Analytical Investigation of Fractional-Order Korteweg–De-Vries-Type Equations under Atangana–Baleanu–Caputo Operator: Modeling Nonlinear Waves in a Plasma and Fluid

Abstract

This article applies the homotopy perturbation transform technique to analyze fractional-order nonlinear fifth-order Korteweg–de-Vries-type (KdV-type)/Kawahara-type equations. This method combines the Zain Ul Abadin Zafar-transform (ZZ-T) and the homotopy perturbation technique (HPT) to show the validation and efficiency of this technique to investigate three examples. It is also shown that the fractional and integer-order solutions have closed contact with the exact result. The suggested technique is found to be reliable, efficient, and straightforward to use for many related models of engineering and several branches of science, such as modeling nonlinear waves in different plasma models.

1. Introduction

Fractional calculus was started in Newton’s period, but recently it has captured the interest of numerous researchers. For the last thirty decades, many interesting and significant steps in applied science have been discovered within the structure of fractional calculus. Because of the intricacy associated with a heterogeneous phenomenon, the fractional derivative was invented [1,2]. Fractional derivative operators are sufficient to capture the attitudes of multidimensional media since they have diffusion procedures. This has been a useful tool, and differential equations of any order can show numerous problems more quickly and accurately. Due to the increased use of mathematical methods that use computer software, many scholars have begun to work on generalized calculus to convey their viewpoints while investigating various complicated phenomena [3,4,5]. Plasma is a multifaceted, quasi-neutral, electrically conducting fluid. It is made up of electrons, ions, and non-magnetic particles. Plasma has electric and magnetic fields as a result of its electrical conductive nature. Plasma can support several types of waves due to particle–field interactions. Wave phenomena are significant in plasma heating, instabilities, and diagnostics, among other things. A magneto-sonic wave is a dispersion-free, longitudinal wave of ions in a magnetized plasma that propagates perpendicular to the stationary magnetic field. In the presence of a low magnetic field, the magneto-acoustic wave behaves like an ion-acoustic wave, and in the presence of a low temperature, the magneto-acoustic wave behaves like an Alfven wave. Magneto-acoustic waves have been discovered in the solar corona.

Nonlinear effects are observed in a wide variety of applied sciences, including particle physics, mathematical biology, quantum optics, quantum field theory, physical chemistry, thermodynamics, fluid mechanics, plasma physics, etc. Nonlinear partial differential equations (PDEs) of various higher orders are used to model these processes. PDEs are frequently used to describe physical processes [6,7,8,9,10]. The nonlinear nature of the majority of fundamental physical systems remains concealed. It may be impossible to determine the exact outcome of such nonlinear events. This phenomenon can be examined solely with the application of helpful tools for solving these nonlinear schemes [11,12,13,14,15,16,17].

In 1895, Korteweg and De Vries developed a Korteweg–De-Vries (KdV) equation that could be used to explain Russell’s soliton phenomenon, such as huge and small waves in water and plasma. Solitons are steady solitary waves, which mean that these solitary waves are particles. The KdV equation and some higher-order nonlinearity-related equations are analyzed for researching dispersive wave processes in different research areas, such as fluid dynamics, quantum mechanics, plasma physics, optics, etc. [18,19,20,21,22,23,24,25]. There is another family to the KdV equation, but with higher-order dispersion and nonlinearity, which is called a fifth-order KdV-type (sometimes called Kawahara-type) equation utilized to investigate many nonlinear particle physics phenomena, especially in plasma physics. This family of higher-order dispersion and nonlinearity is effective and important for describing nonlinear waves in many nonlinear and dispersive media such as the physics of plasmas [26,27,28,29,30,31,32]. Researchers have examined the KdV-type model and found that it has dispersive terms of the 3rd and 5th order that are important to the acoustic waves in a plasma [27,28,29,30]. Dispersive terms appear near the critical angle of propagation [27]. Motivated by theoretical studies and satellite observations of nonlinear waves that propagate in different plasma models, we will focus in this work on finding some solutions for the family of fifth-order KdV-type equations in order to understand the mechanisms of propagation of these waves in different plasma systems [27,28,29,30,33,34].

This is the general model for studying magnetic properties, which includes waves in shallow water and acoustic waves in different plasma models with surface tension. The fifth-order KdV equation is used to describe this model. Researchers have recently analyzed that the solutions of the traveling waves of this equation do not vanish at infinity [35,36]. Consider the three well-known types of the 5th-order KdV equations as [37,38]:

$$D_{\tau }^{\gamma }\mathcal{U}+{\mathcal{U}}_{\zeta }+{\mathcal{U}}^2{\mathcal{U}}_{2\zeta }+{\mathcal{U}}_{\zeta }{\mathcal{U}}_{2\zeta }-20{\mathcal{U}}^2{\mathcal{U}}_{3\zeta }+{\mathcal{U}}_{5\zeta }=0,0<\beta \le 1,$$

(1)

with initial condition $\mathcal{U}(\zeta ,0)=\frac{1}{\zeta }$,

$$D_{\tau }^{\beta }\mathcal{U}+\mathcal{U}{\mathcal{U}}_{\zeta }-\mathcal{U}{\mathcal{U}}_{3\zeta }+{\mathcal{U}}_{5\zeta }=0,0<\beta \le 1,$$

(2)

with initial condition $\mathcal{U}(\zeta ,0)=e^{\zeta }$,

$$D_{\tau }^{\beta }\mathcal{U}+\mathcal{U}{\mathcal{U}}_{\zeta }+{\mathcal{U}}_{3\zeta }-{\mathcal{U}}_{5\zeta }=0,0<\beta \le 1,$$

(3)

with initial condition $\mathcal{U}(\zeta ,0)=\frac{105}{169}sec{h}^4\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)$.

Equations (1) and (2) called fifth-order KdV equations and Equation (3) is called the Kawahara equation. The analytical approaches to these physical problems are very difficult to find since they are extremely nonlinear. In the last decade, several researchers have applied different analytical and computational techniques to solving linear and nonlinear KdV equations, such as the variational iteration method [38], multi-symplectic method [39], exp-function method [40], and He’s homotopy perturbation method [41].

The homotopy perturbation method is combined with the ZZ-transform method to provide a good way of dealing with nonlinear terms. This method is called the homotopy perturbation transform method (HPTM). This method can facilitate obtaining the result quickly in convergent series, which leads to a closed form of the solution. The use of He’s polynomials in nonlinear terms was first presented by Ghorbani [42,43,44]. Later on, numerous investigators utilized the homotopy perturbation transform methodology for linear and nonlinear differential equations such as Navier–Stokes equations [45], heat-like equations [46], the gas dynamic equation [47], the hyperbolic equation, and Fisher’s equation [48].

2. Preliminaries

Definition 1.

The set of functions of the Aboodh transformation is obtained as

$$B=\left\{g(\vartheta ):\exists M,n_1,n_2>0,|g(\vartheta )|<M{e}^{-s\vartheta }\right\}$$

and is expressed as [49,50]

$$A\left\{g(\vartheta )\right\}=\frac{1}{s}{\int }_0^{\infty }g(\vartheta )e^{-s\vartheta }d\vartheta ,\vartheta >0\mathrm{and}{n}_1\le s\le n_2$$

Theorem 1.

Now let us regard F and G as the Laplace and Aboodh transformations of $g(\vartheta )\in B$, then [51,52]

$$G(s)=\frac{F(s)}{s}.$$

(4)

Zain Ul Abadin Zafar [53] was the first to introduce the ZZ transformation. It combined the integral transformations of Aboodh and Laplace. The ZZ transformation is defined in the following.

