & {ϵ}^k:{\mathit{w}}_k(\varkappa ,\tau )=Y^{-1}\left[u^{\epsilon }Y(\mathcal{G}\left[\varkappa \right]{\mathit{w}}_{k-1}(\varkappa ,\tau )+H_{k-1}\left(\mathit{w}\right))\right],\hfill \\ \hfill k>0,k\in N.\end{array}$$

(27)

The final solution can be written as

$$\mathit{w}(\varkappa ,\tau )=\underset{ϵ\to 1}{lim}\sum _{k=0}^{\infty }{ϵ}^k{\mathit{w}}_k(\varkappa ,\tau ).$$

(28)

5. Methodology of YTDM

Take into account the FDE of the form:

$$D_{\tau }^{\epsilon }\mathit{w}(\varkappa ,\tau )=\mathcal{G}(\varkappa ,\tau )+\mathcal{H}(\varkappa ,\tau ),0<\epsilon \le 1,$$

(29)

with

$$\mathit{w}(\varkappa ,0)=\theta \left(\varkappa \right).$$

Here, $D_{\tau }^{\epsilon }={\displaystyle \frac{{\partial }^{\epsilon }}{\partial {\tau }^{\epsilon }}}$ denotes the Caputo operator, $\mathcal{G}$ is a linear operator, and $\mathcal{H}$ is a nonlinear operator.

Taking the YT of Equation (29), we obtain

$$\begin{array}{cc}\hfill & Y\left[D_{\tau }^{\epsilon }\mathit{w}(\varkappa ,\tau )\right]=Y[\mathcal{G}(\varkappa ,\tau )+\mathcal{H}(\varkappa ,\tau )],\hfill \\ \hfill & {\displaystyle \frac{1}{u^{\epsilon }}}\{M\left(u\right)-u\mathit{w}\left(0\right)\}=Y[\mathcal{G}(\varkappa ,\tau )+\mathcal{H}(\varkappa ,\tau )].\hfill \end{array}$$

(30)

After some simplification, we obtain

$$M\left(\mathit{w}\right)=u\mathit{w}\left(0\right)+u^{\epsilon }Y[\mathcal{G}(\varkappa ,\tau )+\mathcal{H}(\varkappa ,\tau )].$$

(31)

Apply the inverse YT to Equation (31):

$$\mathit{w}(\varkappa ,\tau )=\mathit{w}\left(0\right)+Y^{-1}[u^{\epsilon }Y[\mathcal{G}(\varkappa ,\tau )+\mathcal{H}(\varkappa ,\tau )].$$

(32)

Assume a series solution of the form:

$$\mathit{w}(\varkappa ,\tau )=\sum _{m=0}^{\infty }{\mathit{w}}_m(\varkappa ,\tau ).$$

(33)

The nonlinear term is calculated as [48]

$$\mathcal{H}(\varkappa ,\tau )=\sum _{m=0}^{\infty }{\mathcal{A}}_m,$$

(34)

with

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

(35)

Substituting (33) and (34) into (32), we obtain

$$\sum _{m=0}^{\infty }{\mathit{w}}_m(\varkappa ,\tau )=\mathit{w}\left(0\right)+Y^{-1}{u}^{\epsilon }\left[Y\left\{\mathcal{G}(\sum _{m=0}^{\infty }{\varkappa }_m,\sum _{m=0}^{\infty }{\tau }_m)+\sum _{m=0}^{\infty }{\mathcal{A}}_m\right\}\right].$$

(36)

By applying the recursive relationship:

$${\mathit{w}}_0(\varkappa ,\tau )=\mathit{w}\left(0\right),$$

(37)

$${\mathit{w}}_1(\varkappa ,\tau )=Y^{-1}\left[u^{\epsilon }Y\{\mathcal{G}({\varkappa }_0,{\tau }_0)+{\mathcal{A}}_0\}\right].$$

The final solution can be written as

$${\mathit{w}}_{m+1}(\varkappa ,\tau )=Y^{-1}\left[u^{\epsilon }Y\{\mathcal{G}({\varkappa }_m,{\tau }_m)+{\mathcal{A}}_m\}\right].$$

(38)

6. Convergence Analysis

The existence results of the proposed techniques are discussed in this section.

Theorem 1.

Assume the exact result of (17) is $\mathit{v}(\varkappa ,\tau )$ and suppose $\mathit{v}(\varkappa ,\tau )$, ${\mathit{v}}_n(\varkappa ,\tau )\in H$ and $\epsilon \in (0,1)$, where H indicates the Hilbert space. The attained outcome ${\sum }_{q=0}^{\infty }{\mathit{v}}_q(\varkappa ,\tau )$ converges to $\mathit{v}(\varkappa ,\tau )$ if ${\mathit{v}}_q(\varkappa ,\tau )\le {\mathit{v}}_{q-1}(\varkappa ,\tau )\forall q>A$, i.e., for all $\varkappa >0\exists A>0$, with $\left|\right|{\mathit{v}}_{q+n}(\varkappa ,\tau )\left|\right|\le \epsilon ,\forall q,n\in N.$

Proof. 

Assume ${\sum }_{q=0}^{\infty }{\mathit{v}}_q(\varkappa ,\tau ).$

$$\begin{array}{cc}\hfill {\mathit{u}}_0=& {\mathit{v}}_0,\hfill \\ \hfill {\mathit{u}}_1=& {\mathit{v}}_0+{\mathit{v}}_1,\hfill \\ \hfill {\mathit{u}}_2=& {\mathit{v}}_0+{\mathit{v}}_1+{\mathit{v}}_2,\hfill \\ \hfill {\mathit{u}}_3=& {\mathit{v}}_0+{\mathit{v}}_1+{\mathit{v}}_2+{\mathit{v}}_3,\hfill \\ \hfill & ⋮\hfill \\ \hfill {\mathit{u}}_q=& {\mathit{v}}_0+{\mathit{v}}_1+{\mathit{v}}_2+\dots +{\mathit{v}}_q.\hfill \end{array}$$

(39)

We must ascertain that ${\mathit{u}}_q$ forms a Cauchy sequence to achieve the preferred outcome. Let

$$\begin{array}{cc}\hfill \left|\right|{\mathit{u}}_{q+1}-{\mathit{u}}_q\left|\right|& =\left|\right|{\mathit{v}}_{q+1}\left|\right|\le \epsilon \left|\right|{\mathit{v}}_q\left|\right|\le {\epsilon }^2\left|\right|{\mathit{v}}_{q-1}\left|\right|\le {\epsilon }^3\left|\right|{\mathit{v}}_{q-2}\left|\right|\dots \hfill \\ \hfill & \le {\epsilon }_{q+1}\left|\right|{\mathit{v}}_0\left|\right|.\hfill \end{array}$$

