On the Solutions of the Fractional-Order Sawada–Kotera–Ito Equation and Modeling Nonlinear Structures in Fluid Mediums

Abstract

This study investigates the wave solutions of the time-fractional Sawada–Kotera–Ito equation (SKIE) that arise in shallow water and many other fluid mediums by utilizing some of the most flexible and high-precision methods. The SKIE is a nonlinear integrable partial differential equation (PDE) with significant applications in shallow water dynamics and fluid mechanics. However, the traditional numerical methods used for analyzing this equation are often plagued by difficulties in handling the fractional derivatives (FDs), which lead to finding other techniques to overcome these difficulties. To address this challenge, the Adomian decomposition (AD) transform method (ADTM) and homotopy perturbation transform method (HPTM) are employed to obtain exact and numerical solutions for the time-fractional SKIE. The ADTM involves decomposing the fractional equation into a series of polynomials and solving each component iteratively. The HPTM is a modified perturbation method that uses a continuous deformation of a known solution to the desired solution. The results show that both methods can produce accurate and stable solutions for the time-fractional SKIE. In addition, we compare the numerical solutions obtained from both methods and demonstrate the superiority of the HPTM in terms of efficiency and accuracy. The study provides valuable insights into the wave solutions of shallow water dynamics and nonlinear waves in plasma, and has important implications for the study of fractional partial differential equations (FPDEs). In conclusion, the method offers effective and efficient solutions for the time-fractional SKIE and demonstrates their usefulness in solving nonlinear integrable PDEs.

1. Introduction

The study of nonlinear PDEs is of great importance in many areas of science and engineering. In recent years, time-fractional PDEs (TFPDEs) have received increasing attention due to their ability to model anomalous diffusion and memory effects in various fields such as physics, engineering, and finance [1,2]. The seventh-order time-fractional SKIE is an example of a TFPDE used to describe various physical phenomena. This equation is a generalization of the classical Sawada–Kotera equation, which was introduced to describe the dynamics of plasma physics. The fractional order in the time-fractional SKIE represents a memory effect that accounts for the influence of previous values of the system on its current state [3,4,5,6,7]. This feature makes the equation more suitable for modeling systems with long-term memory and non-local interactions. This equation is formulated using fractional derivatives, extensions of the classical derivatives to non-integer orders. These derivatives provide a more flexible and accurate representation of the system behavior than classical derivatives, making the time-fractional SKIE a powerful tool for modeling complex systems with memory and non-local interactions [8,9].

The seventh-order time-fractional SKIE has found applications in a wide range of fields, including finance, where it is used to model the volatility of financial markets, and physics, where it is used to describe the propagation of waves in complex media such as composite materials and porous media. It has also been used in engineering to model the dynamics of electrical and mechanical systems, and study chaotic and nonlinear systems [10,11,12,13]. In summary, the seventh-order time-fractional SKIE is a powerful and versatile mathematical model that has found numerous applications in various fields due to its ability to accurately describe complex physical phenomena with memory and non-local interactions [14,15,16,17]. The Kortweg–de Vries (KdV) equation and related equations with higher-order nonlinearities and dispersions are nonlinear PDEs used to model the behavior of traveling waves in shallow water and plasma physics [18,19,20,21,22,23,24]. Boussinesq first proposed this equation in 1877 and it was later developed by Kortweg and de Vries in 1895. In a study conducted by Pomeau et al. [25], the well-known seventh-order KdV equation was introduced and its stability was analyzed in the presence of a specific type of perturbation. Additionally, the time-fractional SKIE was also explored in this research.

$$\begin{array}{cc}\hfill & D_{\Im }^{ß}\mathcal{V}(\mathbf{Ø},\wp )=-252{\mathcal{V}}^3(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}}^3(\mathbf{Ø},\wp )-378\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp )-126{\mathcal{V}}^2(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )\hfill \\ \hfill & -63{\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-42{\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{4\mathbf{Ø}}(\mathbf{Ø},\wp )-21\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{5\mathbf{Ø}}(\mathbf{Ø},\wp )-{\mathcal{V}}_{7\mathbf{Ø}}(\mathbf{Ø},\wp ),0<ß\le 1,\hfill \end{array}$$

(1)

with the initial condition (IC)

$$\mathcal{V}(\mathbf{Ø},0)=\frac{4}{3}{\rho }^2(2-3{\tanh }^2(\rho \mathbf{Ø})).$$

(2)

Recently, several methods have been proposed for solving the family of the seventh-order time-fractional SKIE. These methods include the Adomian decomposition method (ADM), homotopy analysis scheme (HAT), q-homotopy analysis approach, fractional reduced differential transform technique, $\left(\frac{G^{\prime }}{G}\right)$-expansion method, Lie symmetry analysis, and the exp-function method. These techniques are explained in various references such as [26,27,28,29,30,31,32].

The ADTM is a mathematical approach to solving nonlinear differential equations (NLDEs). It was introduced by George Adomian in the mid-1980s and has since become a popular and well-respected technique for solving a wide range of nonlinear problems. This method decomposes the NLDE into a series of linear terms, each of which can be solved analytically [33,34]. The sum of these solutions provides an approximate solution to the nonlinear equation. The ADM is particularly useful when the traditional numerical or analytical methods are too complex or fail to produce accurate results. The method has been applied successfully in various fields, including physics, engineering, and economics [35,36,37,38,39,40].

The HPTM is a mathematical technique for solving nonlinear problems. He introduced it in 1993 and it has since been widely used in various fields such as engineering, physics, and mathematics [41,42,43]. The method is based on the homotopy perturbation theory and offers a simple and efficient way to obtain approximate solutions to nonlinear problems. The main idea of HPTM is to introduce a homotopy deformation between the original nonlinear problem and a linear problem, which can be easily solved [40,44,45,46,47]. The solutions of the linear problem are then used to obtain the approximate solutions of the original nonlinear problem [48,49,50,51].

Section 2 provides the fundamental definitions of ZZ transformation, the ZZ transform, and its properties. Section 3 introduces the concept of HPTM, whereas Section 4 introduces the concept of ZTDM. In Section 5, it is illustrated how it can be used with the nonlinear Sawada–Kotera–Ito equations. In Section 6, we provide a concluding analysis.

2. Preliminaries

Definition 1.