Definition 2.

(ZZ Transformation) Assume that $g(\vartheta )\forall \vartheta \ge 0$ is a function, then the ZZ transform $Z(v,s)$ of $g(\vartheta )$ is expressed as [53]

$$ZZ(g(\vartheta ))=Z(v,s)=s{\int }_0^{\infty }g(v\vartheta )e^{-s\vartheta }d\vartheta .$$

The ZZ transform is linear, just like the Laplace and Aboodh transformations. The Mittag–Leffler function (MLF) is a function that is defined as an extension of the exponential function.

$$E_{\wp }(z)=\sum _{m=0}^{\infty }\frac{z^m}{\mathsf{\Gamma }(1+m\wp )},Re(\wp )>0.$$

Definition 3.

The Atangana–Baleanu–Caputo derivative of a function $\nu (\phi ,\vartheta )\in H^1(a,b)$, then for $\wp \in$$(0,1)$, it is defined as [54]

$${ABC}_a{D}_{\vartheta }^{\wp }\nu (\phi ,\vartheta )=\frac{\psi (\wp )}{1-\wp }{\int }_a^{\vartheta }{\nu }^{\prime }(\phi ,\vartheta )E_{\wp }\left(\frac{-\wp {(\vartheta -\eta )}^{\wp }}{1-\wp }\right)d\eta .$$

Definition 4.

Let the Atangana–Baleanu Riemann–Liouville derivative $\nu (\phi ,\vartheta )\in H^1(a,b)$, then for $\wp \in (0,1)$, it is given as [54]

$${}_a^{ABR}{D}_{\vartheta }^{\wp }\nu (\phi ,\eta )=\frac{\psi (\wp )}{1-\wp }\frac{d}{d\vartheta }{\int }_a^{\vartheta }\nu (\phi ,\eta )E_{\wp }\left(\frac{-\wp {(\vartheta -\eta )}^{\wp }}{1-\wp }\right)d\eta ,$$

where with the conditions $\psi (0)=\psi (1)=1$, $\psi (\wp )$ is a function and $b>a$.

Theorem 2.

The Laplace transformation of the Atangana–Baleanu–Caputo and Atangana–Baleanu Riemann–Liouville derivative are, respectively, defined as [54]

$$L\left\{{}_a^{ABC}{D}_{\vartheta }^{\wp }\nu (\phi ,\vartheta )\right\}(s)=\frac{\psi (\wp )}{1-\wp }\frac{s^{\wp }L\left\{\nu (\phi ,\vartheta )\right\}-s^{\wp -1}\nu (\phi ,0)}{s^{\wp }+\frac{\wp }{1-\wp }}$$

(5)

and

$$L\left\{\begin{array}{c}ABR\hfill \\ a\hfill \end{array}{D}_{\vartheta }^{\wp }\nu (\phi ,\vartheta )\right\}(s)=\frac{\psi (\wp )}{1-\wp }\frac{s^{\wp }L\left\{\nu (\phi ,\vartheta )\right\}}{s^{\wp }+\frac{\wp }{1-\wp }}$$

(6)

The following theorems have been proposed, with the assumption that $g(\vartheta )\in H^1(a,b),b>a$ and $\wp \in (0,1)$.

Theorem 3.

The Aboodh transformation of the Atangana–Baleanu Riemann–Liouville derivative is defined as [52]

$$G(s)=A\left\{\begin{array}{c}ABR\hfill \\ a\hfill \end{array}{D}_{\vartheta }^{\wp }\nu (\phi ,\vartheta )\right\}(s)=\frac{1}{s}\left[\frac{\psi (\wp )}{1-\wp }\frac{s^{\wp }L\left\{\nu (\phi ,\vartheta )\right\}}{s^{\wp }+\frac{\wp }{1-\wp }}\right]$$

(7)

Proof. 

We arrive at the needed solution using Theorem 1 and Equation (6). In the theorem below, the relationship between the transformations of ZZ and Aboodh is defined. □

Theorem 4.

The Aboodh transformation of the Atangana–Baleanu–Caputo operator is defined as [52]

$$\begin{array}{cc}\hfill G(s)=& A\left\{\begin{array}{c}ABC\hfill \\ a\hfill \end{array}{D}_{\vartheta }^{\wp }\nu (\phi ,\vartheta )\right\}(s)=\frac{1}{s}\left[\frac{\psi (\wp )}{1-\wp }\frac{s^{\wp }L\left\{\nu (\phi ,\vartheta )\right\}-s^{\wp -1}\nu (\phi ,0)}{s^{\wp }+\frac{\wp }{1-\wp }}\right]\hfill \end{array}$$

(8)

Proof. 

We may find the desired solution by using Theorem $2.1$ and Equation (5). □

Theorem 5.

If $G(s)$ and $Z(v,s)$ are the ZZ and Aboodh transformations of $g(\vartheta )\in B$, then we achieve [52]

$$Z(v,s)=\frac{s^2}{v^2}G\left(\frac{s}{v}\right)$$

Proof. 

Using the $ZZ$ transformation definition, we obtain

$$Z(v,s)=s{\int }_0^{\infty }g(v\vartheta )e^{-s\vartheta }d\vartheta$$

(9)

Putting $v\vartheta =\vartheta$ in Equation (9) we obtain

$$Z(v,s)=\frac{s}{v}{\int }_0^{\infty }g(\vartheta )e^{-\frac{\pi z}{v}}d\vartheta$$

(10)

The right-hand side of Equation (10) can be written as

$$Z(v,s)=\frac{s}{v}F\left(\frac{s}{v}\right),$$

(11)

where $F(.)$ expresses the Laplace transformation of $g(\vartheta )$. Using Theorem 1, Equation (11) can be defined as

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

(12)

where $G(.)$ signifies the Aboodh transformation of $g(\vartheta )$. □

Theorem 6.

$ZZ$ transformation of $g(\vartheta )={\vartheta }^{\wp -1}$ is defined as

$$Z(v,s)=\mathsf{\Gamma }(\wp ){\left(\frac{v}{s}\right)}^{\wp -1}$$

(13)

Proof. 

The Aboodh transformation of $g(\vartheta )={\vartheta }^{\wp },\wp \ge 0$ is

$$G(s)=\frac{\mathsf{\Gamma }(\wp )}{s^{\wp +1}}$$

$$\mathrm{Now},G\left(\frac{s}{v}\right)=\frac{\mathsf{\Gamma }(\wp )v^{\wp +1}}{s^{\wp +1}}.$$

Applying Equation (13), we achieve

$$Z(v,s)=\frac{s^2}{v^2}G\left(\frac{s}{v}\right)=\frac{s^2}{v^2}\frac{\mathsf{\Gamma }(\wp )v^{\wp +1}}{s^{\wp +1}}=\mathsf{\Gamma }(\wp ){\left(\frac{v}{s}\right)}^{\wp -1}$$

□

Theorem 7.

Let $\wp ,\omega \in C$ and $Re(\wp )>0$, then the $\mathit{ZZ}$ transformation of $E_{\wp }\left(\omega {\vartheta }^{\wp }\right)$ is defined as [52]

$$ZZ\left\{\left(E_{\wp }\left(\omega {\vartheta }^{\wp }\right)\right)\right\}=Z(v,s)={\left(1-\omega {\left(\frac{v}{s}\right)}^{\wp }\right)}^{-1}$$

(14)

Proof. 