(40)

For $q,n\in N$, then:

$$\begin{array}{cc}\hfill \left|\right|{\mathit{u}}_q-{\mathit{u}}_n\left|\right|=& \left|\right|{\mathit{v}}_{q+n}\left|\right|=\left|\right|{\mathit{u}}_q-{\mathit{u}}_{q-1}+({\mathit{u}}_{q-1}-{\mathit{u}}_{q-2})\hfill \\ \hfill & +({\mathit{u}}_{q-2}-{\mathit{u}}_{q-3})+\dots +({\mathit{u}}_{n+1}-{\mathit{u}}_n)\left|\right|\hfill \\ \hfill \le & \left|\right|{\mathit{u}}_q-{\mathit{u}}_{q-1}\left|\right|+\left|\right|({\mathit{u}}_{q-1}-{\mathit{u}}_{q-2})\left|\right|\hfill \\ \hfill & +\left|\right|({\mathit{u}}_{q-2}-{\mathit{u}}_{q-3})\left|\right|+\dots +\left|\right|({\mathit{u}}_{n+1}-{\mathit{u}}_n)\left|\right|\hfill \\ \hfill \le & {\epsilon }^q\left|\right|{\mathit{v}}_0\left|\right|+{\epsilon }^{q-1}\left|\right|{\mathit{v}}_0\left|\right|+\dots +{\epsilon }^{q+1}\left|\right|{\mathit{v}}_0\left|\right|\hfill \\ \hfill =& \left|\right|{\mathit{v}}_0\left|\right|({\epsilon }^q+{\epsilon }^{q-1}+{\epsilon }^{q+1})\hfill \\ \hfill =& \left|\right|{\mathit{v}}_0\left|\right|{\displaystyle \frac{1-{\epsilon }^{q-n}}{1-{\epsilon }^{q+1}}}{\epsilon }^{n+1}.\hfill \end{array}$$

(41)

Since $0<\epsilon <1$ and ${\mathit{v}}_0$ are bounded, it follows that: $\epsilon =1-\epsilon /(1-{\epsilon }_{q-n}){\epsilon }^{n+1}\left|\right|{\mathit{v}}_0\left|\right|$, and we obtain

$$\left|\right|{\mathit{v}}_{q+n}\left|\right|\le \epsilon ,\forall q,n\in N.$$

(42)

Hence, ${\left\{{\mathit{v}}_q\right\}}_{q=0}^{\infty }$ forms a Cauchy sequence in H. Thus, it is a convergent sequence with ${lim}_{q\to \infty }{\mathit{v}}_q=\mathit{v}$, for $\exists \mathit{v}\in \mathcal{H}$. □

Theorem 2.

Assume that ${\sum }_{h=0}^k{\mathit{v}}_h(\varkappa ,\tau )$ is finite and $\mathit{v}(\varkappa ,\tau )$ is the corresponding series solution. Considering $0<\epsilon <1$ and $\left|\right|{\mathit{v}}_{h+1}(\varkappa ,\tau )\left|\right|\le \left|\right|{\mathit{v}}_h(\varkappa ,\tau )\left|\right|$, the maximum absolute error is given by:

$$\left|\right|\mathit{v}(\varkappa ,\tau )-\sum _{h=0}^k{\mathit{v}}_h(\varkappa ,\tau )\left|\right|<{\displaystyle \frac{{\epsilon }^{k+1}}{1-\epsilon }}\left|\right|{\mathit{v}}_0(\varkappa ,\tau )\left|\right|.$$

(43)

Proof. 

Let ${\sum }_{h=0}^k{\mathit{v}}_h$ be finite, then ${\sum }_{h=0}^k{\mathit{v}}_h<\infty$.

Assume

$$\begin{array}{cc}\hfill \left|\right|\mathit{v}-{\displaystyle \sum _{h=0}^k}{\mathit{v}}_h\left|\right|=& \left|\right|{\displaystyle \sum _{h=k+1}^{\infty }}{\mathit{v}}_h\left|\right|\hfill \\ \hfill \le & {\displaystyle \sum _{h=k+1}^{\infty }}\left|\right|{\mathit{v}}_h\left|\right|\hfill \\ \hfill \le & {\displaystyle \sum _{h=k+1}^{\infty }}{\epsilon }^h\left|\right|{\mathit{v}}_0\left|\right|\hfill \\ \hfill \le & {\epsilon }^{k+1}(1+\epsilon +{\epsilon }^2+\dots )\left|\right|{\mathit{v}}_0\left|\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\epsilon }^{k+1}}{1-\epsilon }}\left|\right|{\mathit{v}}_0\left|\right|,\hfill \end{array}$$

(44)

which completes the proof. □

Theorem 3.

Suppose $(H,||.|\left|\right)$ is Banach space and $P:H\to H$ is a self-map of H. Then, the iteration procedure of YTDM is defined by:

$$P\left({\mathit{w}}_m(\varkappa ,\tau )\right)={\mathit{w}}_{m+1}(\varkappa ,\tau )=\mathit{w}(\varkappa ,0)+Y^{-1}\left[u^{\epsilon }Y\{\mathcal{G}({\varkappa }_m,{\tau }_m)+{\mathcal{A}}_m\}\right],$$

is P-stable if.

• $\left|\right|\mathcal{G}{\mathit{w}}_m(\varkappa ,\tau )-\mathcal{G}{\mathit{w}}_n(\varkappa ,\tau )\left|\right|\le {\mu }_0\left|\right|{\mathit{w}}_m(\varkappa ,\tau )-{\mathit{w}}_n(\varkappa ,\tau )\left|\right|$ for some ${\mu }_0\in R^{+},$
• $\left|\right|{\mathcal{A}}_m\left(\mathit{w}\right)-{\mathcal{A}}_n\left(\mathit{w}\right)\left|\right|\le {\mu }_1\left|\right|{\mathit{w}}_m(\varkappa ,\tau )-{\mathit{w}}_n(\varkappa ,\tau )\left|\right|$ for some ${\mu }_1\in R^{+},$
• $\mu =({\mu }_0+{\mu }_1)\left|\right|{\displaystyle \frac{{\tau }^{\epsilon }}{\Gamma (\epsilon +1)}}\left|\right|\le 1$

Proof. 

For proof, see [49]. □

Theorem 4.

