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Modified homotopy methods for generalized fractional perturbed Zakharov–Kuznetsov equation in dusty plasma

Abstract

We propose a new modification of homotopy perturbation method (HPM) called the δ-homotopy perturbation transform method (δ-HPTM). This modification consists of the Laplace transform method, HPM, and a control parameter δ. This control convergence parameter δ in this new modification helps in adjusting and controlling the convergence region of the series solution and overcome some limitations of HPM and HPTM. The δ-HPTM and q-homotopy analysis transform method (q-HATM) are considered to study the generalized time-fractional perturbed \((3+1)\)-dimensional Zakharov–Kuznetsov equation with Caputo fractional time derivative. This equation describes nonlinear dust-ion-acoustic waves in the magnetized two-ion-temperature dusty plasmas. The selection of an appropriate value of δ in δ-HPTM and the auxiliary parameters n and ħ in q-HATM gives a guaranteed convergence of series solution, but the difference between the two techniques is that the embedding parameter p in δ-HPTM varies from zero to nonzero δ, whereas the embedding parameter q in q-HATM varies from zero to \(\frac{1}{n}, n\geq{1}\). We examine the effect of fractional order on the considered problem and present the error estimate when compared with exact solution. The outcomes reveal complete reliability and efficiency of the proposed algorithm for solving various types of physical models arising in sciences and engineering. Furthermore, we present the convergence and error analysis of the two methods.

Introduction

The study of fractional partial differential equations (FPDEs) has enticed the interest of many researchers in the field of applied sciences and engineering by virtue of its enormous applications in electrodynamics, random walk, biotechnology, viscoelasticity, chaos theory, signal and image processing, nanotechnology, and many other areas [120]. Also, essential properties of fractional calculus were outlined by many researchers (see [2124] for detailed discussion). Nevertheless, solving FPDEs is generally more complex than the classical type since their operators are defined through integrals. There are many techniques proposed by many researchers to handle analytical and approximate solutions of nonlinear FPDEs such as the residual power series method [2528], iterative Shehu transform method [29], Laplace decomposition method [30], q-homotopy analysis method [3134], Adomian decomposition method [35], fractional reduced differential transform method [36, 37], variational iteration method [38, 39], homotopy analysis method [40], and other methods [4144].

The homotopy perturbation method (HPM) was developed by He [4550] by combining the perturbation and standard homotopy for solving numerous physical problems. We refer the reader to He’s works for a clear understanding of HPM, where further insights can be found. Recently, an improved modification of HPM, called the parameterized homotopy perturbation method (PHPM), was proposed in [51, 52]. Another formulation, called the He–Laplace method, was proposed to obtain an exact closed approximate solution of nonlinear models [53, 54]. The HPM and well-known Laplace transformation method were combined to produce a highly effective technique, called the homotopy perturbation transform method (HPTM), for solving many nonlinear problems [55, 56]. It is worth noting that the Laplace transform method alone in some cases is insufficient in handling nonlinear problems because of the difficulties that may arise by the nonlinear terms. In this present study, we propose a new modification of HPM, called the δ-homotopy perturbation transform method (δ-HPTM), which consists of HPM, the Laplace transform method, and a control parameter δ. Similarly to the control parameters n and ħ in q-HATM, the control parameter δ in δ-HPTM also helps in adjusting and controlling the convergence region of the series solutions and can overcome some limitations of HPM, HPTM, and He–Laplace method. It is worth mentioning that the present modification (δ-HPTM) requires neither polynomials like ADM nor Lagrange multipliers like VIM and overcomes the limitations of these methods.

To elucidate the reliability and effectiveness of the proposed modification, we consider the generalized time-fractional perturbed \((3+1)\)-dimensional Zakharov–Kuznetsov (gpZK) equation given by

$$\begin{aligned} &\mathcal{D}_{t}^{\gamma}W+\beta_{1} W^{k} \frac{\partial{W}}{\partial{x}}+\beta_{2} \frac{\partial^{3} W}{\partial{x^{3}}}+ \beta_{3} \biggl( \frac{\partial^{3} W}{\partial x\,\partial y^{2}}+ \frac{\partial^{3} W}{\partial x\,\partial z^{2}} \biggr)+\xi \frac{\partial^{5} W}{\partial x^{5}}=0, \\ &\quad 0< \gamma\leq{1}, t>0, \end{aligned}$$
(1)