We know that the Aboodh transformation of $E_{\wp }\left(\omega {\vartheta }^{\wp }\right)$ is defined as

$$G(s)=\frac{F(s)}{s}=\frac{s^{\wp -1}}{s\left(s^{\wp }-\omega \right)}$$

(15)

So,

$$G\left(\frac{s}{v}\right)=\frac{{\left(\frac{s}{v}\right)}^{\wp -1}}{\left(\frac{s}{v}\right)\left({\left(\frac{s}{v}\right)}^{\wp }-\omega \right)},$$

(16)

Applying Theorem 6, we achieve

$$\begin{array}{c}Z(v,s)={\left(\frac{s}{v}\right)}^{2}G\left(\frac{s}{v}\right)={\left(\frac{s}{v}\right)}^2\frac{{\left(\frac{s}{v}\right)}^{\wp -1}}{\left(\frac{s}{v}\right)\left({\left(\frac{s}{v}\right)}^{\wp }-\omega \right)}\\ =\frac{{\left(\frac{s}{v}\right)}^{\wp }}{{\left(\frac{s}{v}\right)}^{\wp }-\omega }={\left(1-\omega {\left(\frac{v}{s}\right)}^{\wp }\right)}^{-1}\end{array}$$

□

Theorem 8.

If $G(s)$ and $Z(v,s)$ are the ZZ and Aboodh transformations of $g(\vartheta )$, then the ZZ transform of the Atangana–Baleanu–Caputo derivative is defined as [52]

$$ZZ\left\{\begin{array}{c}ABC\hfill \\ 0\hfill \end{array}{D}_{\vartheta }^{\wp }g(\vartheta )\right\}=\left[\frac{\psi (\wp )}{1-\wp }\frac{\frac{s^{a+2}}{v^{\wp +2}}G\left(\frac{s}{v}\right)-\frac{s^{\wp }}{v^{\wp }}f(0)}{\frac{s^{\wp }}{v^{\wp }}+\frac{\wp }{1-\wp }}\right]$$

(17)

Proof. 

Applying Equations (4) and (8), we obtain

$$G\left(\frac{s}{v}\right)=\frac{v}{s}\left[\frac{\psi (\wp )}{1-\wp }\frac{{\left(\frac{s}{v}\right)}^{\wp +1}G\left(\frac{s}{v}\right)-{\left(\frac{s}{v}\right)}^{\wp -1}f(0)}{{\left(\frac{s}{v}\right)}^{\wp }+\frac{\wp }{1-\wp }}\right]$$

(18)

Therefore, the $ZZ$ transformation of Atangana–Baleanu–Caputo is given as

$$\begin{array}{cc}\hfill Z(v,s)=& {\left(\frac{s}{v}\right)}^{2}G\left(\frac{s}{v}\right)={\left(\frac{s}{v}\right)}^2\frac{v}{s}\left[\frac{\psi (\wp )}{1-\wp }\frac{{\left(\frac{s}{v}\right)}^{\wp +1}G\left(\frac{s}{v}\right)-{\left(\frac{s}{v}\right)}^{\wp -1}f(0)}{{\left(\frac{s}{v}\right)}^{\wp }+\frac{\wp }{1-\wp }}\right]\hfill \\ & =\left[\frac{\psi (\wp )}{1-\wp }\frac{{\left(\frac{s}{v}\right)}^{\wp +2}G\left(\frac{s}{v}\right)-{\left(\frac{s}{v}\right)}^{\wp }f(0)}{{\left(\frac{s}{v}\right)}^{\wp }+\frac{\wp }{1-\wp }}\right]\hfill \end{array}$$

□

Theorem 9.

Let us suppose that $G(s)$ and $Z(v,s)$ are the ZZ and Aboodh transformations of $g(\vartheta )$. Then the ZZ transformation of the Atangana–Baleanu Riemann–Liouville derivative is defined as [52]

$$ZZ\left\{\begin{array}{c}ABR\hfill \\ 0\hfill \end{array}{D}_{\vartheta }^{\wp }f(\vartheta )\right\}=\left[\frac{\psi (\wp )}{1-\wp }\frac{\frac{5^{\wp +2}}{v^{\wp +2}}G\left(\frac{s}{v}\right)}{\frac{s^{\mu }}{v^{\mu }}+\frac{\wp }{1-\wp }}\right]$$

(19)

Proof. 

Applying Equations (4) and (7), we obtain

$$G\left(\frac{s}{v}\right)=\frac{v}{s}\left[\frac{\psi (\wp )}{1-\wp }\frac{{\left(\frac{s}{v}\right)}^{\wp +1}G\left(\frac{s}{v}\right)}{{\left(\frac{s}{v}\right)}^{\wp }+\frac{\wp }{1-\wp }}\right]$$

(20)

From Equation (12), the ZZ transformation of Atangana–Baleanu Riemann–Liouville is defined as

$$\begin{array}{cc}\hfill Z(v,s)={\left(\frac{s}{v}\right)}^{2}G\left(\frac{s}{v}\right)={\left(\frac{s}{v}\right)}^2\left(\frac{v}{s}\right)& \left[\frac{\psi (\wp )}{1-\wp }\frac{{\left(\frac{s}{v}\right)}^{\wp +1}G\left(\frac{s}{v}\right)}{{\left(\frac{s}{v}\right)}^{\wp }+\frac{\wp }{1-\wp }}\right]\hfill \\ & =\left[\frac{\psi (\wp )}{1-\wp }\frac{{\left(\frac{s}{v}\right)}^{\wp +2}G\left(\frac{s}{v}\right)}{{\left(\frac{s}{v}\right)}^{\wp }+\frac{\wp }{1-\wp }}\right]\hfill \end{array}$$

□

3. Basic Concept of the Suggested Method

Here, we suggest the arbitrary-order differential equation in order to show the fundamental solution method.

$$\begin{array}{cc}\hfill & {}^{ABC}{D}_{\tau }^{\wp }\mathcal{U}(\zeta ,\tau )+M\mathcal{U}(\zeta ,\tau )+N\mathcal{U}(\zeta ,\tau )=h(\zeta ,\tau ),\tau >0,0<\wp \le 1,\hfill \\ \hfill & \mathcal{U}(\zeta ,0)=g(\zeta ).\hfill \end{array}$$

(21)

Using the ZZ transform of Equation (21), we obtain

$$\begin{array}{cc}\hfill & \mathcal{Z}[D_{\tau }^{\wp }\mathcal{U}(\zeta ,\tau )+M\mathcal{U}(\zeta ,\tau )+N\mathcal{U}(\zeta ,\tau )]=\mathcal{Z}\left[h(\zeta ,\tau )\right],\tau >0,0<\wp \le 1,\hfill \\ \hfill & \mathcal{U}(\zeta ,\tau )=\frac{\nu }{s}g(\zeta )+\frac{1}{s^{\wp }}\mathcal{Z}\left[h(\zeta ,\tau )\right]-\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}[M\mathcal{U}(\zeta ,\tau )+N\mathcal{U}(\zeta ,\tau )].\hfill \end{array}$$

(22)

Now, applying the inverse ZZ transform, we obtain

$$\mathcal{U}(\zeta ,\tau )=F(\zeta ,\tau )-{\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}\{M\mathcal{U}(\zeta ,\tau )+N\mathcal{U}(\zeta ,\tau )\}\right],$$

(23)

where

$$\begin{array}{cc}\hfill F(\zeta ,\tau )& ={\mathcal{Z}}^{-1}\left[\frac{\nu }{s}g(\zeta )+\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}\left[h(\zeta ,\tau )\right]\right]\hfill \\ \hfill & =g(\nu )+{\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}\left[h(\zeta ,\tau )\right]\right].\hfill \end{array}$$

(24)

Now, the perturbation technique is described in terms of a power series with parameter p.

$$\mathcal{U}(\zeta ,\tau )=\sum _{\Im =0}^{\infty }{p}^{\Im }{\mathcal{U}}_{\Im }(\zeta ,\tau ),$$

(25)

where p is the perturbation parameter and $p\in [0,1]$.