Suppose $(H,||.|\left|\right)$ is Banach space and $P:H\to H$ is a mapping associated with the YTDM, defined by (38). Then, P has a unique fixed point, and the sequence ${\left\{{\mathit{w}}_n\right\}}_{n=0}^{\infty }$ converges to that fixed point in H, starting from an initial value ${\mu }_0\in H$, if $0<\mu <1$ such that $\left|\right|{\mathit{w}}_{n+1}\left|\right|\le \mu \left|\right|{\mathit{w}}_n\left|\right|$.

Proof. 

For proof, see [49]. □

7. Error Estimation

To evaluate the performance and accuracy of the proposed approaches, we present the error functions in this portion. Assuming that $\mathit{w}\left(\tau \right)$ is the accurate solution and ${\mathit{w}}_n\left(\tau \right)$ is the $n^{\mathrm{th}}$-order approximation of $\mathit{w}\left(\tau \right)$ generated via the suggested methods, the absolute error $\left(E_n\right)$ in the $n^{\mathrm{th}}$-order approximation is as follows:

$$E_n\left(\tau \right)=|\mathit{w}\left(\tau \right)-{\mathit{w}}_n\left(\tau \right)|.$$

(45)

Then, maximum absolute error $\left(M{E}_n\right)$ is given by:

$$M{E}_n=ma{x}_{\tau \in [0,1]}|\mathit{w}\left(\tau \right)-{\mathit{w}}_n\left(\tau \right)|.$$

(46)

The authors of [50,51,52] discussed error bounds for their numerical approaches based on well-known lemmas and theorems. In our work, we provide an upper bound on the absolute error for the proposed methods using the following theorem.

Theorem 5

(Error bound). Suppose $\mathit{w}\left(\tau \right)\in C^{(n+1)}[0,1]$ and let ${\mathit{w}}_n\left(\tau \right)={\sum }_{i=0}^n{c}_i{\tau }^i$ represent the exact and $n^{\mathrm{th}}$-order approximate solution of (17) and (29). Thus, the upper bound on the absolute error is given by:

$$\left|\right|\mathit{w}\left(\tau \right)-{\mathit{w}}_n{\left(\tau \right)\left|\right|}_{\infty }\le {\displaystyle \frac{M}{(n+1)!}}+ma{x}_{0\le i\le n}\left|a_i\right|,$$

(47)

with $M=ma{x}_{0\le t\le 1}\left|{\mathit{w}}^{n+1}\left(\tau \right)\right|,a_i={\sum }_{i=0}^n\left({\displaystyle \frac{{\mathit{w}}^i\left(0\right)}{i!}}-c_i\right).$

Proof. 

Suppose $\mathit{w}$ is a continuous function on $[0,1]$. Then, the upper bound of $\left|\mathit{w}\right|$ is defined as

$${\left|\right|\mathit{w}\left|\right|}_{\infty }=\underset{\tau \in [0,1]}{sup}\left|\mathit{w}\left(\tau \right)\right|.$$

(48)

By applying the norm property, we obtain

$$\left|\right|\mathit{w}\left(\tau \right)-{\mathit{w}}_n{\left(\tau \right)\left|\right|}_{\infty }\le \left|\right|\mathit{w}\left(\tau \right)-{\mathit{w}}_n^{\ast }{\left(\tau \right)\left|\right|}_{\infty }+\left|\right|{\mathit{w}}_n^{\ast }\left(\tau \right)-\mathit{w}\left(\tau \right){\left|\right|}_{\infty }$$

(49)

Since $\mathit{w}\left(\tau \right)\in C^{(n+1)}[0,1]$, we apply the Taylor expansion to obtain

$$\mathit{w}\left(\tau \right)={\mathit{w}}_n^{\ast }\left(\tau \right)+{\displaystyle \frac{{\mathit{w}}^{n+1}\left({\tau }_0\right)}{(n+1)!}}{\tau }^{n+1},{\tau }_0\in (0,1),$$

(50)

with ${\mathit{w}}_n^{\ast }\left(\tau \right)={\sum }_{i=0}^n{\displaystyle \frac{{\mathit{w}}^i\left(0\right)}{i!}}{\tau }^i.$

From (50), we have

$$\begin{array}{cc}\hfill & \left|\right|\mathit{w}\left(\tau \right)-{\mathit{w}}_n^{\ast }{\left(\tau \right)\left|\right|}_{\infty }=\underset{0\le \tau \le 1}{max}\left|{\displaystyle \frac{{\mathit{w}}^{n+1}\left(\tau \right)}{(n+1)!}}{\tau }^{n+1}\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{(n+1)!}}\underset{0\le \tau \le 1}{max}\left|{\mathit{w}}^{n+1}\left(\tau \right)\right|.\hfill \end{array}$$

(51)

Now, we calculate the value of $\left|\right|\mathit{w}\left(\tau \right)-{\mathit{w}}_n^{\ast }\left(\tau \right){\left|\right|}_{\infty }$.

Let $A=(a_0,a_1,\dots ,a_n),T={(t_0,t_1,\dots ,t_n)}^T,$ with $a_i={\displaystyle \frac{{\mathit{w}}^i\left(0\right)}{i!}}-c_i,i=0,1,\dots ,n$ so

$$\begin{array}{cc}\hfill & \parallel {\mathit{w}}_n^{\ast }\left(\tau \right)-{\mathit{w}}_n\left(\tau \right)\parallel =\parallel {\displaystyle \sum _{i=0}^n}{\displaystyle \frac{{\mathit{w}}^i\left(0\right)}{i!}}{\tau }^i-{\displaystyle \sum _{i=0}^n}{c}_i{\tau }^i\parallel =\parallel {\displaystyle \sum _{i=0}^n}\left({\displaystyle \frac{{\mathit{w}}^i\left(0\right)}{i!}}-c_i\right){\tau }^i\parallel \hfill \\ \hfill & {\parallel A\parallel }_{\infty }.{\parallel T\parallel }_{\infty },\hfill \\ \hfill & \parallel {\mathit{w}}_n^{\ast }\left(\tau \right)-{\mathit{w}}_n{\left(\tau \right)\parallel =\parallel A\parallel }_{\infty }.{\parallel T\parallel }_{\infty }.\hfill \end{array}$$

(52)