The Aboodh transformation on functions is achieved by

$$B=\left\{U\left(\varrho \right):\exists M,n_1,n_2>0,\left|U\left(\varrho \right)\right|<M{e}^{-\epsilon \varrho }\right\},$$

and is expressed as [52,53]

$$A\left\{U\left(\varrho \right)\right\}=\frac{1}{\epsilon }{\int }_0^{\infty }U\left(\varrho \right)e^{-\epsilon \varrho }d\varrho ,\varrho >0\mathit{and}{n}_1\le \epsilon \le n_2.$$

Theorem 1.

Let us examine G and F as the Laplace and Aboodh transformations of $U\left(\varrho \right)$ in the set B [54,55]

$$G\left(\epsilon \right)=\frac{F\left(\epsilon \right)}{\epsilon }.$$

(3)

The ZZ transformation, introduced by Zain Ul Abadin Zafar [56], is a generalization of the Laplace and Aboodh integral transformations. The definition of the ZZ transformation is as follows:

Definition 2.

(ZZ Transformation) The ZZ transformation $Z(\kappa ,\epsilon )$ of the function $U\left(\varrho \right)$ for all values of $\varrho \ge 0$ can be represented as follows [56]

$$ZZ\left(U\left(\varrho \right)\right)=Z(\kappa ,\epsilon )=\epsilon {\int }_0^{\infty }U\left(\kappa \varrho \right)e^{-\epsilon \varrho }d\varrho .$$

Like the Laplace and Aboodh transforms, the ZZ transform is also linear in nature. The MLF, on the other hand, is an expansion of the exponential function.

$$E_{\delta }\left(z\right)=\sum _{m=0}^{\infty }\frac{z^m}{\mathrm{\Gamma }(1+m\delta )},\mathrm{Re}\left(\delta \right)>0.$$

Definition 3.

The Atangana–Baleanu Caputo derivative of a function $U(\phi ,\varrho )$ belonging to the space $H^1(a,b)$ is defined as follows for $ß\in$ $(0,1)$ [57]

$${ABC}_a{D}_{\varrho }^{ß}U(\phi ,\varrho )=\frac{B(ß)}{1-ß}{\int }_a^{\varrho }{U}^{\prime }(\phi ,\varrho )E_{ß}\left(\frac{-ß{(\varrho -\eta )}^{ß}}{1-ß}\right)d\eta .$$

Definition 4.

The Atangana–Baleanu Riemann–Liouville derivative, denoted as $U(\phi ,\varrho )$, is a member of the space $H^1(a,b)$. For any value of $ß$ within the interval $(0,1)$, the derivative can be expressed as [57]

$${}_a^{ABR}{D}_{\varrho }^{ß}U(\phi ,\eta )=\frac{B(ß)}{1-ß}\frac{d}{d\varrho }{\int }_a^{\varrho }U(\phi ,\eta )E_{ß}\left(\frac{-ß{(\varrho -\eta )}^{ß}}{1-ß}\right)d\eta ,$$

The function $B(ß)$ has the property that it evaluates to 1 for both 0 and 1. Furthermore, the value of $B(ß)$ is always greater than a when $ß$ is greater than 0.

Theorem 2.

The Laplace transformation for the Atangana–Baleanu Caputo derivative and the Atangana–Baleanu Riemann–Liouville derivative are defined as follows [57]

$$\mathrm{L}\left\{{}_a^{ABC}{D}_{\varrho }^{ß}U(\phi ,\varrho )\right\}\left(\epsilon \right)=\frac{B(ß)}{1-ß}\frac{{\epsilon }^{ß}L\left\{U(\phi ,\varrho )\right\}-{\epsilon }^{ß-1}U(\phi ,0)}{{\epsilon }^{ß}+\frac{ß}{1-ß}},$$

(4)

and

$$L\left\{\begin{array}{c}ABR\hfill \\ a\hfill \end{array}{D}_{\varrho }^{ß}U(\phi ,\varrho )\right\}\left(\epsilon \right)=\frac{B(ß)}{1-ß}\frac{{\epsilon }^{ß}L\left\{U(\phi ,\varrho )\right\}}{{\epsilon }^{ß}+\frac{ß}{1-ß}}.$$

(5)

The following theorems have been proposed with the assumption that $U\left(\varrho \right)$ belongs to the Sobolev space $H^1(a,b)$, where $b>a$, and ß is a value in the interval $(0,1)$.

Theorem 3.

The Atangana–Baleanu Riemann–Liouville derivative is given a new form through the Aboodh transform, which is referred to as the Aboodh transformed Atangana–Baleanu Riemann–Liouville derivative [55]

$$G\left(\epsilon \right)=A\left\{\begin{array}{c}ABR\hfill \\ a\hfill \end{array}{D}_{\varrho }^{ß}U(\phi ,\varrho )\right\}\left(\epsilon \right)=\frac{1}{\epsilon }\left[\frac{B(ß)}{1-ß}\frac{{\epsilon }^{ß}L\left\{U(\phi ,\varrho )\right\}}{{\epsilon }^{ß}+\frac{ß}{1-ß}}\right].$$

(6)

Proof. 

By utilizing Theorem 1 and Equation (5), we obtain the necessary solution. The relationship between the ZZ and Aboodh transformations is outlined in the following theorem. □

Theorem 4.

The definition of the AT of the Atangana–Baleanu Caputo derivative reads [55]

$$\begin{array}{cc}\hfill G\left(\epsilon \right)=& A\left\{\begin{array}{c}ABC\hfill \\ a\hfill \end{array}{D}_{\varrho }^{ß}U(\phi ,\varrho )\right\}\left(\epsilon \right)=\frac{1}{\epsilon }\left[\frac{B(ß)}{1-ß}\frac{{\epsilon }^{ß}L\left\{U(\phi ,\varrho )\right\}-{\epsilon }^{ß-1}U(\phi ,0)}{{\epsilon }^{ß}+\frac{ß}{1-ß}}\right].\hfill \end{array}$$

(7)

Proof. 

By utilizing Theorem 2.1 and Equation (4), the solution can be obtained. □

Theorem 5.

We attain the ZZ and Aboodh transformations of $U\left(\varrho \right)\in B$, represented as $G\left(\epsilon \right)$ and $Z(\kappa ,\epsilon )$, respectively [55]

$$Z(\kappa ,\epsilon )=\frac{{\epsilon }^2}{{\kappa }^2}G\left(\frac{\epsilon }{\kappa }\right).$$

Proof. 