The non-linear term can be defined as

$$N\mathcal{U}(\zeta ,\tau )=\sum _{\Im =0}^{\infty }{p}^{\Im }{H}_{\Im }({\mathcal{U}}_{\Im }),$$

(26)

where $H_{\Im }$ are He’s polynomials in terms of ${\mathcal{U}}_0,{\mathcal{U}}_1,{\mathcal{U}}_2,\cdots ,{\mathcal{U}}_{\Im },$ and can be calculated as

$$H_{\Im }({\mathcal{U}}_0,{\mathcal{U}}_1,\cdots ,{\mathcal{U}}_{\Im })=\frac{1}{\wp (\Im +1)}{D}_p^{\Im }{\left[N\left(\sum _{\Im =0}^{\infty }{p}^{\Im }{\mathcal{U}}_{\Im }\right)\right]}_{p=0},$$

(27)

where $D_p^{\Im }=\frac{{\partial }^{\Im }}{\partial p^{\Im }}.$

Substituting Equations (26) and (27) into Equation (23), we obtain

$$\begin{array}{cc}\hfill \sum _{\Im =0}^{\infty }{p}^{\Im }{\mathcal{U}}_{\Im }(\zeta ,\tau )& =F(\zeta ,\tau )-p\times \left[{\mathcal{Z}}^{-1}\left\{\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}\{M\sum _{\Im =0}^{\infty }{p}^{\Im }{\mathcal{U}}_{\Im }(\zeta ,\tau )\right.\right.\hfill \\ \hfill & \left.\left.+\sum _{\Im =0}^{\infty }{p}^{\Im }{H}_{\Im }({\mathcal{U}}_{\Im })\}\right\}\right].\hfill \end{array}$$

(28)

Both sides have a comparison coefficient of p, and we have

$$\begin{array}{cc}\hfill & p^0:{\mathcal{U}}_0(\zeta ,\tau )=F(\zeta ,\tau ),\hfill \\ \hfill & p^1:{\mathcal{U}}_1(\zeta ,\tau )={\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}(M{\mathcal{U}}_0(\zeta ,\tau )+H_0(\mathcal{U}))\right],\hfill \\ \hfill & p^2:{\mathcal{U}}_2(\zeta ,\tau )={\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}(M{\mathcal{U}}_1(\zeta ,\tau )+H_1(\mathcal{U}))\right],\hfill \\ \hfill & ⋮\hfill \\ \hfill & p^{\Im }:{\mathcal{U}}_{\Im }(\zeta ,\tau )={\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}(M{\mathcal{U}}_{\Im -1}(\zeta ,\tau )+H_{\Im -1}(\mathcal{U}))\right],\Im >0,\Im \in N.\hfill \end{array}$$

(29)

The ${\mathcal{U}}_{\Im }(\zeta ,\tau )$ component can be determined easily, which quickly leads us to the convergent series. We can obtain $p\to 1,$

$$\mathcal{U}(\zeta ,\tau )=\underset{M\to \infty }{lim}\sum _{\Im =1}^M{\mathcal{U}}_{\Im }(\zeta ,\tau ).$$

(30)

4. Numerical Implementations

This section uses HPTM to analyze magnetic-acoustic plasma waves in the nonlinear fifth-order model KdV equations.

Example 1.

Firstly, the following nonlinear fifth-order KdV equation is considered

$${}^{ABC}{D}_{\tau }^{\wp }\mathcal{U}+{\mathcal{U}}_{\zeta }+{\mathcal{U}}^2{\mathcal{U}}_{2\zeta }-{\mathcal{U}}_{\zeta }{\mathcal{U}}_{2\zeta }-20{\mathcal{U}}^2{\mathcal{U}}_{3\zeta }+{\mathcal{U}}_{5\zeta }=0,0<\wp \le 1,$$

(31)

with the initial condition

$$\mathcal{U}(\zeta ,\tau )=\frac{1}{\zeta }.$$

(32)

Using the ZZ transformation on Equation (31), we obtain

$$\mathcal{Z}\mathcal{U}(\zeta ,\tau )]=\frac{\nu }{s\zeta }-\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}{\mathcal{U}}_{\zeta }+{\mathcal{U}}^2{\mathcal{U}}_{2\zeta }+{\mathcal{U}}_{\zeta }{\mathcal{U}}_{2\zeta }-20{\mathcal{U}}^2{\mathcal{U}}_{3\zeta }+{\mathcal{U}}_{5\zeta }].$$

(33)

Next, applying the inverse of the ZZ transformation of Equation (33),

$$\left[\mathcal{U}(\zeta ,\tau )\right]=\frac{1}{\zeta }-L_{\rho }^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}[{\mathcal{U}}_{\zeta }+{\mathcal{U}}^2{\mathcal{U}}_{2\zeta }+{\mathcal{U}}_{\zeta }{\mathcal{U}}_{2\zeta }-20{\mathcal{U}}^2{\mathcal{U}}_{3\zeta }+{\mathcal{U}}_{5\zeta }]\right].$$

Now, we apply HPM

$$\begin{array}{cc}\hfill \sum _{\Im =0}^{\infty }{p}^{\Im }{\mathcal{U}}_{\Im }(\zeta ,\tau )& =\frac{1}{\zeta }-p\left[{\mathcal{Z}}^{-1}\right[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}[\left(\sum _{\Im =0}^{\infty }{p}^{\Im }{H}_{\Im }(\mathcal{U})\right)\hfill \\ \hfill & +{\left(\sum _{\Im =0}^{\infty }{p}^{\Im }{\mathcal{U}}_{\Im }(\zeta ,\tau )\right)}_{\zeta }+{\left(\sum _{\Im =0}^{\infty }{p}^{\Im }{\mathcal{U}}_{\Im }(\zeta ,\tau )\right)}_{5\zeta }\left]\right]].\hfill \end{array}$$

(34)

In Equation (34), $H_{\Im }(\zeta )$ represents the non-linear terms of He’s polynomial. He’s polynomials of the first few components are presented by

$$H_0(\mathcal{U})={\mathcal{U}}_0^2{({\mathcal{U}}_0)}_{2\zeta }+{({\mathcal{U}}_0)}_{\zeta }{({\mathcal{U}}_0)}_{2\zeta }-20{\mathcal{U}}_0^2{({\mathcal{U}}_0)}_{3\zeta },$$

$$H_1(\mathcal{U})={\mathcal{U}}_0^2{({\mathcal{U}}_1)}_{2\zeta }+2{\mathcal{U}}_0{\mathcal{U}}_1{({\mathcal{U}}_o)}_{2\zeta }+{({\mathcal{U}}_0)}_{\zeta }{({\mathcal{U}}_1)}_{2\zeta }+{({\mathcal{U}}_0)}_{2\zeta }{({\mathcal{U}}_1)}_{\zeta }-20{\mathcal{U}}_0^2{({\mathcal{U}}_1)}_{3\zeta }-40{\mathcal{U}}_0{\mathcal{U}}_1{({\mathcal{U}}_0)}_{3\zeta },$$