From (49), (51), and (52), we obtain

$$\begin{array}{cc}\hfill & \parallel \mathit{w}\left(\tau \right)-{\mathit{w}}_n\left(\tau \right)\parallel \le {\displaystyle \frac{1}{(n+1)!}}\underset{0\le \tau \le 1}{max}|{\mathit{w}}^{n+1}{\left(\tau \right)|+\parallel A\parallel }_{\infty }.{\parallel T\parallel }_{\infty },\hfill \\ \hfill & \parallel \mathit{w}\left(\tau \right)-{\mathit{w}}_n\left(\tau \right)\parallel \le {\displaystyle \frac{M}{(n+1)!}}+\underset{0\le i\le n}{max}\left|a_i\right|,\hfill \end{array}$$

(53)

which completes the proof. □

8. Application

8.1. Case I

Examine the fractional KdV–Burgers equation:

$${\displaystyle \frac{{\partial }^{\epsilon }\mathit{w}}{\partial {\tau }^{\epsilon }}}=6\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau ),$$

(54)

with

$$\mathit{w}(\varkappa ,0)=-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^2}{{(1+e^{\rho \varkappa })}^2}}.$$

Application of HPTM

Taking the YT of Equation (54), we obtain

$$Y\left[{\displaystyle \frac{{\partial }^{\epsilon }\mathit{w}}{\partial {\tau }^{\epsilon }}}\right]=Y\left[6\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )\right].$$

(55)

After some simplification, we obtain

$${\displaystyle \frac{1}{u^{\epsilon }}}\{M\left(u\right)-u\mathit{w}\left(0\right)\}=Y\left[6\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )\right],$$

(56)

$$M\left(u\right)=u\mathit{w}\left(0\right)+u^{\epsilon }\left[6\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )\right].$$

(57)

Apply the inverse YT to Equation (57):

$$\begin{array}{cc}\hfill & \mathit{w}(\varkappa ,\tau )=\mathit{w}\left(0\right)+Y^{-1}\left[u^{\epsilon }\left\{Y\left(6\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )\right)\right\}\right],\hfill \\ \hfill & \mathit{w}(\varkappa ,\tau )=\left(-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^2}{{(1+e^{\rho \varkappa })}^2}}\right)+Y^{-1}\left[u^{\epsilon }\left\{Y\left(6\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )\right)\right\}\right].\hfill \end{array}$$

(58)

By implementing the homotopy perturbation method, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{k=0}^{\infty }}{ϵ}^k{\mathit{w}}_k(\varkappa ,\tau )=\left(-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^2}{{(1+e^{\rho \varkappa })}^2}}\right)+ϵ\left[Y^{-1}\right[u^{\epsilon }Y[6\left({\displaystyle \sum _{k=0}^{\infty }}{ϵ}^k{H}_k^0(\varkappa ,\tau )\right)-\hfill \\ \hfill & {\left({\displaystyle \sum _{k=0}^{\infty }}{ϵ}^k{\mathit{w}}_k(\varkappa ,\tau )\right)}_{\varkappa \varkappa \varkappa }\left]\right]].\hfill \end{array}$$

(59)

Using the recursive relationship:

$$\begin{array}{cc}\hfill & {ϵ}^0:{\mathit{w}}_0(\varkappa ,\tau )=-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^2}{{(1+e^{\rho \varkappa })}^2}},\hfill \\ \hfill & {ϵ}^1:{\mathit{w}}_1(\varkappa ,\tau )=-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^5(-1+e^{\rho \varkappa })}{{(1+e^{\rho \varkappa })}^3}}{\displaystyle \frac{{\tau }^{\epsilon }}{\Gamma (\epsilon +1)}},\hfill \\ \hfill & {ϵ}^2:{\mathit{w}}_2(\varkappa ,\tau )={\displaystyle \frac{2e^{\rho \varkappa }{\rho }^8(-1-e^{2\rho \varkappa }+4e^{\rho \varkappa })}{{(1+e^{\rho \varkappa })}^4}}{\displaystyle \frac{{\tau }^{2\epsilon }}{\Gamma (2\epsilon +1)}},\hfill \\ \hfill & ⋮\hfill \end{array}$$

The final solution can be written as

$$\begin{array}{cc}\hfill & \mathit{w}(\varkappa ,\tau )={\mathit{w}}_0(\varkappa ,\tau )+{\mathit{w}}_1(\varkappa ,\tau )+{\mathit{w}}_2(\varkappa ,\tau )+\dots \hfill \\ \hfill & \mathit{w}(\varkappa ,\tau )=-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^2}{{(1+e^{\rho \varkappa })}^2}}-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^5(-1+e^{\rho \varkappa })}{{(1+e^{\rho \varkappa })}^3}}{\displaystyle \frac{{\tau }^{\epsilon }}{\Gamma (\epsilon +1)}}+{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^8(-1-e^{2\rho \varkappa }+4e^{\rho \varkappa })}{{(1+e^{\rho \varkappa })}^4}}{\displaystyle \frac{{\tau }^{2\epsilon }}{\Gamma (2\epsilon +1)}}+\dots \hfill \end{array}$$

(60)

Application of YTDM

Taking the YT of Equation (54), we obtain

$$Y\left[{\displaystyle \frac{{\partial }^{\epsilon }\mathit{w}}{\partial {\tau }^{\epsilon }}}\right]=Y\left[6\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )\right].$$

(61)

After some simplification, we obtain

$${\displaystyle \frac{1}{u^{\epsilon }}}\{M\left(u\right)-u\mathit{w}\left(0\right)\}=Y\left[6\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )\right],$$

(62)

$$M\left(u\right)=u\mathit{w}\left(0\right)+u^{\epsilon }\left[6\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )\right].$$

(63)

Apply the inverse YT to Equation (63):

$$\begin{array}{cc}\hfill & \mathit{w}(\varkappa ,\tau )=\mathit{w}\left(0\right)+Y^{-1}\left[u^{\epsilon }\left\{Y\left(6\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )\right)\right\}\right],\hfill \\ \hfill & \mathit{w}(\varkappa ,\tau )=\left(-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^2}{{(1+e^{\rho \varkappa })}^2}}\right)+Y^{-1}\left[u^{\epsilon }\left\{Y\left(6\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )\right)\right\}\right].\hfill \end{array}$$

(64)

Assume a series solution of the form:

$$\mathit{w}(\varkappa ,\tau )=\sum _{m=0}^{\infty }{\mathit{w}}_m(\varkappa ,\tau ),$$

(65)

The nonlinear term is calculated as $\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )={\sum }_{m=0}^{\infty }{\mathcal{A}}_m.$ Thus,