According to the Z transform definition,

$$Z(\kappa ,\epsilon )=\epsilon {\int }_0^{\infty }U\left(\kappa \varrho \right)e^{-\epsilon \varrho }d\varrho .$$

(8)

By substituting $\kappa \varrho =\varrho$ into Equation (8), we obtain

$$Z(\kappa ,\epsilon )=\frac{\epsilon }{\kappa }{\int }_0^{\infty }U\left(\varrho \right)e^{-\frac{\pi z}{\kappa }}d\varrho .$$

(9)

The expression on the right side of Equation (9) can be rephrased as

$$Z(\kappa ,\epsilon )=\frac{\epsilon }{\kappa }F\left(\frac{\epsilon }{\kappa }\right),$$

(10)

By utilizing Theorem 1 Equation (10) can be reinterpreted as the Laplace transformation of $U\left(\varrho \right)$, represented as $F(.)$.

$$Z(\kappa ,\epsilon )=\frac{\epsilon }{\kappa }\frac{F\left(\frac{\epsilon }{\kappa }\right)}{\left(\frac{\epsilon }{\kappa }\right)}\times \left(\frac{\epsilon }{\kappa }\right)={\left(\frac{\epsilon }{\kappa }\right)}^{2}G\left(\frac{\epsilon }{\kappa }\right),$$

(11)

The Aboodh transformation, represented by $G(.)$, transforms the function $U\left(\varrho \right)$. □

Theorem 6.

The $\mathrm{ZZ}$ transformation of $U\left(\varrho \right)={\varrho }^{ß-1}$ is defined as

$$Z(\kappa ,\epsilon )=\mathrm{\Gamma }(ß){\left(\frac{\kappa }{\epsilon }\right)}^{ß-1}.$$

(12)

Proof. 

The Aboodh transformation of $U\left(\varrho \right)={\varrho }^{ß},ß\ge 0$ is

$$G\left(\epsilon \right)=\frac{\mathrm{\Gamma }(ß)}{{\epsilon }^{ß+1}},$$

$$\mathrm{Now},G\left(\frac{\epsilon }{\kappa }\right)=\frac{\mathrm{\Gamma }(ß){\kappa }^{ß+1}}{{\epsilon }^{ß+1}}.$$

Applying Equation (12), we achieve

$$Z(\kappa ,\epsilon )=\frac{{\epsilon }^2}{{\kappa }^2}G\left(\frac{\epsilon }{\kappa }\right)=\frac{{\epsilon }^2}{{\kappa }^2}\frac{\mathrm{\Gamma }(ß){\kappa }^{ß+1}}{{\epsilon }^{ß+1}}=\mathrm{\Gamma }(ß){\left(\frac{\kappa }{\epsilon }\right)}^{ß-1}.$$

□

Theorem 7.

Let ß and ω be complex numbers and assume that the real part of ß is greater than 0. The $\mathrm{ZZ}$ transformation of $E_{ß}\left(\omega {\varrho }^{ß}\right)$ can be defined as [55]

$$ZZ\left\{\left(E_{ß}\left(\omega {\varrho }^{ß}\right)\right)\right\}=Z(\kappa ,\epsilon )={\left(1-\omega {\left(\frac{\kappa }{\epsilon }\right)}^{ß}\right)}^{-1}.$$

(13)

Proof. 

The Aboodh transformation of $E_{ß}\left(\omega {\varrho }^{ß}\right)$ is defined as

$$G\left(\epsilon \right)=\frac{F\left(\epsilon \right)}{\epsilon }=\frac{{\epsilon }^{ß-1}}{\epsilon \left({\epsilon }^{ß}-\omega \right)},$$

(14)

So,

$$G\left(\frac{\epsilon }{\kappa }\right)=\frac{{\left(\frac{\epsilon }{\kappa }\right)}^{ß-1}}{\left(\frac{\epsilon }{\kappa }\right)\left({\left(\frac{\epsilon }{\kappa }\right)}^{ß}-\omega \right)}.$$

(15)

Applying theorem we achieve

$$\begin{array}{c}Z(\kappa ,\epsilon )={\left(\frac{\epsilon }{\kappa }\right)}^{2}G\left(\frac{\epsilon }{\kappa }\right)={\left(\frac{\epsilon }{\kappa }\right)}^2\frac{{\left(\frac{\epsilon }{\kappa }\right)}^{ß-1}}{\left(\frac{\epsilon }{\kappa }\right)\left({\left(\frac{\epsilon }{\kappa }\right)}^{ß}-\omega \right)}\\ =\frac{{\left(\frac{\epsilon }{\kappa }\right)}^{ß}}{{\left(\frac{\epsilon }{\kappa }\right)}^{ß}-\omega }={\left(1-\omega {\left(\frac{\kappa }{\epsilon }\right)}^{ß}\right)}^{-1}.\end{array}$$

□

Theorem 8.

The ZZ transformation of the Atangana–Baleanu Caputo derivative can be defined as follows: If $G\left(\epsilon \right)$ and $Z(\kappa ,\epsilon )$ are the ZZ and Aboodh transformations of $U\left(\varrho \right)$, respectively [55]

$$ZZ\left\{\begin{array}{c}ABC\hfill \\ 0\hfill \end{array}{D}_{\varrho }^{ß}U\left(\varrho \right)\right\}=\left[\frac{B(ß)}{1-ß}\frac{\frac{{\epsilon }^{a+2}}{{\kappa }^{ß+2}}G\left(\frac{\epsilon }{\kappa }\right)-\frac{{\epsilon }^{ß}}{{\kappa }^{ß}}f\left(0\right)}{\frac{{\epsilon }^{ß}}{{\kappa }^{ß}}+\frac{ß}{1-ß}}\right].$$

(16)

Proof. 