$$\begin{array}{cc}\hfill H_2(\mathcal{U})=& {\mathcal{U}}_0^2{({\mathcal{U}}_2)}_{2\zeta }+2{\mathcal{U}}_0{\mathcal{U}}_1{({\mathcal{U}}_1)}_{2\zeta }+2{\mathcal{U}}_0{\mathcal{U}}_2{({\mathcal{U}}_0)}_{2\zeta }+{\mathcal{U}}_1^2{({\mathcal{U}}_0)}_{2\zeta }+{({\mathcal{U}}_0)}_{\zeta }{({\mathcal{U}}_2)}_{2\zeta }+{({\mathcal{U}}_1)}_{\zeta }{({\mathcal{U}}_1)}_{2\zeta }\hfill \\ \hfill & +{({\mathcal{U}}_0)}_{2\zeta }{({\mathcal{U}}_2)}_{\zeta }-20{\mathcal{U}}_0^2{({\mathcal{U}}_2)}_{3\zeta }-40{\mathcal{U}}_0{\mathcal{U}}_1{({\mathcal{U}}_1)}_{3\zeta }-40{\mathcal{U}}_0{\mathcal{U}}_2{({\mathcal{U}}_0)}_{3\zeta }-20{\mathcal{U}}_1^2{({\mathcal{U}}_0)}_{3\zeta },\hfill \end{array}$$

$$\begin{array}{c}\hfill H_3(\mathcal{U})={\mathcal{U}}_0^2{({\mathcal{U}}_3)}_{2\zeta }+2{\mathcal{U}}_0{\mathcal{U}}_1{({\mathcal{U}}_2)}_{2\zeta }+2{\mathcal{U}}_0{\mathcal{U}}_2{({\mathcal{U}}_1)}_{2\zeta }+2{\mathcal{U}}_0{\mathcal{U}}_3{({\mathcal{U}}_0)}_{2\zeta }+{\mathcal{U}}_1^2{({\mathcal{U}}_1)}_{2\zeta }+2{\mathcal{U}}_1{\mathcal{U}}_2{({\mathcal{U}}_0)}_{2\zeta }\\ \hfill +{({\mathcal{U}}_0)}_{\zeta }{({\mathcal{U}}_3)}_{2\zeta }+{({\mathcal{U}}_1)}_{\zeta }{({\mathcal{U}}_2)}_{2\zeta }+{({\mathcal{U}}_1)}_{2\zeta }{({\mathcal{U}}_2)}_{\zeta }+{({\mathcal{U}}_0)}_{2\zeta }{({\mathcal{U}}_3)}_{\zeta }-20{\mathcal{U}}_0^2{({\mathcal{U}}_3)}_{3\zeta }-40{\mathcal{U}}_0{\mathcal{U}}_1{({\mathcal{U}}_2)}_{3\zeta }\\ \hfill -40{\mathcal{U}}_0{\mathcal{U}}_2{({\mathcal{U}}_1)}_{3\zeta }-40{\mathcal{U}}_0{\mathcal{U}}_3{({\mathcal{U}}_0)}_{3\zeta }-20{\mathcal{U}}_1^2{({\mathcal{U}}_1)}_{3\zeta }-40{\mathcal{U}}_1{\mathcal{U}}_2{({\mathcal{U}}_0)}_{3\zeta },\end{array}$$

$$\begin{array}{c}\hfill H_4(\mathcal{U})={\mathcal{U}}_0^2{(\mathcal{U})}_{2\zeta }+2{\mathcal{U}}_0{\mathcal{U}}_1{({\mathcal{U}}_3)}_{2\zeta }+2{\mathcal{U}}_0{\mathcal{U}}_2{({\mathcal{U}}_2)}_{2\zeta }+2{\mathcal{U}}_0{\mathcal{U}}_3{({\mathcal{U}}_1)}_{2\zeta }+2{\mathcal{U}}_0{\mathcal{U}}_4{({\mathcal{U}}_o)}_{2\zeta }+{\mathcal{U}}_1^2{({\mathcal{U}}_2)}_{2\zeta }\\ \hfill +2{\mathcal{U}}_1{\mathcal{U}}_2{({\mathcal{U}}_1)}_{2\zeta }+2{\mathcal{U}}_1{\mathcal{U}}_3{({\mathcal{U}}_0)}_{2\zeta }+{\mathcal{U}}_2^2{({\mathcal{U}}_0)}_{2\zeta }+{({\mathcal{U}}_0)}_{\zeta }{({\mathcal{U}}_4)}_{2\zeta }+{({\mathcal{U}}_1)}_{\zeta }({({\mathcal{U}}_3)}_{2\zeta }+{({\mathcal{U}}_2)}_{2\zeta }{({\mathcal{U}}_2)}_{\zeta }\\ \hfill +{({\mathcal{U}}_1)}_{2\zeta }{({\mathcal{U}}_3)}_{\zeta }+{({\mathcal{U}}_0)}_{2\zeta }{({\mathcal{U}}_4)}_{\zeta }-20{\mathcal{U}}_0^2{({\mathcal{U}}_4)}_{3\zeta }-40{\mathcal{U}}_0{\mathcal{U}}_1{({\mathcal{U}}_3)}_{3\zeta }-40{\mathcal{U}}_0{\mathcal{U}}_2{({\mathcal{U}}_2)}_{3\zeta }-40{\mathcal{U}}_0{\mathcal{U}}_3{({\mathcal{U}}_1)}_{3\zeta }\\ \hfill -40{\mathcal{U}}_0{\mathcal{U}}_4{({\mathcal{U}}_0)}_{3\zeta }-20{\mathcal{U}}_1^2{({\mathcal{U}}_2)}_{3\zeta }-40{\mathcal{U}}_1{\mathcal{U}}_2{({\mathcal{U}}_1)}_{3\zeta }-40{\mathcal{U}}_1{\mathcal{U}}_3{({\mathcal{U}}_0)}_{3\zeta }-20{\mathcal{U}}_2^2{({\mathcal{U}}_0)}_{2\zeta }.\\ ⋮\end{array}$$

Comparing the P-like coefficient, we obtain

$$\begin{array}{cc}\hfill P^0:{\mathcal{U}}_0(\zeta ,\tau )=& \frac{1}{\zeta },\hfill \\ \hfill P^1:{\mathcal{U}}_1(\zeta ,\tau )=& -{\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}[H_0(\mathcal{U})+{({\mathcal{U}}_0)}_{\zeta }+{({\mathcal{U}}_0)}_{5\zeta }]\right]\hfill \\ \hfill & =\frac{1}{{\zeta }^2\psi (\wp )}\left[1-\wp +\frac{\wp {\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill P^2:{\mathcal{U}}_2(\zeta ,\tau )=& -{\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}[H_1(\mathcal{U})+{({\mathcal{U}}_1)}_{\zeta }+{({\mathcal{U}}_1)}_{5\zeta }]\right]\hfill \\ \hfill =& \frac{1}{{\zeta }^3{(\psi (\wp ))}^2}\left[{(1-\wp )}^2+\frac{2\wp (1-\wp ){\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}+\frac{{\wp }^2{\tau }^{2\wp }}{\mathsf{\Gamma }(2\wp +1)}\right],\hfill \end{array}$$

$$\begin{array}{c}\hfill P^3:{\mathcal{U}}_3(\zeta ,\tau )=-{\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}[H_2(\mathcal{U})+{({\mathcal{U}}_2)}_{\zeta }+{({\mathcal{U}}_2)}_{5\zeta }]\right]\\ \hfill =\frac{1}{{\zeta }^4{(\psi (\alpha ))}^3}\left[{(1-\wp )}^3+\frac{3\wp {(1-\wp )}^2{\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}+\frac{{\wp }^2(1-\wp ){\tau }^{2\wp +1}}{\mathsf{\Gamma }(2\wp +2)}+\frac{2{\wp }^2(1-\wp ){\tau }^{2\wp }}{\mathsf{\Gamma }(2\wp +1)}+\frac{{\wp }^3{\tau }^{2\wp +1}}{\mathsf{\Gamma }(2\wp +2)}\right],\\ \hfill ⋮\end{array}$$