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{m=0}^{\infty }}{\mathit{w}}_m(\varkappa ,\tau )=\mathit{w}(\varkappa ,0)+Y^{-1}\left[u^{\epsilon }Y\left[6{\displaystyle \sum _{m=0}^{\infty }}{\mathcal{A}}_m-{\mathit{w}}_{\varkappa }(\varkappa ,\tau )\right]\right],\hfill \\ \hfill & {\displaystyle \sum _{m=0}^{\infty }}{\mathit{w}}_m(\varkappa ,\tau )=\left(-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^2}{{(1+e^{\rho \varkappa })}^2}}\right)+Y^{-1}\left[u^{\epsilon }Y\left[6{\displaystyle \sum _{m=0}^{\infty }}{\mathcal{A}}_m-{\mathit{w}}_{\varkappa }(\varkappa ,\tau )\right]\right].\hfill \end{array}$$

(66)

Using the recursive relationship:

$${\mathit{w}}_0(\varkappa ,\tau )=-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^2}{{(1+e^{\rho \varkappa })}^2}},$$

On $m=0$

$${\mathit{w}}_1(\varkappa ,\tau )=-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^5(-1+e^{\rho \varkappa })}{{(1+e^{\rho \varkappa })}^3}}{\displaystyle \frac{{\tau }^{\epsilon }}{\Gamma (\epsilon +1)}},$$

On $m=1$

$$\begin{array}{cc}\hfill & {\mathit{w}}_2(\varkappa ,\tau )={\displaystyle \frac{2e^{\rho \varkappa }{\rho }^8(-1-e^{2\rho \varkappa }+4e^{\rho \varkappa })}{{(1+e^{\rho \varkappa })}^4}}{\displaystyle \frac{{\tau }^{2\epsilon }}{\Gamma (2\epsilon +1)}}.\hfill \end{array}$$

The final solution can be written as

$$\begin{array}{cc}\hfill & \mathit{w}(\varkappa ,\tau )={\mathit{w}}_0(\varkappa ,\tau )+{\mathit{w}}_1(\varkappa ,\tau )+{\mathit{w}}_2(\varkappa ,\tau )+\dots \hfill \\ \hfill & \mathit{w}(\varkappa ,\tau )=-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^2}{{(1+e^{\rho \varkappa })}^2}}-{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^5(-1+e^{\rho \varkappa })}{{(1+e^{\rho \varkappa })}^3}}{\displaystyle \frac{{\tau }^{\epsilon }}{\Gamma (\epsilon +1)}}+{\displaystyle \frac{2e^{\rho \varkappa }{\rho }^8(-1-e^{2\rho \varkappa }+4e^{\rho \varkappa })}{{(1+e^{\rho \varkappa })}^4}}{\displaystyle \frac{{\tau }^{2\epsilon }}{\Gamma (2\epsilon +1)}}+\dots \hfill \end{array}$$

(67)

Hence, we obtain the exact solution for $\epsilon =1$ as shown below:

$$\mathit{w}(\varkappa ,\tau )=-{\displaystyle \frac{2e^{\rho (\varkappa -{\rho }^2\tau )}{\rho }^2}{{(1+e^{\rho (\varkappa -{\rho }^2\tau )})}^2}},$$

(68)

8.2. Case II

Examine the fractional Sawada–Kotera equation.

$${\displaystyle \frac{{\partial }^{\epsilon }\mathit{w}}{\partial {\tau }^{\epsilon }}}+45{\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )+15{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )+15\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )+{\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )=0,$$

(69)

with

$$\mathit{w}(\varkappa ,0)=2{\rho }^{22}\left(\rho (\varkappa -\varsigma )\right).$$

Application of HPTM

Taking the YT of Equation (69), we obtain

$$Y\left[{\displaystyle \frac{{\partial }^{\epsilon }\mathit{w}}{\partial {\tau }^{\epsilon }}}\right]=Y\left[-45{\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-15{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )-15\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )\right].$$

(70)

After some simplification, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{u^{\epsilon }}}\{M\left(u\right)-u\mathit{w}\left(0\right)\}=Y[-45{\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-15{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )-15\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )-\hfill \\ \hfill & {\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )],\hfill \end{array}$$

(71)

$$\begin{array}{cc}\hfill & M\left(u\right)=u\mathit{w}\left(0\right)+u^{\epsilon }[-45{\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-15{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )-15\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )-\hfill \\ \hfill & {\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )].\hfill \end{array}$$

(72)

Apply the inverse YT to Equation (72):

$$\begin{array}{cc}\hfill & \mathit{w}(\varkappa ,\tau )=\mathit{w}\left(0\right)-Y^{-1}\left[u^{\epsilon }\right\{Y(45{\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )+15{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )+15\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )+\hfill \\ \hfill & {\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )\left)\right\}],\hfill \\ \hfill & \mathit{w}(\varkappa ,\tau )=\left(2{\rho }^{22}\left(\rho (\varkappa -\varsigma )\right)\right)-Y^{-1}\left[u^{\epsilon }\right\{Y(45{\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )+15{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )+\hfill \\ \hfill & 15\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )+{\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )\left)\right\}].\hfill \end{array}$$

(73)

By implementing the homotopy perturbation method, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{k=0}^{\infty }}{ϵ}^k{\mathit{w}}_k(\varkappa ,\tau )=\left(2{\rho }^{22}\left(\rho (\varkappa -\varsigma )\right)\right)+ϵ\left[Y^{-1}\right[u^{\epsilon }Y[45\left({\displaystyle \sum _{k=0}^{\infty }}{ϵ}^k{H}_k^0(\varkappa ,\tau )\right)+15\left({\displaystyle \sum _{k=0}^{\infty }}{ϵ}^k{H}_k^1(\varkappa ,\tau )\right)+\hfill \\ \hfill & 15\left({\displaystyle \sum _{k=0}^{\infty }}{ϵ}^k{H}_k^2(\varkappa ,\tau )\right)+{\left({\displaystyle \sum _{k=0}^{\infty }}{ϵ}^k{\mathit{w}}_k(\varkappa ,\tau )\right)}_{\varkappa \varkappa \varkappa \varkappa \varkappa }\left]\right]].\hfill \end{array}$$

(74)