Applying Equations (3) and (7), we get

$$G\left(\frac{\epsilon }{\kappa }\right)=\frac{\kappa }{\epsilon }\left[\frac{B(ß)}{1-ß}\frac{{\left(\frac{\epsilon }{\kappa }\right)}^{ß+1}G\left(\frac{\epsilon }{\kappa }\right)-{\left(\frac{\epsilon }{\kappa }\right)}^{ß-1}f\left(0\right)}{{\left(\frac{\epsilon }{\kappa }\right)}^{ß}+\frac{ß}{1-ß}}\right].$$

(17)

The Atangana–Baleanu Caputo Z transformation is represented by

$$\begin{array}{cc}\hfill Z(\kappa ,\epsilon )=& {\left(\frac{\epsilon }{\kappa }\right)}^{2}G\left(\frac{\epsilon }{\kappa }\right)={\left(\frac{\epsilon }{\kappa }\right)}^2\frac{\kappa }{\epsilon }\left[\frac{B(ß)}{1-ß}\frac{{\left(\frac{\epsilon }{\kappa }\right)}^{ß+1}G\left(\frac{\epsilon }{\kappa }\right)-{\left(\frac{\epsilon }{\kappa }\right)}^{ß-1}f\left(0\right)}{{\left(\frac{\epsilon }{\kappa }\right)}^{ß}+\frac{ß}{1-ß}}\right]\hfill \\ & =\left[\frac{B(ß)}{1-ß}\frac{{\left(\frac{\epsilon }{\kappa }\right)}^{ß+2}G\left(\frac{\epsilon }{\kappa }\right)-{\left(\frac{\epsilon }{\kappa }\right)}^{ß}f\left(0\right)}{{\left(\frac{\epsilon }{\kappa }\right)}^{ß}+\frac{ß}{1-ß}}\right].\hfill \end{array}$$

□

Theorem 9.

Let’s assume that the ZZ transformation of $U\left(\varrho \right)$ is represented by $G\left(\epsilon \right)$ and the Aboodh transformation of $U\left(\varrho \right)$ is represented by $Z(\kappa ,\epsilon )$. Then, the ZZ transformation of the Atangana–Baleanu Riemann–Liouville derivative is defined as [55]

$$ZZ\left\{\begin{array}{c}ABR\hfill \\ 0\hfill \end{array}{D}_{\varrho }^{ß}f\left(\varrho \right)\right\}=\left[\frac{B(ß)}{1-ß}\frac{\frac{{\epsilon }^{ß+2}}{{\kappa }^{ß+2}}G\left(\frac{\epsilon }{\kappa }\right)}{\frac{{\epsilon }^{\mu }}{{\kappa }^{\mu }}+\frac{ß}{1-ß}}\right].$$

(18)

Proof. 

Applying Equations (3) and (6), we get

$$G\left(\frac{\epsilon }{\kappa }\right)=\frac{\kappa }{\epsilon }\left[\frac{B(ß)}{1-ß}\frac{{\left(\frac{\epsilon }{\kappa }\right)}^{ß+1}G\left(\frac{\epsilon }{\kappa }\right)}{{\left(\frac{\epsilon }{\kappa }\right)}^{ß}+\frac{ß}{1-ß}}\right]$$

(19)

The ZZ transformation of the Atangana–Baleanu Riemann–Liouville is defined in Equation (11).

$$\begin{array}{cc}\hfill Z(\kappa ,\epsilon )={\left(\frac{\epsilon }{\kappa }\right)}^{2}G\left(\frac{\epsilon }{\kappa }\right)={\left(\frac{\epsilon }{\kappa }\right)}^2\left(\frac{\kappa }{\epsilon }\right)& \left[\frac{B(ß)}{1-ß}\frac{{\left(\frac{\epsilon }{\kappa }\right)}^{ß+1}G\left(\frac{\epsilon }{\kappa }\right)}{{\left(\frac{\epsilon }{\kappa }\right)}^{ß}+\frac{ß}{1-ß}}\right]\hfill \\ & =\left[\frac{B(ß)}{1-ß}\frac{{\left(\frac{\epsilon }{\kappa }\right)}^{ß+2}G\left(\frac{\epsilon }{\kappa }\right)}{{\left(\frac{\epsilon }{\kappa }\right)}^{ß}+\frac{ß}{1-ß}}\right]\hfill \end{array}$$

□

3. Fundamental Concept of HPTM

Here, the general methodology of HPTM is given to solve the following FPDE

$$D_{\wp }^{ß}\mathcal{V}(\mathbf{Ø},\wp )={\mathcal{P}}_1[\mathbf{Ø}]\mathcal{V}(\mathbf{Ø},\wp )+{\mathcal{Q}}_1[\mathbf{Ø}]\mathcal{V}(\mathbf{Ø},\wp ),1<ß\le 2,$$

(20)

subjected to initial sources

$$\mathcal{V}(\mathbf{Ø},0)=\xi (\mathbf{Ø}),\frac{\partial }{\partial \wp }\mathcal{V}(\mathbf{Ø},0)={\xi }^{{}^{\prime }}(\mathbf{Ø}).$$

Here $D_{\wp }^{ß}=\frac{{\partial }^{ß}}{\partial {\wp }^{ß}}$ is the Caputo type operator, ${\mathcal{P}}_1[\mathbf{Ø}]$ is the linear and ${\mathcal{Q}}_1[\mathbf{Ø}]$ is the nonlinear function.

By utilizing ZT, we get

$$Z\left[D_{\wp }^{ß}\mathcal{V}(\mathbf{Ø},\wp )\right]=Z[{\mathcal{P}}_1[\mathbf{Ø}]\mathcal{V}(\mathbf{Ø},\wp )+{\mathcal{Q}}_1[\mathbf{Ø}]\mathcal{V}(\mathbf{Ø},\wp )],$$

(21)

On simplifying the above equation, we get

$$M\left(\mathcal{V}\right)=u\mathcal{V}\left(0\right)+u^2{\mathcal{V}}^{{}^{\prime }}\left(0\right)+\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}Z[{\mathcal{P}}_1[\mathbf{Ø}]\mathcal{V}(\mathbf{Ø},\wp )+{\mathcal{Q}}_1[\mathbf{Ø}]\mathcal{V}(\mathbf{Ø},\wp )].$$

(22)

On utilizing inverse ZT, we get

$$\mathcal{V}(\mathbf{Ø},\wp )=\mathcal{V}\left(0\right)+{\mathcal{V}}^{{}^{\prime }}\left(0\right)+Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}Z[{\mathcal{P}}_1[\mathbf{Ø}]\mathcal{V}(\mathbf{Ø},\wp )+{\mathcal{Q}}_1[\mathbf{Ø}]\mathcal{V}(\mathbf{Ø},\wp )]\right].$$

(23)

According to standard HPM [44,45,46], the solution $\mathcal{V}(\mathbf{Ø},\wp )$ can be expanded into infinite series as

$$\mathcal{V}(\mathbf{Ø},\wp )=\sum _{k=0}^{\infty }{ϵ}^k{\mathcal{V}}_k(\mathbf{Ø},\wp ).$$

(24)

with parameter $ϵ\in [0,1]$.