Therefore, the approximate result of $\mathcal{U}(\zeta ,\tau )$ is given by

$$\begin{array}{cc}\hfill \mathcal{U}(\zeta ,\tau )& =\sum _{\Im =0}^{\infty }{\mathcal{U}}_{\Im }(\zeta ,\tau )=\frac{1}{\zeta }+\frac{1}{{\zeta }^2\psi (\wp )}\left[1-\wp +\frac{\wp {\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}\right]+\frac{1}{{\zeta }^3{(\psi (\wp ))}^2}ł[{(1-\wp )}^2+\hfill \\ \hfill & \frac{2\wp (1-\wp ){\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}+\frac{{\wp }^2{\tau }^{2\wp }}{\mathsf{\Gamma }(2\wp +1)}]+\frac{1}{{\zeta }^4{(\psi (\alpha ))}^3}[{(1-\wp )}^3+\frac{3\wp {(1-\wp )}^2{\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}+\frac{{\wp }^2(1-\wp ){\tau }^{2\wp +1}}{\mathsf{\Gamma }(2\wp +2)}\hfill \\ \hfill & +\frac{2{\wp }^2(1-\wp ){\tau }^{2\wp }}{\mathsf{\Gamma }(2\wp +1)}+\frac{{\wp }^3{\tau }^{2\wp +1}}{\mathsf{\Gamma }(2\wp +2)}]+\cdots .\hfill \end{array}$$

(35)

Then, with $\wp =1$ put into Equation (35)

$$\mathcal{U}(\zeta ,\tau )=\sum _{\Im =0}^{\infty }{\mathcal{U}}_{\Im }(\zeta ,\tau )=\frac{1}{\zeta }+\frac{\tau }{{\zeta }^2}+\frac{{\tau }^2}{{\zeta }^3}+\frac{{\tau }^3}{{\zeta }^4}+\cdots .$$

As $\wp =1$, this series has the closed form $\mathcal{U}(\zeta ,\tau )=\frac{1}{\zeta -\tau }$, which is an exact result of the given Korteweg–de Vries Equation (31) for standard value $\wp =1$ and this solution is closed, in agreement with Goswami et al. [16].

In Figure 1, the 3-dimensional graphs of actual and HPTM results are shown in Figure 1a and Figure 1b, respectively, at $\wp =1$, and the close relation of the HPTM and exact results is studied. In Figure 2, we show different fractional-orders of HPTM solutions at $\wp =1,0.8,0.6,0.4$. The fractional outcomes are examined to be convergent to an integer-order solution of the model.

Example 2.

The following nonlinear fifth-order KdV equation is considered

$${}^{ABC}{D}_{\tau }^{\wp }\mathcal{U}+\mathcal{U}{\mathcal{U}}_{\zeta }-\mathcal{U}{\mathcal{U}}_{3\zeta }+{\mathcal{U}}_{5\zeta }=0,0<\wp \le 1,$$

(36)

with the initial condition

$$\mathcal{U}(\zeta ,\tau )=e^{\zeta }.$$

(37)

Using the ZZ transformation on Equation (36), we obtain

$$\mathcal{Z}\left[\mathcal{U}(\zeta ,\tau )\right]=\frac{\nu }{s}{e}^{\zeta }+\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}[\mathcal{U}{\mathcal{U}}_{3\zeta }-\mathcal{U}{\mathcal{U}}_{\zeta }-{\mathcal{U}}_{5\zeta }].$$

(38)

Next, applying the inverse of the ZZ transformation of Equation (38),

$$\mathcal{U}(\zeta ,\tau )=e^{\zeta }+{\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}\left\{\mathcal{U}{\mathcal{U}}_{3\zeta }-\mathcal{U}{\mathcal{U}}_{\zeta }-{\mathcal{U}}_{5\zeta }\right\}\right].$$

Now, we apply HPM

$$\begin{array}{cc}\hfill \sum _{\Im =0}^{\infty }{p}^{\Im }{\mathcal{U}}_{\Im }(\zeta ,\tau )& =e^{\zeta }+p\left[{\mathcal{Z}}^{-1}\right\{\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}(\left(\sum _{\Im =0}^{\infty }{p}^{\Im }{H}_{\Im }(\mathcal{U})\right)\hfill \\ \hfill & -\left(\sum _{\Im =0}^{\infty }{p}^{\Im }{\mathcal{U}}_{\Im }(\zeta ,\tau )\right){)}_{5\zeta }\left\}\right].\hfill \end{array}$$

(39)

In Equation (39), $H_n(\zeta )$ represents the non-linear terms of He’s polynomial. He’s polynomials of the first few components are presented by

$$H_0(\mathcal{U})={\mathcal{U}}_o{({\mathcal{U}}_0)}_{3\zeta }-{\mathcal{U}}_0{({\mathcal{U}}_0)}_{\zeta }.$$

$$H_1(\mathcal{U})={\mathcal{U}}_1{({\mathcal{U}}_0)}_{3\zeta }+{\mathcal{U}}_0{({\mathcal{U}}_1)}_{3\zeta }-{\mathcal{U}}_1{({\mathcal{U}}_0)}_{\zeta }-{\mathcal{U}}_0{({\mathcal{U}}_1)}_{\zeta }.$$

$$H_2(\mathcal{U})={\mathcal{U}}_2{({\mathcal{U}}_0)}_{3\zeta }+{\mathcal{U}}_1{({\mathcal{U}}_1)}_{3\zeta }+{\mathcal{U}}_0{({\mathcal{U}}_2)}_{3\zeta }-{\mathcal{U}}_2{({\mathcal{U}}_0)}_{\zeta }-{\mathcal{U}}_1{({\mathcal{U}}_1)}_{\zeta }-{\mathcal{U}}_0{({\mathcal{U}}_2)}_{\zeta }.$$

$$H_3(\mathcal{U})={\mathcal{U}}_3{({\mathcal{U}}_0)}_{3\zeta }+{\mathcal{U}}_2{({\mathcal{U}}_1)}_{3\zeta }+{\mathcal{U}}_1{({\mathcal{U}}_2)}_{3\zeta }+{\mathcal{U}}_0{({\mathcal{U}}_3)}_{3\zeta }-{\mathcal{U}}_3{({\mathcal{U}}_0)}_{\zeta }-{\mathcal{U}}_2{({\mathcal{U}}_1)}_{\zeta }-{\mathcal{U}}_1{({\mathcal{U}}_2)}_{\zeta }-{\mathcal{U}}_0{({\mathcal{U}}_3)}_{\zeta }.$$

$$\begin{array}{cc}\hfill H_4(\mathcal{U})=& {\mathcal{U}}_4{({\mathcal{U}}_0)}_{3\zeta }+{\mathcal{U}}_3{({\mathcal{U}}_1)}_{3\zeta }+{\mathcal{U}}_2{({\mathcal{U}}_2)}_{3\zeta }+{\mathcal{U}}_1{({\mathcal{U}}_3)}_{3\zeta }+{\mathcal{U}}_0{({\mathcal{U}}_4)}_{3\zeta }-{\mathcal{U}}_4{({\mathcal{U}}_0)}_{\zeta }-{\mathcal{U}}_3{({\mathcal{U}}_1)}_{\zeta }\hfill \\ \hfill & -{\mathcal{U}}_2{({\mathcal{U}}_2)}_{\zeta }-{\mathcal{U}}_1{({\mathcal{U}}_3)}_{\zeta }-{\mathcal{U}}_0{({\mathcal{U}}_4)}_{\zeta }.\hfill \end{array}$$