Using the recursive relationship

$$\begin{array}{cc}\hfill & {ϵ}^0:{\mathit{w}}_0(\varkappa ,\tau )=2{\rho }^{22}\left(\rho (\varkappa -\varsigma )\right),\hfill \\ \hfill & {ϵ}^1:{\mathit{w}}_1(\varkappa ,\tau )=-{\displaystyle \frac{64{\rho }^{7}sinh\left(\rho (-\varkappa +\varsigma )\right)}{cosh{\left(\rho (-\varkappa +\varsigma )\right)}^3}}{\displaystyle \frac{{\tau }^{\epsilon }}{\Gamma (\epsilon +1)}},\hfill \\ \hfill & {ϵ}^2:{\mathit{w}}_2(\varkappa ,\tau )={\displaystyle \frac{1024{\rho }^{12}(2cosh{\left(\rho (-\varkappa +\varsigma )\right)}^2-3)}{cosh{\left(\rho (-\varkappa +\varsigma )\right)}^4}}{\displaystyle \frac{{\tau }^{2\epsilon }}{\Gamma (2\epsilon +1)}},\hfill \\ \hfill & ⋮\hfill \end{array}$$

The final solution can be written as

$$\begin{array}{cc}\hfill & \mathit{w}(\varkappa ,\tau )={\mathit{w}}_0(\varkappa ,\tau )+{\mathit{w}}_1(\varkappa ,\tau )+{\mathit{w}}_2(\varkappa ,\tau )+\dots \hfill \\ \hfill & \mathit{w}(\varkappa ,\tau )=2{\rho }^{22}\left(\rho (\varkappa -\varsigma )\right)-{\displaystyle \frac{64{\rho }^{7}sinh\left(\rho (-\varkappa +\varsigma )\right)}{cosh{\left(\rho (-\varkappa +\varsigma )\right)}^3}}{\displaystyle \frac{{\tau }^{\epsilon }}{\Gamma (\epsilon +1)}}+{\displaystyle \frac{1024{\rho }^{12}(2cosh{\left(\rho (-\varkappa +\varsigma )\right)}^2-3)}{cosh{\left(\rho (-\varkappa +\varsigma )\right)}^4}}\hfill \\ \hfill & {\displaystyle \frac{{\tau }^{2\epsilon }}{\Gamma (2\epsilon +1)}}+\dots \hfill \end{array}$$

(75)

Application of YTDM

Taking the YT of Equation (69), we obtain

$$Y\left[{\displaystyle \frac{{\partial }^{\epsilon }\mathit{w}}{\partial {\tau }^{\epsilon }}}\right]=Y\left[-45{\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-15{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )-15\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )-{\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )\right].$$

(76)

After some simplification, we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{u^{\epsilon }}}\{M\left(u\right)-u\mathit{w}\left(0\right)\}=Y[-45{\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-15{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )-15\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )-\hfill \\ \hfill & {\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )],\hfill \end{array}$$

(77)

$$\begin{array}{cc}\hfill & M\left(u\right)=u\mathit{w}\left(0\right)+u^{\epsilon }[-45{\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )-15{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )-15\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )-\hfill \\ \hfill & {\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )].\hfill \end{array}$$

(78)

Apply the inverse YT to Equation (78):

$$\begin{array}{cc}\hfill & \mathit{w}(\varkappa ,\tau )=\mathit{w}\left(0\right)-Y^{-1}\left[u^{\epsilon }\right\{Y(45{\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )+15{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )+15\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )\hfill \\ \hfill & +{\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )\left)\right\}],\hfill \\ \hfill & \mathit{w}(\varkappa ,\tau )=\left(2{\rho }^{22}\left(\rho (\varkappa -\varsigma )\right)\right)-Y^{-1}\left[u^{\epsilon }\right\{Y(45{\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )+15{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )+\hfill \\ \hfill & 15\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )+{\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )\left)\right\}].\hfill \end{array}$$

(79)

Assume a series solution of the form:

$$\mathit{w}(\varkappa ,\tau )=\sum _{m=0}^{\infty }{\mathit{w}}_m(\varkappa ,\tau ).$$

(80)

The nonlinear term is calculated as ${\mathit{w}}^2(\varkappa ,\tau ){\mathit{w}}_{\varkappa }(\varkappa ,\tau )={\sum }_{m=0}^{\infty }{\mathcal{A}}_m,{\mathit{w}}_{\varkappa }(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa }(\varkappa ,\tau )={\sum }_{m=0}^{\infty }{\mathcal{B}}_m,\mathit{w}(\varkappa ,\tau ){\mathit{w}}_{\varkappa \varkappa \varkappa }(\varkappa ,\tau )={\sum }_{m=0}^{\infty }{\mathcal{C}}_m.$ Thus,

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{m=0}^{\infty }}{\mathit{w}}_m(\varkappa ,\tau )=\mathit{w}(\varkappa ,0)-Y^{-1}\left[u^{\epsilon }Y\left[45{\displaystyle \sum _{m=0}^{\infty }}{\mathcal{A}}_m+15{\displaystyle \sum _{m=0}^{\infty }}{\mathcal{B}}_m+15{\displaystyle \sum _{m=0}^{\infty }}{\mathcal{C}}_m+{\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )\right]\right],\hfill \\ \hfill & {\displaystyle \sum _{m=0}^{\infty }}{\mathit{w}}_m(\varkappa ,\tau )=\left(2{\rho }^{22}\left(\rho (\varkappa -\varsigma )\right)\right)-Y^{-1}\left[u^{\epsilon }Y\left[45{\displaystyle \sum _{m=0}^{\infty }}{\mathcal{A}}_m+15{\displaystyle \sum _{m=0}^{\infty }}{\mathcal{B}}_m+15{\displaystyle \sum _{m=0}^{\infty }}{\mathcal{C}}_m+{\mathit{w}}_{\varkappa \varkappa \varkappa \varkappa \varkappa }(\varkappa ,\tau )\right]\right].\hfill \end{array}$$

(81)

Using the recursive relationship:

$${\mathit{w}}_0(\varkappa ,\tau )=2{\rho }^{22}\left(\rho (\varkappa -\varsigma )\right),$$

On $m=0$

$${\mathit{w}}_1(\varkappa ,\tau )=-{\displaystyle \frac{64{\rho }^{7}sinh\left(\rho (-\varkappa +\varsigma )\right)}{cosh{\left(\rho (-\varkappa +\varsigma )\right)}^3}}{\displaystyle \frac{{\tau }^{\epsilon }}{\Gamma (\epsilon +1)}}.$$

On $m=1$

$$\begin{array}{cc}\hfill & {\mathit{w}}_2(\varkappa ,\tau )={\displaystyle \frac{1024{\rho }^{12}(2cosh{\left(\rho (-\varkappa +\varsigma )\right)}^2-3)}{cosh{\left(\rho (-\varkappa +\varsigma )\right)}^4}}{\displaystyle \frac{{\tau }^{2\epsilon }}{\Gamma (2\epsilon +1)}}.\hfill \end{array}$$