The nonlinear term is considered as

$${\mathcal{Q}}_1[\mathbf{Ø}]\mathcal{V}(\mathbf{Ø},\wp )=\sum _{k=0}^{\infty }{ϵ}^k{H}_n\left(\mathcal{V}\right).$$

(25)

Also, $H_k\left(\mathcal{V}\right)$ represents He’s polynomials

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

(26)

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

By inserting Equations (24) and (25) into Equation (23), we have

$$\sum _{k=0}^{\infty }{ϵ}^k{\mathcal{V}}_k(\mathbf{Ø},\wp )=\mathcal{V}\left(0\right)+{\mathcal{V}}^{{}^{\prime }}\left(0\right)+ϵ\times \left(Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}Z\{{\mathcal{P}}_1\sum _{k=0}^{\infty }{ϵ}^k{\mathcal{V}}_k(\mathbf{Ø},\wp )+\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left(\mathcal{V}\right)\}\right]\right).$$

(27)

By comparing the coefficient of $ϵ$, we obtain

$$\begin{array}{cc}\hfill & {ϵ}^0:{\mathcal{V}}_0(\mathbf{Ø},\wp )=\mathcal{V}\left(0\right)+{\mathcal{V}}^{{}^{\prime }}\left(0\right),\hfill \\ \hfill & {ϵ}^1:{\mathcal{V}}_1(\mathbf{Ø},\wp )=Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}Z({\mathcal{P}}_1[\mathbf{Ø}]{\mathcal{V}}_0(\mathbf{Ø},\wp )+H_0\left(\mathcal{V}\right))\right],\hfill \\ \hfill & {ϵ}^2:{\mathcal{V}}_2(\mathbf{Ø},\wp )=Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}Z({\mathcal{P}}_1[\mathbf{Ø}]{\mathcal{V}}_1(\mathbf{Ø},\wp )+H_1\left(\mathcal{V}\right))\right],\hfill \\ \hfill & .\hfill \\ \hfill & .\hfill \\ \hfill & .\hfill \\ \hfill & {ϵ}^k:{\mathcal{V}}_k(\mathbf{Ø},\wp )=Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}Z({\mathcal{P}}_1[\mathbf{Ø}]{\mathcal{V}}_{k-1}(\mathbf{Ø},\wp )+H_{k-1}\left(\mathcal{V}\right))\right],\hfill \\ \hfill k>0,k\in N.\end{array}$$

(28)

Lastly, the solution of ${\mathcal{V}}_k(\mathbf{Ø},\wp )$ is stated as

$$\mathcal{V}(\mathbf{Ø},\wp )=\underset{ϵ\to 1}{\lim }\sum _{k=0}^{\infty }{ϵ}^k{\mathcal{V}}_k(\mathbf{Ø},\wp ).$$

(29)

4. Fundamental Concept of ZTDM

Here, the general methodology of ZTDM is given to solve the following FPDE

$$D_{\wp }^{ß}\mathcal{V}(\mathbf{Ø},\wp )={\mathcal{P}}_1(\mathbf{Ø},\wp )+{\mathcal{Q}}_1(\mathbf{Ø},\wp ),0<ß\le 1,$$

(30)

subjected to initial sources

$$\mathcal{V}(\mathbf{Ø},0)=\xi (\mathbf{Ø}),\frac{\partial }{\partial \wp }\mathcal{V}(\mathbf{Ø},0)={\xi }^{{}^{\prime }}(\mathbf{Ø}).$$

Here $D_{\wp }^{ß}=\frac{{\partial }^{ß}}{\partial {\wp }^{ß}}$ is the Caputo type operator, ${\mathcal{P}}_1$ is the linear and ${\mathcal{Q}}_1$ is the nonlinear function.

On simplifying the above equation, we get

$$M\left(\mathcal{V}\right)=u\mathcal{V}\left(0\right)+u^2{\mathcal{V}}^{{}^{\prime }}\left(0\right)+\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}Z[{\mathcal{P}}_1(\mathbf{Ø},\wp )+{\mathcal{Q}}_1(\mathbf{Ø},\wp )],$$

(31)

On utilizing inverse ZT, we get

$$\mathcal{V}(\mathbf{Ø},\wp )=\mathcal{V}\left(0\right)+{\mathcal{V}}^{{}^{\prime }}\left(0\right)+Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}Z\right[{\mathcal{P}}_1(\mathbf{Ø},\wp )+{\mathcal{Q}}_1(\mathbf{Ø},\wp )\left]\right].$$

(32)

The ADM defines the unknown function $\mathcal{V}(\mathbf{Ø},\wp )$ by an infinite series. Thus by ZTDM

$$\mathcal{V}(\mathbf{Ø},\wp )=\sum _{m=0}^{\infty }{\mathcal{V}}_m(\mathbf{Ø},\wp ).$$

(33)

The nonlinear term is considered as

$${\mathcal{Q}}_1(\mathbf{Ø},\wp )=\sum _{m=0}^{\infty }{\mathcal{A}}_m.$$

(34)

with

$${\mathcal{A}}_m=\frac{1}{m!}{\left[\frac{{\partial }^m}{\partial {\ell }^m}\left\{{\mathcal{Q}}_1\left(\sum _{k=0}^{\infty }{\ell }^k{\mathbf{Ø}}_k,\sum _{k=0}^{\infty }{\ell }^k{\wp }_k\right)\right\}\right]}_{\ell =0},$$

(35)

By putting Equations (33) and (34) into Equation (32), we get

$$\sum _{m=0}^{\infty }{\mathcal{V}}_m(\mathbf{Ø},\wp )=\mathcal{V}\left(0\right)+{\mathcal{V}}^{{}^{\prime }}\left(0\right)+Z^{-1}\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}\left[Z\left\{{\mathcal{P}}_1(\sum _{m=0}^{\infty }{\mathbf{Ø}}_m,\sum _{m=0}^{\infty }{\wp }_m)+\sum _{m=0}^{\infty }{\mathcal{A}}_m\right\}\right].$$

(36)