where W represents the electrostatic potential, k is a positive number, γ is the fractional order, ξ represents a smallness parameter, and the physical quantities \(\beta_{1}, \beta_{2}\), and \(\beta_{3}\) are constants. Zhen et al. [57] and Seadawy et al. [58, 59] have outlined these physical quantities. This equation is used to describe the nonlinear dust-ion-acoustic waves in the magnetized two-ion-temperature dusty plasmas [60, 61]. The study of ion-acoustic waves and structures in dense quantum plasmas has attracted a lot of consideration in recent years. The ZK equation comprises the nonlinear term \(W\frac{\partial{W}}{\partial{x}}\) and third-order dispersion term \(\frac{\partial^{3} W}{\partial{x^{3}}}\):

$$\begin{aligned} \frac{\partial{W}}{\partial{t}}+\beta_{1} W \frac{\partial{W}}{\partial{x}}+ \beta_{2} \frac{\partial^{3} W}{\partial{x^{3}}}+\beta_{3} \biggl( \frac{\partial^{3} W}{\partial x\,\partial y^{2}}+ \frac{\partial^{3} W}{\partial x\,\partial z^{2}} \biggr). \end{aligned}$$
(2)

Equation (2) is limited to the waves of small amplitudes only. The width of the soliton and its velocity deviate from the predictions of this equation as the amplitude of the wave increases. The pZK equation (1) with fractional order \(\gamma=1\) and \(k=1\) includes an of extra fifth-order dispersion term \(\xi\frac{\partial^{5} W}{\partial x^{5}}\) was proposed to overcome this problem (see [5759, 62], for more detail). The proposed δ-HPTM and q-HATM are employed to compute numerical solutions of Eq. (1). The two algorithms provide the solutions in a rapid convergent series, which can lead the solutions to a closed form. To the author’s knowledge, the approximate solutions of the gpZK (1) was not addressed in the literature before.

The rest of the paper is structured as follows. Useful notations and definitions are provided in Sect. 2. The essential idea of the two methods with convergence and error analysis are presented in Sect. 3. The applications of δ-HPTM and q-HATM on the generalized time-fractional pZK equation are detailed in Sect. 4. Numerical comparison and discussion are provided in Sect. 5. Lastly, Sect. 6 concludes the paper.

Preliminaries

This section contains some helpful notations and definitions.

Definition 1

Let \(\omega\in\mathbb{R}\) and \(m\in\mathbb{N}\). A function W is said to be in the space \(\mathbb{C}_{\omega}\) if there exists \(\eta\in\mathbb{R}\), \(\eta>\omega\), and \(Z\in C{[0, \infty)}\) such that \(W(t)=t^{\eta}Z(t)\) \(t\in\mathbb{R}^{+}\). Furthermore, \(W\in\mathbb{C}^{m}_{\omega}\) if \(W^{(m)}\in\mathbb{C}_{\omega}\) [63].

Definition 2

The Riemann–Liouville (RL) fractional integral of order γ of a function \(W(t)\in C_{\omega}, \omega\geq{-1}\), is given as [23, 6365]

$$ J^{\gamma}W(t)=\frac{1}{\Gamma(\gamma)} \int_{0}^{t}(t-\zeta)^{ \gamma-1} W(\zeta) \,d \zeta,\quad \gamma, t>0, $$
(3)

where \(J^{0}W(t)=W(t)\), and Γ is the classical gamma function.

Definition 3

The fractional derivative of \(W(t)\) (denoted by \(\mathcal{D}^{\gamma}W(t)\)) in the Caputo sense for \(m-1<\gamma<m\), \(m\in\mathbb{N}\), is defined as [23, 65]

$$\begin{aligned} \mathcal{D}^{\gamma}W(t)= \textstyle\begin{cases} W^{(m)}(t),& \gamma=m, \\ J^{m-\gamma}W^{(m)}(t),& m-1< \gamma< m, \end{cases}\displaystyle \end{aligned}$$
(4)

where

$$ J^{m-\gamma}W^{(\gamma)}(t)=\frac{1}{\Gamma(m-\gamma)} \int_{0}^{t}(t- \zeta)^{m-\gamma-1}W^{(\gamma)}( \zeta) \,d\zeta,\quad \gamma, t>0, $$
(5)

with the following properties:

  1. a.

    \(\mathcal{D}^{\gamma} (\tau_{1}W(t)+\tau_{2}V(t) )=\tau_{1} \mathcal{D}^{\gamma}W(t)+\tau_{2} \mathcal{D}^{\gamma}V(t)\), \(\tau_{1},\tau_{2}\in\mathbb{R}\),

  2. b.

    \(\mathcal{D}^{\gamma}J^{\gamma}W(t)=W(t)\),

  3. c.