The final solution can be expressed as

$$\begin{array}{cc}\hfill & \mathit{w}(\varkappa ,\tau )={\mathit{w}}_0(\varkappa ,\tau )+{\mathit{w}}_1(\varkappa ,\tau )+{\mathit{w}}_2(\varkappa ,\tau )+\dots \hfill \\ \hfill & \mathit{w}(\varkappa ,\tau )=2{\rho }^{22}\left(\rho (\varkappa -\varsigma )\right)-{\displaystyle \frac{64{\rho }^{7}sinh\left(\rho (-\varkappa +\varsigma )\right)}{cosh{\left(\rho (-\varkappa +\varsigma )\right)}^3}}{\displaystyle \frac{{\tau }^{\epsilon }}{\Gamma (\epsilon +1)}}+{\displaystyle \frac{1024{\rho }^{12}(2cosh{\left(\rho (-\varkappa +\varsigma )\right)}^2-3)}{cosh{\left(\rho (-\varkappa +\varsigma )\right)}^4}}\hfill \\ \hfill & {\displaystyle \frac{{\tau }^{2\epsilon }}{\Gamma (2\epsilon +1)}}+\dots \hfill \end{array}$$

(82)

Hence, we obtain the exact solution for $\epsilon =1$ as follows

$$\mathit{w}(\varkappa ,\tau )=2{\rho }^{22}\left(\rho (\varkappa -16{\rho }^4\varsigma \tau )\right).$$

(83)

• Results and Discussion

The first numerical solutions to the Caputo fractional problems using the newly proposed techniques, HPTM and YTDM, are presented in this article. The inclusion of the Caputo fractional operator in the models is crucial for capturing anomalous diffusion and memory effects. MATHEMATICA 13.3 software was used to generate 2D and 3D graphs to visualize the numerical results. These graphical representations illustrate that the surface structures vary significantly with different values of $\epsilon$. The solution $\mathit{w}(\varkappa ,\tau )$ for different fractional orders $\epsilon$ is graphically compared in Figure 1, which is divided into four sections. The 3D surface of the suggested methods solution at $\epsilon =0.4$ is shown in Figure 1a. The behavior at $\epsilon =0.7$ is depicted in Figure 1b, and the solution at $\epsilon =1$ is shown in Figure 1c. A 2D comparison of the solution $\mathit{w}(\varkappa ,\tau )$ at $\tau =0.1$ is shown in Figure 1d, highlighting the similarity of the proposed approaches. The exact and approximate solutions $\mathit{w}(\varkappa ,\tau )$ for $\tau =0.1$ of Equation (54) are shown in both 3D and 2D in Figure 2. Similarly, Figure 3 shows the error comparison between the exact and approximate solutions for Case (I). The obtained approximations are shown in Figure 1, Figure 2 and Figure 3 for a range of $ϵ$ values, with $-1\le \varkappa \le 1$ and temporal variable $0\le \tau \le 0.1$.

Table 1. Comparison between the exact and HPTM/YTDM solutions for $\mathit{w}(\varkappa ,\tau )$.

$\mathit{\tau }$	$\mathit{\varkappa }$	$\mathit{\epsilon }=0.85$	$\mathit{\epsilon }=0.90$	$\mathit{\epsilon }=0.95$	$\mathit{\epsilon }=1(\mathit{HPTM}/\mathit{YTDM})$	$\mathit{\epsilon }=1\left(\mathit{exact}\right)$
	0.0	−0.0049999999	−0.0050000000	−0.0050000000	−0.0050000000	−0.0050000000
	0.2	−0.0049995010	−0.0049995008	−0.0049995006	−0.0049995005	−0.0049995005
0.01	0.4	−0.0049980026	−0.0049980021	−0.0049980018	−0.0049980015	−0.0049980015
	0.6	−0.0049955058	−0.0049955051	−0.0049955046	−0.0049955041	−0.0049955041
	0.8	−0.0049920127	−0.0049920118	−0.0049920110	−0.0049920105	−0.0049920105
	1.0	−0.0049875260	−0.0049875249	−0.0049875240	−0.0049875232	−0.0049875232
	0.0	−0.0049999999	−0.0049999999	−0.0049999999	−0.0050000000	−0.0050000000
	0.2	−0.0049995027	−0.0049995022	−0.0049995018	−0.0049995015	−0.0049995015
0.03	0.4	−0.0049980058	−0.0049980049	−0.0049980041	−0.0049980035	−0.0049980035
	0.6	−0.0049955107	−0.0049955093	−0.0049955081	−0.0049955071	−0.0049955071
	0.8	−0.0049920192	−0.0049920173	−0.0049920158	−0.0049920145	−0.0049920145
	1	−0.0049875341	−0.0049875318	−0.0049875298	−0.0049875282	−0.0049875282
	0.0	−0.0049999999	−0.0049999999	−0.0049999999	−0.0049999999	−0.0049999999
	0.2	−0.0049995041	−0.0049995035	−0.0049995029	−0.0049995025	−0.0049995025
0.05	0.4	−0.0049980088	−0.0049980075	−0.0049980064	−0.0049980055	−0.0049980055
	0.6	−0.0049955151	−0.0049955132	−0.0049955115	−0.0049955101	−0.0049955101
	0.8	−0.0049920250	−0.0049920225	−0.0049920203	−0.0049920185	−0.0049920185
	1	−0.0049875414	−0.0049875382	−0.0049875355	−0.0049875332	−0.0049875332

displays both the exact solution and the approximate solutions derived using the proposed methods.

Table 2. Comparison among HPTM/YTDM, NITM, and OFAM solutions for $\mathit{w}(\varkappa ,\tau )$.

$\mathit{\varkappa }$	Exact	$\mathit{HPTM}/\mathit{YTDM}$ Solution	$\mathit{HPTM}/\mathit{YTDM}$ Error	$\mathit{NTIM}$ Error	$\mathit{OAFM}$ Error
0.25	−0.0049992250	−0.0049992250	1.0000000000 $\times {10}^{-12}$	1.24945 $\times {10}^{-8}$	1.25899 $\times {10}^{-9}$
0.50	−0.0049968887	−0.0049968887	2.0000000000 $\times {10}^{-12}$	2.49558 $\times {10}^{-8}$	2.97366 $\times {10}^{-9}$
0.75	−0.0049929940	−0.0049929940	1.0000000000 $\times {10}^{-12}$	3.73509 $\times {10}^{-8}$	1.15346 $\times {10}^{-9}$
1.0	−0.0049875457	−0.0049875457	1.0000000000 $\times {10}^{-12}$	4.96471 $\times {10}^{-8}$	1.26266 $\times {10}^{-8}$