Thus, we get

$${\mathcal{V}}_0(\mathbf{Ø},\wp )=\mathcal{V}\left(0\right)+\wp {\mathcal{V}}^{{}^{\prime }}\left(0\right),$$

(37)

$${\mathcal{V}}_1(\mathbf{Ø},\wp )=Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}{Z}^{+}\{{\mathcal{P}}_1({\mathbf{Ø}}_0,{\wp }_0)+{\mathcal{A}}_0\}\right],$$

Hence, in general for $m\ge 1$, we have

$${\mathcal{V}}_{m+1}(\mathbf{Ø},\wp )=Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}{Z}^{+}\{{\mathcal{P}}_1({\mathbf{Ø}}_m,{\wp }_m)+{\mathcal{A}}_m\}\right].$$

5. Application

Example

Let’s suppose the seventh order SLIE, which has the following form

$$\begin{array}{cc}\hfill & D_{\wp }^{ß}\mathcal{V}(\mathbf{Ø},\wp )=-252{\mathcal{V}}^3(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}}^3(\mathbf{Ø},\wp )-378\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp )-126{\mathcal{V}}^2(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )\hfill \\ \hfill & -63{\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-42{\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{4\mathbf{Ø}}(\mathbf{Ø},\wp )-21\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{5\mathbf{Ø}}(\mathbf{Ø},\wp )-{\mathcal{V}}_{7\mathbf{Ø}}(\mathbf{Ø},\wp ),0<ß\le 1,\hfill \end{array}$$

(38)

subjected to initial source

$$\mathcal{V}(\mathbf{Ø},0)=\frac{4}{3}{\rho }^2(2-3{\tanh }^2(\rho \mathbf{Ø})).$$

By utilizing ZT, we get

$$\begin{array}{cc}\hfill & Z\left[\frac{{\partial }^{ß}\mathcal{V}}{\partial {\wp }^{ß}}\right]=Z[-252{\mathcal{V}}^3(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}}^3(\mathbf{Ø},\wp )-378\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp )-126{\mathcal{V}}^2(\mathbf{Ø},\wp )\hfill \\ \hfill & {\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-42{\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{4\mathbf{Ø}}(\mathbf{Ø},\wp )-21\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{5\mathbf{Ø}}(\mathbf{Ø},\wp )-{\mathcal{V}}_{7\mathbf{Ø}}(\mathbf{Ø},\wp )],\hfill \end{array}$$

(39)

On simplifying the above equation, we get

$$\begin{array}{cc}\hfill & M\left(u\right)=u\mathcal{V}\left(0\right)+\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}[-252{\mathcal{V}}^3(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}}^3(\mathbf{Ø},\wp )-378\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp )\hfill \\ \hfill & -126{\mathcal{V}}^2(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-42{\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{4\mathbf{Ø}}(\mathbf{Ø},\wp )-21\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{5\mathbf{Ø}}(\mathbf{Ø},\wp )-{\mathcal{V}}_{7\mathbf{Ø}}(\mathbf{Ø},\wp )].\hfill \end{array}$$

(40)

Applying the inverse of the ZT, we get

$$\begin{array}{cc}\hfill & \mathcal{V}(\mathbf{Ø},\wp )=\mathcal{V}\left(0\right)+Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}\right\{Z(-252{\mathcal{V}}^3(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}}^3(\mathbf{Ø},\wp )-378\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp )\hfill \\ \hfill & -126{\mathcal{V}}^2(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-42{\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{4\mathbf{Ø}}(\mathbf{Ø},\wp )-21\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{5\mathbf{Ø}}(\mathbf{Ø},\wp )-{\mathcal{V}}_{7\mathbf{Ø}}(\mathbf{Ø},\wp )\left)\right\}],\hfill \\ \hfill & \mathcal{V}(\mathbf{Ø},\wp )=\frac{4}{3}{\rho }^2(2-3{\tanh }^2(\rho \mathbf{Ø}))+Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}\right\{Z(-252{\mathcal{V}}^3(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}}^3(\mathbf{Ø},\wp )\hfill \\ \hfill & -378\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp )-126{\mathcal{V}}^2(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-42{\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{4\mathbf{Ø}}(\mathbf{Ø},\wp )\hfill \\ \hfill & -21\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{5\mathbf{Ø}}(\mathbf{Ø},\wp )-{\mathcal{V}}_{7\mathbf{Ø}}(\mathbf{Ø},\wp )\left)\right\}].\hfill \end{array}$$

(41)

Using the HPM, the nonlinear term is taken in the form of He’s polynomial $H_k\left(\mathcal{V}\right)$ as

$$\begin{array}{cc}\hfill & \sum _{k=0}^{\infty }{ϵ}^k{\mathcal{V}}_k(\mathbf{Ø},\wp )=\frac{4}{3}{\rho }^2(2-3{\tanh }^2(\rho \mathbf{Ø}))+ϵ\left(Z^{-1}\right[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}Z[-252\left(\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left(\mathcal{V}\right)\right)-63\left(\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left(\mathcal{V}\right)\right)-\hfill \\ \hfill & 378\left(\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left(\mathcal{V}\right)\right)-126\left(\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left(\mathcal{V}\right)\right)-63\left(\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left(\mathcal{V}\right)\right)-42\left(\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left(\mathcal{V}\right)\right)-21\left(\sum _{k=0}^{\infty }{ϵ}^k{H}_k\left(\mathcal{V}\right)\right)-\hfill \\ \hfill & {\left(\sum _{k=0}^{\infty }{ϵ}^k{\mathcal{V}}_k(\mathbf{Ø},\wp )\right)}_{7\mathbf{Ø}}\left]\right]).\hfill \end{array}$$

(42)

On comparing the coefficient of $ϵ$, we have

$$\begin{array}{cc}\hfill & {ϵ}^0:{\mathcal{V}}_0(\mathbf{Ø},\wp )=\frac{4}{3}{\rho }^2(2-3{\tanh }^2(\rho \mathbf{Ø})),\hfill \\ \hfill & {ϵ}^1:{\mathcal{V}}_1(\mathbf{Ø},\wp )=-2048{\rho }^9\tanh (\rho \mathbf{Ø}){\mathrm{sech}}^2(\rho \mathbf{Ø})\left(1-ß+\frac{ß{\wp }^{ß}}{\Gamma (ß+1)}\right),\hfill \\ \hfill & {ϵ}^2:{\mathcal{V}}_2(\mathbf{Ø},\wp )=524288{\rho }^{16}(\cosh (2\rho \mathbf{Ø})-2){\mathrm{sech}}^4(\rho \mathbf{Ø})\left[\frac{{ß}^2{\wp }^{2ß}}{\Gamma (2ß+1)}+2ß(1-ß)\frac{{\wp }^{ß}}{\Gamma (ß+1)}+{(1-ß)}^2\right],\hfill \\ \hfill & ⋮\hfill \end{array}$$