    \(J^{\gamma}\mathcal{D}^{\gamma}W(t)=W(t)-\sum_{j=0}^{m-1}W^{j}_{0}(t) \frac{t^{j}}{j!}\).

Definition 4

The Laplace transform (denoted by \(\mathscr{L}\)) of a Riemann–Liouville fractional integral \((J_{t}^{\gamma}W(t) )\) and Caputo fractional derivative \((\mathcal{D}_{t}^{\gamma}W(t) )\) of a function \(W\in\mathbb{C}_{\omega} (\omega\geq{-1})\) are given respectively as [21, 65]

$$ \begin{aligned} &\mathscr{L} \bigl[J_{t}^{\gamma}W(t) \bigr]=s^{-\gamma} \mathscr{L}\bigl[W(t)\bigr], \\ &\mathscr{L} \bigl[\mathcal{D}_{t}^{\gamma}W(t) \bigr]=s^{\gamma} \mathscr{L}\bigl[W(t)\bigr]-\sum _{r=0}^{m-1}s^{\gamma-r-1}{W^{(r)}} \bigl(0^{+}\bigr),\quad {m-1}< \gamma\leq{m}, \end{aligned} $$
(6)

where s is a parameter.

Analysis of the proposed methods

Here we give the general idea of the δ-HPTM and q-HATM. We also present some convergence and error analysis of the two methods. Consider the general nonlinear FPDE of the form

$$\begin{aligned} \mathcal{D}_{t}^{\gamma}W+\mathcal{M} (W )+ \mathcal {N}(W )= \Phi,\quad m-1< \gamma\leq{m}, \end{aligned}$$
(7)

with initial conditions

$$\begin{aligned} W^{(r)}(x,y,z,0)=\frac{\partial^{r} W(x,y,z,0)}{\partial t^{r}}=f_{r}(x,y,z),\quad r=0, 1, 2, \ldots, m-1, \end{aligned}$$
(8)

where \(\mathcal{D}_{t}^{\gamma}\) represents the Caputo fractional derivative, \(\mathcal{M}\) and \(\mathcal{N}\) denote, respectively, the linear and nonlinear differential operators, \(W=W(x,y,z,t)\) specifies the unknown function, and \(\Phi=\Phi(x,y,z,t)\) is the provided source term. Applying the Laplace transform (denoted by \(\mathscr{L}\)) to both sides of Eq. (7), we have

$$\begin{aligned} s^{\gamma}\mathscr{L}[W]-\sum_{r=0}^{m-1} s^{\gamma-r-1}W^{r}(x,y,z,0)+ \mathscr{L} \bigl[\mathcal{M} (W ) \bigr]+\mathscr{L} \bigl[ \mathcal{N} (W ) \bigr]=\mathscr{L} [\Phi ]. \end{aligned}$$
(9)

Using the differentiation property of the Laplace transform with the initial conditions (8), upon simplification and the inverse Laplace transform (denoted by \(\mathscr{L}^{-1}\)), we obtain

$$\begin{aligned} W={}&\mathscr{L}^{-1} \Biggl[\frac{1}{s^{\gamma}} \Biggl( \sum_{r=0}^{m-1}s^{ \gamma-r-1}W^{(r)}(x,y,z,0)+ \mathscr{L} [\Phi ] \Biggr) \Biggr] \\ &{}- \mathscr{L}^{-1} \biggl[ \frac{1}{s^{\gamma}}\mathscr{L} \bigl[\mathcal{M}(W)+ \mathcal{N}(W) \bigr] \biggr]. \end{aligned}$$
(10)

The δ-homotopy perturbation transform method (δ-HPTM)

We employ the concept of HPM [4550] to Eq. (10) as follows:

$$\begin{aligned} W=\sum_{r=0}^{\infty}p^{r} W_{r}. \end{aligned}$$
(11)

We decompose the nonlinear term as

$$\begin{aligned} \mathcal{N}(W)=\sum_{r=0}^{\infty}p^{r} \mathcal{H}_{r}(W), \end{aligned}$$
(12)

where \(\mathcal{H}_{r}(W)\) are the He’s polynomials expressed in the form

$$\begin{aligned} \mathcal{H}_{r}(W_{0},W_{1}, \ldots,W_{r})=\frac{1}{r!} \frac{\partial^{r}}{\partial p^{r}} \Biggl[\mathcal{N} \Biggl(\sum_{j=0}^{ \infty}p^{j}W_{j} \Biggr) \Biggr]_{p=0},\quad r=0,1,2,\ldots. \end{aligned}$$
(13)