$$\begin{array}{c}\hfill ⋮\hfill \end{array}$$

Comparing the P-like coefficient, we obtain

$$\begin{array}{cc}\hfill p^0:{\mathcal{U}}_0(\zeta ,\tau )=& e^{\zeta },\hfill \\ \hfill p^1:{\mathcal{U}}_1(\zeta ,\tau )=& {\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}\{H_0(\mathcal{U})-{({\mathcal{U}}_0)}_{5\zeta }\}\right]=-\frac{1}{\psi (\wp )}\left[1-\wp +\frac{\wp {\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}\right]e^{\zeta },\hfill \\ \hfill p^2:{\mathcal{U}}_2(\zeta ,\tau )=& {\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}\{H_1(\mathcal{U})-{({\mathcal{U}}_1)}_{5\zeta }\}\right]\hfill \\ & =\frac{1}{{(\psi (\wp ))}^2}\left[{(1-\wp )}^2+\frac{2\wp (1-\wp ){\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}+\frac{{\wp }^2{\tau }^{2\wp }}{\mathsf{\Gamma }(2\wp +1)}\right]e^{\zeta },\hfill \\ \hfill p^3:{\mathcal{U}}_3(\zeta ,\tau )=& {\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}\{H_2(\mathcal{U})-{({\mathcal{U}}_2)}_{5\zeta }\}\right]\hfill \\ \hfill & =-\frac{1}{{(\psi (\alpha ))}^3}[{(1-\wp )}^3+\frac{3\wp {(1-\wp )}^2{\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}+\frac{{\wp }^2(1-\wp ){\tau }^{2\wp +1}}{\mathsf{\Gamma }(2\wp +2)}\hfill \\ \hfill & +\frac{2{\wp }^2(1-\wp ){\tau }^{2\wp }}{\mathsf{\Gamma }(2\wp +1)}+\frac{{\wp }^3{\tau }^{2\wp +1}}{\mathsf{\Gamma }(2\wp +2)}]e^{\zeta },\hfill \\ \hfill & ⋮\hfill \end{array}$$

Therefore, the approximate result of $\mathcal{U}(\zeta ,\tau )$ is given by

$$\begin{array}{cc}\hfill \mathcal{U}(\zeta ,\tau )& =\sum _{\Im =0}^{\infty }{\mathcal{U}}_{\Im }(\zeta ,\tau )=e^{\zeta }(1-\frac{1}{\psi (\wp )}\left[1-\wp +\frac{\wp {\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}\right]+\frac{1}{{(\psi (\wp ))}^2}\left[{(1-\wp )}^2\right.\hfill \\ \hfill & \left.+\frac{2\wp (1-\wp ){\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}+\frac{{\wp }^2{\tau }^{2\wp }}{\mathsf{\Gamma }(2\wp +1)}\right]-\frac{1}{{(\psi (\alpha ))}^3}[{(1-\wp )}^3+\frac{3\wp {(1-\wp )}^2{\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}+\frac{{\wp }^2(1-\wp ){\tau }^{2\wp +1}}{\mathsf{\Gamma }(2\wp +2)}\hfill \\ \hfill & +\frac{2{\wp }^2(1-\wp ){\tau }^{2\wp }}{\mathsf{\Gamma }(2\wp +1)}+\frac{{\wp }^3{\tau }^{2\wp +1}}{\mathsf{\Gamma }(2\wp +2)}]+\cdots ).\hfill \end{array}$$

(40)

Then, with $\wp =1$ for Equation (40), we obtain

$$\mathcal{U}(\zeta ,\tau )=\sum _{\Im =0}^{\infty }{\mathcal{U}}_{\Im }(\zeta ,\tau )=e^{\zeta }\left(1-\tau +\frac{{\tau }^2}{2!}-\frac{{\tau }^3}{3!}+\frac{{\tau }^4}{4!}-\frac{{\tau }^5}{5!}+\cdots \right).$$

As $\wp =1$, this series has the closed form $\mathcal{U}(\zeta ,\tau )=e^{\zeta -\tau }$, which is an exact result of the given KdV Equation (36) for standard value $\wp =1$, and this solution is closed in agreement with Goswami et al. [16].

In Figure 3, the 3-dimensional graphs of the actual and HPTM results are shown in Figure 3a and Figure 3b, respectively, at $\wp =1$, and the close relation of the HPTM and exact results is studied. In Figure 4, we show different fractional-orders of HPTM solutions at $\wp =1,0.8,0.6,0.4$. The fractional outcomes are examined to be convergent to an integer-order solution of the model.

Example 3.

Consider the following non-linear Kawahara equation

$${}^{ABC}{D}_{\tau }^{\wp }\mathcal{U}+\mathcal{U}{\mathcal{U}}_{\zeta }+{\mathcal{U}}_{3\zeta }-{\mathcal{U}}_{5\zeta }=0,0<\wp \le 1,$$

(41)

with the initial condition

$$\mathcal{U}(\zeta ,\tau )=\frac{105}{169}sec{h}^4\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right).$$

(42)

Using the ZZ transformation on Equation (41), we obtain

$$\mathcal{Z}\left[\mathcal{U}(\zeta ,\tau )\right]=\frac{\nu }{s}\frac{105}{169}sec{h}^4\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)+\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}[{\mathcal{U}}_{5\zeta }-{\mathcal{U}}_{3\zeta }-\mathcal{U}{\mathcal{U}}_{\zeta }].$$

(43)

Next, applying the inverse of ZZ transformation of Equation (43)

$$\mathcal{U}(\zeta ,\tau )=\frac{105}{169}sec{h}^4\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)+{\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}[{\mathcal{U}}_{5\zeta }-{\mathcal{U}}_{3\zeta }-\mathcal{U}{\mathcal{U}}_{\zeta }]\right].$$

Now, we apply HPM

$$\begin{array}{cc}\hfill \sum _{\Im =0}^{\infty }{p}^{\Im }{\mathcal{U}}_{\Im }(\zeta ,\tau )=& \frac{105}{169}sec{h}^4\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)+p\left[{\mathcal{Z}}^{-1}\right\{\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}({\left(\sum _{\Im =0}^{\infty }{p}^{\Im }{\mathcal{U}}_{\Im }(\zeta ,\tau )\right)}_{5\zeta }\hfill \\ \hfill & -{\left(\sum _{\Im =0}^{\infty }{p}^{\Im }{\mathcal{U}}_{\Im }(\zeta ,\tau )\right)}_{3\zeta }-\left(\sum _{\Im =0}^{\infty }{p}^{\Im }{H}_{\Im }(\mathcal{U})\right))\}].\hfill \end{array}$$

(44)