The obtained solution can be taken in series form as

$$\begin{array}{cc}\hfill & \mathcal{V}(\mathbf{Ø},\wp )={\mathcal{V}}_0(\mathbf{Ø},\wp )+{\mathcal{V}}_1(\mathbf{Ø},\wp )+{\mathcal{V}}_2(\mathbf{Ø},\wp )+\cdots \hfill \\ \hfill & \mathcal{V}(\mathbf{Ø},\wp )=\frac{4}{3}{\rho }^2(2-3{\tanh }^2(\rho \mathbf{Ø}))-2048{\rho }^9\tanh (\rho \mathbf{Ø}){\mathrm{sech}}^2(\rho \mathbf{Ø})\left(1-ß+\frac{ß{\wp }^{ß}}{\mathrm{\Gamma }(ß+1)}\right)\hfill \\ \hfill & +524288{\rho }^{16}(\cosh (2\rho \mathbf{Ø})-2){\mathrm{sech}}^4(\rho \mathbf{Ø})\left[\frac{{ß}^2{\wp }^{2ß}}{\mathrm{\Gamma }(2ß+1)}+2ß(1-ß)\frac{{\wp }^{ß}}{\mathrm{\Gamma }(ß+1)}+{(1-ß)}^2\right]+\cdots \hfill \end{array}$$

• Implementation of ZTDM

Now we present the solution of TFSKIE by utilizing ZTDM is given as

By utilizing ZT, we get

$$\begin{array}{cc}\hfill & Z\left\{\frac{{\partial }^{ß}\mathcal{V}}{\partial {\wp }^{ß}}\right\}=Z[-252{\mathcal{V}}^3(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}}^3(\mathbf{Ø},\wp )-378\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp )-126{\mathcal{V}}^2(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )\hfill \\ \hfill & -63{\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-42{\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{4\mathbf{Ø}}(\mathbf{Ø},\wp )-21\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{5\mathbf{Ø}}(\mathbf{Ø},\wp )-{\mathcal{V}}_{7\mathbf{Ø}}(\mathbf{Ø},\wp )],\hfill \end{array}$$

(43)

On simplifying the above equation, we get

$$\begin{array}{cc}\hfill & M\left(u\right)=u\mathcal{V}\left(0\right)+\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}Z[-252{\mathcal{V}}^3(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}}^3(\mathbf{Ø},\wp )-378\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp )-126\hfill \\ \hfill & {\mathcal{V}}^2(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-42{\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{4\mathbf{Ø}}(\mathbf{Ø},\wp )-21\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{5\mathbf{Ø}}(\mathbf{Ø},\wp )-{\mathcal{V}}_{7\mathbf{Ø}}(\mathbf{Ø},\wp )].\hfill \end{array}$$

(44)

Applying the inverse of the ZT, we get

$$\begin{array}{cc}\hfill & \mathcal{V}(\mathbf{Ø},\wp )=\mathcal{V}\left(0\right)+Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}\right\{Z[-252{\mathcal{V}}^3(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}}^3(\mathbf{Ø},\wp )-378\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp )\hfill \\ \hfill & -126{\mathcal{V}}^2(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-42{\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{4\mathbf{Ø}}(\mathbf{Ø},\wp )-21\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{5\mathbf{Ø}}(\mathbf{Ø},\wp )-{\mathcal{V}}_{7\mathbf{Ø}}(\mathbf{Ø},\wp )\left]\right\}],\hfill \\ \hfill & \mathcal{V}(\mathbf{Ø},\wp )=\frac{4}{3}{\rho }^2(2-3{\tanh }^2(\rho \mathbf{Ø}))+Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}\right\{Z[-252{\mathcal{V}}^3(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}}^3(\mathbf{Ø},\wp )\hfill \\ \hfill & -378\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp )-126{\mathcal{V}}^2(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )-63{\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )\hfill \\ \hfill & -42{\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{4\mathbf{Ø}}(\mathbf{Ø},\wp )-21\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{5\mathbf{Ø}}(\mathbf{Ø},\wp )-{\mathcal{V}}_{7\mathbf{Ø}}(\mathbf{Ø},\wp )\left]\right\}].\hfill \end{array}$$

(45)

Thus the solution in series form reads

$$\mathcal{V}(\mathbf{Ø},\wp )=\sum _{m=0}^{\infty }{\mathcal{V}}_m(\mathbf{Ø},\wp ).$$

(46)

Let us solve the nonlinear terms by Adomian polynomial as ${\mathcal{V}}^3(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp )={\sum }_{m=0}^{\infty }{\mathcal{A}}_m$, ${\mathcal{V}}_{\mathbf{Ø}}^3(\mathbf{Ø},\wp )={\sum }_{m=0}^{\infty }{\mathcal{B}}_m,\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp )={\sum }_{m=0}^{\infty }{\mathcal{C}}_m$, ${\mathcal{V}}^2(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )={\sum }_{m=0}^{\infty }{\mathcal{D}}_m$, ${\mathcal{V}}_{\mathbf{Ø}\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{3\mathbf{Ø}}(\mathbf{Ø},\wp )={\sum }_{m=0}^{\infty }{\mathcal{E}}_m$, ${\mathcal{V}}_{\mathbf{Ø}}(\mathbf{Ø},\wp ){\mathcal{V}}_{4\mathbf{Ø}}(\mathbf{Ø},\wp )={\sum }_{m=0}^{\infty }{\mathcal{V}}_m,\mathcal{V}(\mathbf{Ø},\wp ){\mathcal{V}}_{5\mathbf{Ø}}(\mathbf{Ø},\wp )={\sum }_{m=0}^{\infty }{\mathcal{G}}_m$. Hence, we have