In view of δ-HPM [66], we derive the propose δ-HPTM as

$$\begin{aligned} \sum_{r=0}^{\infty}p^{r}W_{r}={}& \mathscr{L}^{-1} \Biggl[ \frac{1}{s^{\gamma}} \Biggl(\sum _{r=0}^{m-1}s^{\gamma-r-1}W^{(r)}(x,y,z,0)+ \mathscr{L} [\Phi ] \Biggr) \Biggr]\\ &{}-p \biggl(1-\frac{1}{\delta } \biggr) \Biggl( \sum_{r=0}^{\infty}p^{r}W_{r}-W_{0} \Biggr) \\ &{}- p \Biggl\{ \mathscr{L}^{-1} \Biggl[ \frac{1}{s^{\gamma}}\mathscr{L} \Biggl[\mathcal{M} \Biggl(\sum _{r=0}^{\infty}p^{r}W_{r} \Biggr)+ \sum_{r=0}^{ \infty}p^{r} \mathcal{H}_{r}(W) \Biggr] \Biggr] \Biggr\} . \end{aligned}$$
(14)

By equating the identical power terms of p in Eq. (14), we generate the sequence of δ-HPTM as

$$\begin{aligned} & p^{(0)}: W_{0}=\mathscr{L}^{-1} \Biggl[ \frac{1}{s^{\gamma}} \Biggl(\sum_{r=0}^{m-1}s^{ \gamma-r-1}W^{(r)}(x,y,z,0)+ \mathscr{L} [\Phi ] \Biggr) \Biggr], \\ &p^{(1)}: W_{1}=-\mathscr{L}^{-1} \biggl[ \frac{1}{s^{\gamma}} \mathscr{L} \bigl[\mathcal{M}(W_{0})+ \mathcal{H}_{0}(W) \bigr] \biggr], \\ &p^{(2)}:W_{2}=- \biggl(1-\frac{1}{\delta} \biggr)W_{1}-\mathscr{L}^{-1} \biggl[\frac{1}{s^{\gamma}} \mathscr{L} \bigl[\mathcal{M}(W_{1})+ \mathcal{H}_{1}(W) \bigr] \biggr], \\ &\vdots \\ &p^{(r)}: W_{r}=- \biggl(1-\frac{1}{\delta} \biggr)W_{r-1}-\mathscr{L}^{-1} \biggl[\frac{1}{s^{\gamma}} \mathscr{L} \bigl[\mathcal{M}(W_{r-1})+ \mathcal{H}_{r-1}(W) \bigr] \biggr],\quad r=2,3,4,\ldots. \end{aligned}$$
(15)

The solution of Eq. (7) is given as

$$\begin{aligned} W=\lim_{p\to\delta}\sum_{r=0}^{\infty} p^{r}W_{r}=\sum_{r=0}^{ \infty} \mathcal{W}_{r}(x,y,z,t;\delta)=\sum_{r=0}^{\infty}W_{r} \delta^{r}. \end{aligned}$$
(16)

Remark 1

The particular case where \(\delta=1\) is the standard HPTM [55, 56].

Convergence and error analysis

Theorem 1

Let \(W=W(x,y,z,t)\) be defined in a Banach space \(\mathcal{B}\) [67]. Then the series solution

$$\begin{aligned} \sum_{r=0}^{\infty} \mathcal{W}_{r}(x,y,z,t;\delta)=\sum_{r=0}^{\infty}W_{r} \delta^{r} \end{aligned}$$
(17)

is convergent for a prescribed value of δ if

$$\begin{aligned} \Vert \mathcal{W}_{r+1} \Vert \leq\frac{\varrho}{ \vert \delta \vert } \Vert \mathcal{W}_{r} \Vert , \quad\forall \mathcal{W}_{0}\in \mathcal{B}, \end{aligned}$$
(18)

where \(0<\varrho<\vert\delta\vert\).

Proof

Let \(W_{0}=\mathcal{W}_{0}\in\mathcal{B}\). Define the sequence of partial sums \(\{S_{r}\}\) of Eq. (16) as

$$ \begin{aligned} &S_{0}=W_{0}, \\ &S_{1}=W_{0}+W_{1}\delta, \\ &S_{2}=W_{0}+W_{1}\delta+W_{2} \delta^{2}, \\ &\vdots \\ &S_{r}=W_{0}+W_{1}\delta+W_{2} \delta^{2}+W_{3}\delta^{3}+\cdots+W_{r} \delta^{r}. \end{aligned} $$
(19)