compares the absolute error of the proposed techniques with the natural transform iterative method (NTIM) and optimal auxiliary function method (OAFM). The comparison of the exact and analytical results reveals a high level of agreement. Accordingly, the proposed methods offer reliable and innovative approaches that require less computational time and are more straightforward and flexible than NTIM and OAFM. Figure 4 illustrates the solution $\mathit{w}(\varkappa ,\tau )$ for different fractional orders $\epsilon$. The result at $\epsilon =0.4$ is shown in Figure 4a, while Figure 4b examines the behavior at $\epsilon =0.7$, demonstrating improved wave characteristics compared to lower orders. Figure 4c corresponds to $\epsilon =1$, representing the classical integer-order scenario with different waveforms. A 2D comparison of the solution $\mathit{w}(\varkappa ,\tau )$ at $\tau =0.1$ is shown in Figure 4d, emphasizing the similarity of the various approaches. Both 3D and 2D visualizations of the exact and estimated solutions $\mathit{w}(\varkappa ,\tau )$ for $\tau =0.1$ of Equation (69) are shown in Figure 5. Similarly, Figure 6 shows the error comparison of the exact and approximate results of Case (II). Figure 4, Figure 5 and Figure 6 cover a range of $ϵ$ values with $-1\le \varkappa \le 1$ and temporal variable $0\le \tau \le 0.1$. The exact solution and the approximate solutions generated via the suggested methods are displayed in

Table 3. Comparison between the exact and HPTM/YTDM solutions for $\mathit{w}(\varkappa ,\tau )$.

$\mathit{\tau }$	$\mathit{\varkappa }$	$\mathit{\epsilon }=0.85$	$\mathit{\epsilon }=0.90$	$\mathit{\epsilon }=0.95$	$\mathit{\epsilon }=1(\mathit{HPTM}/\mathit{YTDM})$	$\mathit{\epsilon }=1\left(\mathit{exact}\right)$
	0.0	0.0199979987	0.0199979990	0.0199979993	0.0199979994	0.0200000000
	0.2	0.0199980014	0.0199980011	0.0199980009	0.0199980007	0.0199920022
0.01	0.4	0.0199820148	0.0199820139	0.0199820132	0.0199820127	0.0199680343
	0.6	0.0199500899	0.0199500884	0.0199500873	0.0199500864	0.0199281728
	0.8	0.0199023286	0.0199023265	0.0199023249	0.0199023237	0.0198725446
	1.0	0.0198388828	0.0198388801	0.0198388781	0.0198388765	0.0198013264
	0.0	0.0199979966	0.0199979973	0.0199979978	0.0199979982	0.0200000000
	0.2	0.0199980035	0.0199980029	0.0199980024	0.0199980020	0.0199920025
0.03	0.4	0.0199820210	0.0199820192	0.0199820177	0.0199820165	0.0199680348
	0.6	0.0199501003	0.0199500973	0.0199500948	0.0199500927	0.0199281736
	0.8	0.0199023431	0.0199023389	0.0199023354	0.0199023326	0.0198725456
	1	0.0198389013	0.0198388960	0.0198388915	0.0198388878	0.0198013277
	0.0	0.0199979948	0.0199979956	0.0199979963	0.0199979969	0.0200000000
	0.2	0.0199980054	0.0199980046	0.0199980039	0.0199980033	0.0199920027
0.05	0.4	0.0199820266	0.0199820242	0.0199820221	0.0199820203	0.0199680353
	0.6	0.0199501096	0.0199501055	0.0199501021	0.0199500991	0.0199281743
	0.8	0.0199023561	0.0199023504	0.0199023456	0.0199023415	0.0198725467
	1	0.0198389180	0.0198389107	0.0198389045	0.0198388992	0.0198013289

. In addition, the comparison of the proposed techniques with NTIM and OAFM in terms of absolute error is shown in

Table 4. Comparative analysis of HPTM/YTDM with NITM and OFAM for $\mathit{w}(\varkappa ,\tau )$.

$\mathit{\varkappa }$	Exact	$\mathit{HPTM}/\mathit{YTDM}$ Solution	$\mathit{HPTM}/\mathit{YTDM}$ Error	$\mathit{NTIM}$ Error	$\mathit{OAFM}$ Error
0.1	0.0199980007	0.0200000000	1.99922000 $\times {10}^{-6}$	1.99922 $\times {10}^{-6}$	1.4687 $\times {10}^{-5}$
0.2	0.0199980014	0.0199999999	1.99857000 $\times {10}^{-6}$	1.99857 $\times {10}^{-6}$	2.93741 $\times {10}^{-5}$
0.3	0.0199980020	0.0199999999	1.99790000 $\times {10}^{-6}$	1.9979 $\times {10}^{-6}$	4.40611 $\times {10}^{-5}$
0.4	0.0199980026	0.0199999999	1.99723000 $\times {10}^{-6}$	1.99723 $\times {10}^{-6}$	5.87481 $\times {10}^{-5}$
0.5	0.0199980033	0.0199999998	1.99654000 $\times {10}^{-6}$	1.99654 $\times {10}^{-6}$	7.34351 $\times {10}^{-5}$

. The graphical and tabular representations confirm that HPTM and YTDM are not only simple to implement but also capable of producing solutions that are highly accurate and closely aligned with the exact solution.

9. Conclusions

Finding exact solutions to fractional partial differential equations (FPDEs), especially when applied to complex models such as the Sawada–Kotera equation and the KdV–Burgers equations, is often a challenging task. The YTDM and HPTM approaches offer innovative techniques for solving FPDEs with high accuracy and efficacy. These approaches offer a robust framework for addressing complex nonlinear FDEs, particularly when applied with the Caputo fractional operator. The findings enhance our understanding of diffusion and wave dynamics in non-integer systems by demonstrating the significant impact of the fractional parameter on solution behavior. The characteristics of the resulting series solutions were illustrated using 2D and 3D graphs for various fractional orders. The proposed solutions, as confirmed by the presented tables and figures, are highly accurate. To support the theoretical results, graphical illustrations were used to confirm the practical relevance of the solutions in modeling physical phenomena. This study demonstrates the efficiency and reliability of YTDM and HPTM as powerful tools for analyzing fractional-order equations. These methods not only simplify the solution process but also offer new perspectives into the dynamics of fractional systems. They hold significant potential for applications in diverse fields, such as nonlinear dynamics, fluid mechanics, and wave propagation. Future studies could explore the application of these methodologies in interdisciplinary contexts and extend them to even more complex fractional models.