$$\begin{array}{cc}\hfill \sum _{m=0}^{\infty }{\mathcal{V}}_m(\mathbf{Ø},\wp )=& \mathcal{V}(\mathbf{Ø},0)+Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}Z\right[-252\sum _{m=0}^{\infty }{\mathcal{A}}_m-63\sum _{m=0}^{\infty }{\mathcal{B}}_m-378\sum _{m=0}^{\infty }{\mathcal{C}}_m-126\sum _{m=0}^{\infty }{\mathcal{D}}_m-\hfill \\ \hfill & 63\sum _{m=0}^{\infty }{\mathcal{E}}_m-42\sum _{m=0}^{\infty }{\mathcal{V}}_m-21\sum _{m=0}^{\infty }{\mathcal{G}}_m-{\mathcal{V}}_{7\mathbf{Ø}}(\mathbf{Ø},\wp )\left]\right],\hfill \\ \hfill \sum _{m=0}^{\infty }{\mathcal{V}}_m(\mathbf{Ø},\wp )=& \frac{4}{3}{\rho }^2(2-3{\tanh }^2(\rho \mathbf{Ø}))+Z^{-1}\left[\frac{\left(1-ß+ß{\left(\frac{\kappa }{u}\right)}^{ß}\right)}{B(ß)}Z\right[-252\sum _{m=0}^{\infty }{\mathcal{A}}_m-63\sum _{m=0}^{\infty }{\mathcal{B}}_m-378\sum _{m=0}^{\infty }{\mathcal{C}}_m-\hfill \\ \hfill & 126\sum _{m=0}^{\infty }{\mathcal{D}}_m-63\sum _{m=0}^{\infty }{\mathcal{E}}_m-42\sum _{m=0}^{\infty }{\mathcal{V}}_m-21\sum _{m=0}^{\infty }{\mathcal{G}}_m-{\mathcal{V}}_{7\mathbf{Ø}}(\mathbf{Ø},\wp )\left]\right].\hfill \end{array}$$

(47)

On comparing both sides, we have

$${\mathcal{V}}_0(\mathbf{Ø},\wp )=\frac{4}{3}{\rho }^2(2-3{\tanh }^2(\rho \mathbf{Ø})),$$

on $m=0$

$${\mathcal{V}}_1(\mathbf{Ø},\wp )=-2048{\rho }^9\tanh (\rho \mathbf{Ø}){\mathrm{sech}}^2(\rho \mathbf{Ø})\left(1-ß+\frac{ß{\wp }^{ß}}{\mathrm{\Gamma }(ß+1)}\right),$$

on $m=1$

$${\mathcal{V}}_2(\mathbf{Ø},\wp )=524288{\rho }^{16}(\cosh (2\rho \mathbf{Ø})-2){\mathrm{sech}}^4(\rho \mathbf{Ø})\left[\frac{{ß}^2{\wp }^{2ß}}{\mathrm{\Gamma }(2ß+1)}+2ß(1-ß)\frac{{\wp }^{ß}}{\mathrm{\Gamma }(ß+1)}+{(1-ß)}^2\right],$$

So in the same sense the other terms for $(m\ge 3)$ are easy to obtain

$$\mathcal{V}(\mathbf{Ø},\wp )=\sum _{m=0}^{\infty }{\mathcal{V}}_m(\mathbf{Ø},\wp )={\mathcal{V}}_0(\mathbf{Ø},\wp )+{\mathcal{V}}_1(\mathbf{Ø},\wp )+{\mathcal{V}}_2(\mathbf{Ø},\wp )+\cdots$$

$$\begin{array}{cc}\hfill & \mathcal{V}(\mathbf{Ø},\wp )=\frac{4}{3}{\rho }^2(2-3{\tanh }^2(\rho \mathbf{Ø}))-2048{\rho }^9\tanh (\rho \mathbf{Ø}){\mathrm{sech}}^2(\rho \mathbf{Ø})\left(1-ß+\frac{ß{\wp }^{ß}}{\mathrm{\Gamma }(ß+1)}\right)\hfill \\ \hfill & +524288{\rho }^{16}(\cosh (2\rho \mathbf{Ø})-2){\mathrm{sech}}^4(\rho \mathbf{Ø})\left[\frac{{ß}^2{\wp }^{2ß}}{\mathrm{\Gamma }(2ß+1)}+2ß(1-ß)\frac{{\wp }^{ß}}{\mathrm{\Gamma }(ß+1)}+{(1-ß)}^2\right]+\cdots \hfill \end{array}$$

By taking $ß=1$, we get

$$\mathcal{V}(\mathbf{Ø},\wp )=\frac{4}{3}{\rho }^2(2-3{\tanh }^2\left(\rho (\frac{256{\rho }^6\wp }{3}+\mathbf{Ø})\right))$$

(48)

• Numerical Simulation Studies

In this section, we present an analytical solution for $\mathcal{V}(\mathbf{Ø},\wp )$ that is both simple and efficient to implement. We validate the accuracy of our solution through numerical results, comparing it to the exact solution. We find that our method provides a good approximation to the exact solution, as shown in Figure 1. Additionally, we demonstrate the behavior of $\mathcal{V}(\mathbf{Ø},\wp )$ for various values of $ß$ in Figure 2 the proposed techniques solution graphical depiction at $ß=0.8,0.6$ and Figure 3 the graphical depiction of the proposed techniques solution for various orders of ß. Finally, we explore the effect of increasing the order of the approximation in Figure 4, which can improve the accuracy of our solution. Overall, our approach offers a practical and effective solution for analyzing $\mathcal{V}(\mathbf{Ø},\wp )$ in a range of contexts.

6. Conclusions

In conclusion, the time-fractional Sawada–Kotera–Ito equation (SKIE) has been successfully solved using both the Adomian decomposition transform method and homotopy perturbation transform method. Both methods have proven to be effective in obtaining accurate numerical solutions for the equation, and they have been compared and analyzed in terms of their efficiency and accuracy. Both methods have demonstrated good results, but it is important to note that the choice of method depends on the specific application and the requirements of the problem. Regardless of the method used, it is clear that the solution of the time-fractional SKIE is important in a variety of fields, including physics, engineering, and mathematics, and has the potential to provide valuable insights into a range of complex phenomena. The SKIE also has potential applications in engineering. For example, the equation could be used to model fluid flow in certain systems, and the solutions could provide information about the flow behavior and help optimize the design of these systems. Overall, the time-fractional SKIE is an important equation with many potential applications in a variety of fields. There is still much to be learned about this equation and its solutions, and future work could help to further our understanding of its properties and behavior.