We need to show that \(\{S_{r}\}_{r=0}^{\infty}\) is a Cauchy sequence in the Banach space \(\mathcal{B}\). For \(\delta\neq{0}\), we have

$$\begin{aligned} \lVert S_{r+1}-S_{r} \rVert= \Vert \mathcal{W}_{r+1} \Vert \leq\frac{\varrho}{ \vert \delta \vert } \Vert \mathcal{W}_{r} \Vert \leq \biggl( \frac{\varrho}{ \vert \delta \vert } \biggr)^{2} \Vert \mathcal {W}_{r-1} \Vert \leq \cdots\leq \biggl(\frac{\varrho}{ \vert \delta \vert } \biggr)^{r+1} \Vert \mathcal{W}_{0} \Vert . \end{aligned}$$
(20)

For all \(r,k\in\mathbb{N}\) with \(r\geq{k}\), applying the triangle inequality, we obtain

$$ \begin{aligned} \Vert S_{r}-S_{k} \Vert &= \bigl\Vert (S_{r}-S_{r-1})+(S_{r-1}-S_{r-2})+ \cdots+(S_{k+1}-S_{k}) \bigr\Vert \\ &\leq \Vert S_{r}-S_{r-1} \Vert + \Vert S_{r-1}-S_{r-2} \Vert +\cdots+ \Vert S_{k+1}-S_{k} \Vert \\ &\leq \biggl(\frac{\varrho}{ \vert \delta \vert } \biggr)^{r} \Vert \mathcal{W}_{0} \Vert + \biggl(\frac{\varrho}{ \vert \delta \vert } \biggr)^{r-1} \Vert \mathcal{W}_{0} \Vert +\cdots+ \biggl( \frac{\varrho}{ \vert \delta \vert } \biggr)^{k+1} \Vert \mathcal{W}_{0} \Vert \\ &\leq \biggl(\frac{\varrho}{ \vert \delta \vert } \biggr)^{k+1} \biggl( \biggl( \frac{\varrho}{ \vert \delta \vert } \biggr)^{r-k-1}+ \biggl( \frac{\varrho}{ \vert \delta \vert } \biggr)^{r-k-2}+\cdots+ \biggl( \frac{\varrho}{ \vert \delta \vert } \biggr)+1 \biggr) \Vert \mathcal {W}_{0} \Vert \\ &\leq \biggl(\frac{\varrho}{ \vert \delta \vert } \biggr)^{k+1} \biggl( \frac{1- (\frac{\varrho}{ \vert \delta \vert } )^{r-k}}{1-\frac{\varrho}{ \vert \delta \vert }} \biggr) \Vert \mathcal{W}_{0} \Vert . \end{aligned} $$
(21)

Since \(0<\varrho<\vert\delta\vert\) and \(\delta\neq{0}\), we have \(1- (\frac{\varrho}{\vert\delta\vert} )^{r-k}<1\). Then

$$\begin{aligned} \Vert S_{r}-S_{k} \Vert \leq \frac{ (\frac{\varrho}{ \vert \delta \vert } )^{k+1}}{1-\frac {\varrho}{ \vert \delta \vert }} \Vert \mathcal{W}_{0} \Vert . \end{aligned}$$
(22)

Since \(\Vert\mathcal{W}_{0}\Vert<\infty\), we have

$$\begin{aligned} \lim_{r\to\infty} \Vert S_{r}-S_{k} \Vert =0. \end{aligned}$$
(23)

Therefore \(\{S_{r}\}_{r=0}^{\infty}\) is a Cauchy sequence in the Banach space \(\mathcal{B}\), so the series solution Eq. (16) converges. □

Theorem 2

If the truncated series \(\sum_{r=0}^{K}\mathcal{W}_{r}(x,y,z,t;\delta)=\sum_{r=0}^{K} W_{r}\delta^{r}\) is employed as an approximate solution of Eq. (7), then the maximum absolute truncation error is estimated as

$$\begin{aligned} \Biggl\lVert {W}-\sum_{r=0}^{K} \mathcal{W}_{r}(x,y,z,t;\delta) \Biggr\rVert \leq \frac{ (\frac{\varrho}{ \vert \delta \vert } )^{K+1}}{1-\frac {\varrho}{ \vert \delta \vert }} \Vert W_{0} \Vert . \end{aligned}$$
(24)

Proof

It follows from inequality (21) in Theorem 1. For \(M\geq{K}\), we have

$$\begin{aligned} \lVert S_{M}-S_{K} \rVert\leq \biggl( \frac{\varrho}{ \vert \delta \vert } \biggr)^{K+1} \biggl( \frac{1- (\frac{\varrho}{ \vert \delta \vert } )^{M-K}}{1-\frac{\varrho}{ \vert \delta \vert }} \biggr) \Vert \mathcal{W}_{0} \Vert . \end{aligned}$$
(25)