In Equation (44), $H_n(\mathcal{U})$ represents the non-linear terms of He’s polynomial. He’s polynomials of the first few components are presented by

$$\begin{array}{cc}\hfill H_0(\mathcal{U})=& {\mathcal{U}}_0{({\mathcal{U}}_0)}_{\zeta },\hfill \\ \hfill H_1(\mathcal{U})=& {\mathcal{U}}_0{({\mathcal{U}}_1)}_{\zeta }+{\mathcal{U}}_1{({\mathcal{U}}_0)}_{\zeta },\hfill \\ \hfill H_2(\mathcal{U})=& {\mathcal{U}}_0{({\mathcal{U}}_2)}_{\zeta }+{\mathcal{U}}_1{({\mathcal{U}}_1)}_{\zeta }+{\mathcal{U}}_2{({\mathcal{U}}_0)}_{\zeta },\hfill \\ \hfill H_3(\mathcal{U})=& {\mathcal{U}}_0{({\mathcal{U}}_3)}_{\zeta }+{\mathcal{U}}_1{({\mathcal{U}}_2)}_{\zeta }+{\mathcal{U}}_2{({\mathcal{U}}_1)}_{\zeta }+{\mathcal{U}}_3{({\mathcal{U}}_0)}_{\zeta },\hfill \\ \hfill H_4(\mathcal{U})=& {\mathcal{U}}_0{({\mathcal{U}}_4)}_{\zeta }+{\mathcal{U}}_1{({\mathcal{U}}_3)}_{\zeta }+{\mathcal{U}}_2{({\mathcal{U}}_2)}_{\zeta }+{\mathcal{U}}_3{({\mathcal{U}}_1)}_{\zeta }+{\mathcal{U}}_4{({\mathcal{U}}_0)}_{\zeta }.\hfill \\ & ⋮\hfill \end{array}$$

Comparing the P-like coefficient, we obtain

$$p^0:{\mathcal{U}}_0(\zeta ,\tau )=\frac{105}{169}sec{h}^4\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right).$$

$$\begin{array}{cc}\hfill p^1:{\mathcal{U}}_1(\zeta ,\tau )& ={\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}[{({\mathcal{U}}_0)}_{5\zeta }-{({\mathcal{U}}_0)}_{5\zeta }-H_0(\mathcal{U})]\right]\hfill \\ \hfill & =-\frac{100}{377\sqrt{13}}sec{h}^4\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)tanh\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)\frac{1}{\psi (\wp )}\left[1-\wp +\frac{\wp {\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & p^2:{\mathcal{U}}_2(\zeta ,\tau )={\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}[{({\mathcal{U}}_1)}_{5\zeta }-{({\mathcal{U}}_1)}_{3\zeta }-H_1(\mathcal{U})\right]\hfill \\ \hfill & =-\frac{21687}{10\times {10}^7\sqrt{13}}sec{h}^6\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)\left[-3+2cosh\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)\right]\frac{1}{{(\psi (\wp ))}^2}\left[{(1-\wp )}^2\right.\hfill \\ \hfill & \left.+\frac{2\wp (1-\wp ){\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}+\frac{{\wp }^2{\tau }^{2\wp }}{\mathsf{\Gamma }(2\wp +1)}\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & p^3:{\mathcal{U}}_3(\zeta ,\tau )={\mathcal{Z}}^{-1}\left[\frac{\left(1-\wp +\wp {\left(\frac{\nu }{s}\right)}^{\wp }\right)}{\psi (\wp )}\mathcal{Z}\{{({\mathcal{U}}_2)}_{5\zeta }-{({\mathcal{U}}_2)}_{3\zeta }-H_2(\mathcal{U})\}\right]\hfill \\ \hfill & =-\frac{461962}{10\times {10}^7\sqrt{13}}sec{h}^7\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)\times \left[-13sinh\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)+2sinh\left(\frac{3(\zeta -\varphi )}{2\sqrt{‘13}}\right)\right]\hfill \\ \hfill & \frac{1}{{(\psi (\alpha ))}^3}\left[{(1-\wp )}^3+\frac{3\wp {(1-\wp )}^2{\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}+\frac{{\wp }^2(1-\wp ){\tau }^{2\wp +1}}{\mathsf{\Gamma }(2\wp +2)}+\frac{2{\wp }^2(1-\wp ){\tau }^{2\wp }}{\mathsf{\Gamma }(2\wp +1)}+\frac{{\wp }^3{\tau }^{2\wp +1}}{\mathsf{\Gamma }(2\wp +2)}\right],\hfill \end{array}$$

$$\begin{array}{c}\hfill ⋮\hfill \end{array}$$

Therefore, the approximate result $\mathcal{U}(\zeta ,\tau )$ is obtained by

$$\mathcal{U}(\zeta ,\tau )=\sum _{\Im =0}^{\infty }{\mathcal{U}}_{\Im }(\zeta ,\tau ),$$

$$\begin{array}{cc}\hfill \mathcal{U}(\zeta ,\tau )=& \frac{105}{169}sec{h}^4\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)-\frac{100}{377\sqrt{13}}sec{h}^4\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)tanh\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)\frac{1}{\psi (\wp )}\left[1-\wp +\frac{\wp {\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}\right]\hfill \\ \hfill & -\frac{21687}{10\times {10}^7\sqrt{13}}sec{h}^6\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)\left[-3+2cosh\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)\right]\frac{1}{{(\psi (\wp ))}^2}[{(1-\wp )}^2\hfill \\ \hfill & +\frac{2\wp (1-\wp ){\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}+\frac{{\wp }^2{\tau }^{2\wp }}{\mathsf{\Gamma }(2\wp +1)}]-\frac{461962}{10\times {10}^7\sqrt{13}}sec{h}^7\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)\times \hfill \\ \hfill & \left[-13sinh\left(\frac{\zeta -\varphi }{2\sqrt{13}}\right)+2sinh\left(\frac{3(\zeta -\varphi )}{2\sqrt{‘13}}\right)\right]\frac{1}{{(\psi (\alpha ))}^3}[{(1-\wp )}^3+\frac{3\wp {(1-\wp )}^2{\tau }^{\wp }}{\mathsf{\Gamma }(\wp +1)}\hfill \\ \hfill & +\frac{{\wp }^2(1-\wp ){\tau }^{2\wp +1}}{\mathsf{\Gamma }(2\wp +2)}+\frac{2{\wp }^2(1-\wp ){\tau }^{2\wp }}{\mathsf{\Gamma }(2\wp +1)}+\frac{{\wp }^3{\tau }^{2\wp +1}}{\mathsf{\Gamma }(2\wp +2)}]+\cdots \hfill \end{array}$$

As $\wp =1$, this series has the closed form $\mathcal{U}(\zeta ,\tau )=\frac{105}{169}sec{h}^4\left[\frac{1}{2\sqrt{13}}(\zeta +\frac{36\tau }{169}-\varphi )\right]$, which is an exact result of the given non-linear Kawahara Equation (41) for standard value $\wp =1$, and this solution is closed in agreement with Goswami et al. [16].

In Figure 5, the 3-dimensional graphs of actual and HPTM results are shown in Figure 5a and Figure 5b, respectively, at $\wp =1$, and the close relation of the HPTM and exact results is studied. In Figure 6, (a) we show different fractional-orders of HPTM solutions at $\wp =1,0.8,0.6,0.4$ and (b) show that the error plot. The fractional outcomes are examined to be convergent to an integer-order solution of the model.

5. Conclusions

This article evaluated the fractional-order fifth-order KdV/Kawahara-type equations, using the HPTM. The results for particular examples were explained by applying the current method. The HPTM result is in close relation with the exact result of the proposed problems. The suggested method also calculated the results of the examples with the fractional-order derivatives. Graphical investigation of the fractional-order results obtained confirmed the convergence towards integer-order results. Moreover, the HPTM technique is straightforward, simple, and carries less computational cost; the proposed method can be modified to solve other fractional-order partial differential equations. This investigation could help researchers to gain a deep understanding of the mechanisms of generation and propagation of nonlinear waves in different plasma systems in addition to the various fields of science. The methods used in this study can also be used for analyzing both the modified Kawahara equation and the Gardner–Kawahara equation, which are used for describing the nonlinear structures in a critical plasma composition and near a critical plasma composition, respectively.