For a prescribed value of \(\delta\neq{0}\), \(S_{M}\to W\) as \(M\to\infty\), and \(1- (\frac{\varrho}{\vert\delta\vert} )^{M-K}<1\) (since \(0<\frac{\varrho}{\vert\delta\vert}<1\)). Thus

$$\begin{aligned} \Biggl\lVert {W}-\sum_{r=0}^{K} \mathcal{W}_{r}(x,y,z,t;\delta) \Biggr\rVert \leq \frac{ (\frac{\varrho}{ \vert \delta \vert } )^{K+1}}{1-\frac {\varrho}{ \vert \delta \vert }} \Vert W_{0} \Vert , \end{aligned}$$
(26)

where \(\mathcal{W}_{0}=W_{0}\). □

The q-homotopy analysis transform method (q-HATM)

To exemplify the idea of q-HATM [6875], we construct the zeroth-order deformation equation for \(0\leq q \leq\frac{1}{n}, n\geq{1}\), as

$$\begin{aligned} (1-nq)\mathscr{L} (\phi-W_{0} )=\hbar q\mathcal{H} \mathcal{N}[ \phi], \end{aligned}$$
(27)

where \(\phi=\phi(x,y,z,t;q)\), and \(\mathcal{N} [\phi ]\) from Eq. (9) is defined as

$$\begin{aligned} \mathcal{N} [ \phi ]=\mathscr{L}[\phi]-\frac{1}{s^{\gamma}} \sum _{r=0}^{m-1} s^{\gamma-r-1} \phi^{(r)}\bigl(0^{+}\bigr)+ \frac{1}{s^{\gamma}} \bigl( \mathscr{L} \bigl[\mathcal{M}(\phi)+ \mathcal{N}(\phi)-\Phi \bigr] \bigr), \end{aligned}$$
(28)

where q indicates the embedded parameter, the nonzero ħ represents an auxiliary parameter, and \(\mathcal{H}\neq0\) is an auxiliary function. From Eq. (27) with \(q=0, \frac{1}{n}\) we get

$$\begin{aligned} \phi (x,y,z,t;0 )=W_{0},\qquad \phi \biggl(x,y,z,t; \frac{1}{n} \biggr)=W. \end{aligned}$$
(29)

As q rises from 0 to \(\frac{1}{n}\), the solutions ϕ ranges from the initial guess \(W_{0}\) to the solution W. In case that \(W_{0}\), ħ, and \(\mathcal{H}\) are all selected appropriately the solutions ϕ in Eq. (27) hold for \(0\leq q \leq\frac{1}{n}\). Hence application of Taylor series expansion [76] to ϕ gives

$$\begin{aligned} \phi=W_{0}+\sum_{r=1}^{\infty}W_{r}{q^{r}}, \end{aligned}$$
(30)

where

$$\begin{aligned} W_{r}=\frac{1}{r!}\frac{\partial^{r}\phi}{\partial q^{r}} \bigg|_{q=0}. \end{aligned}$$
(31)

If we choose \(W_{0}\), ħ, and \(\mathcal{H}\) adequately, then Eq. (30) converges at \(q=\frac{1}{n}\). From Eq. (29) we obtain

$$\begin{aligned} W=W_{0}+\sum_{r=1}^{\infty}W_{r} \biggl(\frac{1}{n} \biggr)^{r}. \end{aligned}$$
(32)

Differentiating Eq. (27) r times with respect to q, setting \(q=0\), and multiplying by \(\frac{1}{r!}\) give

$$\begin{aligned} \mathscr{L} \bigl[W_{r}-\Upsilon^{*}_{r} W_{r-1} \bigr]=\hbar \mathcal{H}\mathcal{R}_{r} ( \vec{W}_{r-1} ). \end{aligned}$$
(33)

The vector \(\vec{W}_{r}\) is expressed as

$$\begin{aligned} \vec{W}_{r}= \{ W_{0},W_{1}, \ldots,W_{r} \}. \end{aligned}$$
(34)

Taking the inverse LT of Eq. (33), we obtain

$$\begin{aligned} W_{r}=\Upsilon^{*}_{r} W_{r-1}+\hbar\mathscr{L}^{-1} \bigl[ \mathcal{H} \mathcal{R}_{r} (\vec{W}_{r-1} ) \bigr], \end{aligned}$$
(35)

where

$$\begin{aligned} \mathcal{R}_{r} (\vec{W}_{r-1} )={}&\mathscr{L} [W ]- \biggl(1- \frac{\Upsilon^{*}_{r}}{n} \biggr) \Biggl(\sum_{r=0}^{m-1} s^{\gamma -r-1}W^{r}(x,y,z,0)+ \frac{1}{s^{\gamma}}\mathscr{L} [\Phi ] \Biggr) \\ &{}+\frac {1}{s^{\gamma}} \mathscr{L} \bigl[\mathcal{M} (W )+ \mathrm{H}_{r-1} \bigr], \end{aligned}$$
(36)

and

$$\begin{aligned} \Upsilon^{*}_{r}= \textstyle\begin{cases} 0& r\leqslant1, \\ n &\text{otherwise}. \end{cases}\displaystyle \end{aligned}$$
(37)

In Eq. (36), \(\mathrm{H}_{r}\) denotes the homotopy polynomial defined as

$$\begin{aligned} \mathrm{H}_{r}=\frac{1}{r!} \frac{\partial^{r}\phi}{\partial q^{r}}\bigg|_{q=0},\quad \phi= \phi_{0}+q\phi_{1}+q^{2} \phi_{2}+q^{3}\phi_{3}+\cdots. \end{aligned}$$
(38)

Convergence and error analysis

Here we present some helpful theorems with detailed proofs in [74, 75] for the purpose of completeness.

Theorem 3

(Convergence theorem [74, 75])

Let \(\mathcal{B}\) be a Banach space, and let \(\mathrm{F}:\mathcal{B}\to\mathcal{B}\) be a nonlinear mapping. Suppose that

$$\begin{aligned} \bigl\lVert \mathrm{F}(W)-\mathrm{F}(\widehat{W}) \bigr\rVert \leq\varrho \lVert{W} - \widehat{W} \rVert, \quad\forall W,\widehat{W}\in{\mathcal{B}}, \end{aligned}$$
(39)

where \(0<\varrho<1\). Then ϱ has a fixed point in light of Banach’s fixed point theory [77]. Furthermore, for arbitrary choice of \(W_{0},\widehat{W}_{0}\in\mathcal{B}\), the sequence generated by the q-HATM converges to a fixed point of ϱ, and

$$\begin{aligned} \lVert{W}_{k} - W_{r} \rVert\leq \frac{\varrho ^{r}}{1-\varrho} \lVert{W}_{1}-W_{0} \rVert,\quad \forall W, \hat{W} \in{\mathcal{B}}. \end{aligned}$$
(40)

Theorem 4

([75])

Suppose that the series solution defined in Eq. (32) converges to the solution W for prescribed values of n and ħ and that there is a real number \(0<\Theta<1\) satisfying

$$\begin{aligned} \Vert W_{j+1} \Vert \leq\Theta \Vert W_{j} \Vert ,\quad \forall j. \end{aligned}$$
(41)

If the truncated series

$$\begin{aligned} \mathcal{W}_{(K)}=\mathcal{W}_{(K)}(x,t;n; \hbar)=\sum_{r=0}^{K} W_{r} \biggl(\frac{1}{n} \biggr)^{r} \end{aligned}$$
(42)

is utilized as an approximation to the solution of problem (7), then the maximum absolute truncated error is evaluated as

$$\begin{aligned} \lVert{W}-\mathcal{W}_{(K)} \rVert\leq \frac{\Theta^{K+1}}{1-\Theta} \lVert{W}_{0} \rVert. \end{aligned}$$
(43)

Application of the proposed methods

We have carefully chosen the generalized time-fractional perturbed \((3+1)\)-dimensional Zakharov–Kuznetsov (gpZK) equation and apply δ-HPTM and q-HATM to obtain analytical approximate solutions in the form of convergent series. Consider

$$\begin{aligned} & \mathcal{D}_{t}^{\gamma}W+\beta_{1} W^{k} \frac{\partial{W}}{\partial{x}}+\beta_{2} \frac{\partial^{3} W}{\partial{x^{3}}}+ \beta_{3} \biggl( \frac{\partial^{3} W}{\partial x\,\partial y^{2}}+ \frac{\partial^{3} W}{\partial x\,\partial z^{2}} \biggr)+\xi \frac{\partial^{5} W}{\partial x^{5}}=0, \\ &\quad 0< \gamma\leq{1}, t>0, \end{aligned}$$
(44)

with initial condition

$$\begin{aligned} W(x,y,z,0)=f(x,y,z). \end{aligned}$$
(45)

Example 1

Consider Eq. (44) with \(k=1\) given as

$$\begin{aligned} &\mathcal{D}_{t}^{\gamma}W+\beta_{1}W \frac{\partial{W}}{\partial x}+ \beta_{2}\frac{\partial^{3} W}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} W}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W}{\partial x^{5}}=0, \\ &\quad 0< \gamma\leq{1}, t>0, \end{aligned}$$
(46)

with initial condition

$$\begin{aligned} W(x,y,z,0)=e_{0}- \frac{1680\xi p^{4}}{\beta_{1} (p x+q y- (\sqrt{-\frac {\beta_{2}p^{2}}{\beta_{3}}-q^{2}})z+\phi )^{4}}, \end{aligned}$$
(47)

where \(e_{0}, p, q\), and ϕ are arbitrary constants. The exact solution for \(\gamma=1\) is given by

$$\begin{aligned} W(x,y,z,t)=e_{0}- \frac{1680 \xi{p^{4}}}{\beta_{1} (p x+q y- (\sqrt{-\frac {\beta_{2}p^{2}}{\beta_{3}}-q^{2}} )z-\beta_{1} e_{0} p t+\phi )^{4}}. \end{aligned}$$
(48)

δ-HPTM Solution:

Application of δ-HPTM to Eq. (44) with Eq. (45) gives

$$\begin{aligned} \sum_{r=0}^{\infty}p^{r}W_{r}={}& \mathscr{L}^{-1} \biggl[\frac{1}{s} \bigl(W(x,y,z,0) \bigr) \biggr]-p \biggl(1-\frac{1}{\delta} \biggr) \Biggl(\sum _{r=0}^{ \infty}p^{r}W_{r}-W_{0} \Biggr) \\ &{}- p \Biggl\{ \mathscr{L}^{-1} \Biggl[ \frac{1}{s^{\gamma}}\mathscr{L} \Biggl[ \biggl(\beta_{2} \frac{\partial^{3}}{\partial x^{3}}+\beta_{3} \frac{\partial^{3}}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} }{\partial x\,\partial z^{2}} \\ &{}+ \xi\frac{\partial^{5}}{\partial x^{5}} \biggr)\sum_{r=0}^{\infty }p^{r}W_{r}+ \beta_{1}\sum_{r=0}^{\infty}p^{r} \sum_{i=0}^{r}W_{i} \frac{\partial W_{r-i}}{\partial{x}} \Biggr] \Biggr] \Biggr\} . \end{aligned}$$
(49)

By equating the identical power terms of p in Eq. (49) we generate the sequence of δ-HPTM as

$$\begin{aligned} &p^{(0)}: W_{0}=W(x,y,z,0), \\ &p^{(1)}: W_{1}=-\mathscr{L}^{-1} \biggl[ \frac{1}{s^{\gamma}} \mathscr{L} \biggl[\beta_{1}W_{0} \frac{\partial{W_{0}}}{\partial x}+ \beta_{2}\frac{\partial^{3} W_{0}}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W_{0}}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} W_{0}}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W_{0}}{\partial x^{5}} \biggr] \biggr], \\ & p^{(2)}: W_{2}=- \biggl(1-\frac{1}{\delta} \biggr)W_{1}-\mathscr{L}^{-1} \biggl[\frac{1}{s^{\gamma}} \mathscr{L} \biggl[\beta_{1}W_{0} \frac{\partial{W_{1}}}{\partial x}+ \beta_{1}W_{1} \frac{\partial{W_{0}}}{\partial x} \\ &\phantom{p^{(2)}: W_{2}=}{} + \beta_{2}\frac{\partial^{3} W_{1}}{\partial x^{3}}+ \beta_{3} \frac{\partial^{3} W_{1}}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} W_{1}}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W_{1}}{\partial x^{5}} \biggr] \biggr], \\ &\vdots \\ &p^{(r)}: W_{r}=- \biggl(1-\frac{1}{\delta} \biggr)W_{r-1}-\mathscr{L}^{-1} \Biggl[\frac{1}{s^{\gamma}} \mathscr{L} \Biggl[\beta_{1}\sum_{i=0}^{r-1}W_{i} \frac{\partial W_{(r-i-1)}}{\partial{x}} \\ &\phantom{p^{(r)}: W_{r}=}{} + \beta_{2}\frac{\partial^{3} W_{r-1}}{\partial x^{3}}+ \beta_{3} \frac{\partial^{3} W_{r-1}}{\partial x\,\partial y^{2}}+ \beta_{3}\frac{\partial^{3} W_{r-1}}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W_{r-1}}{\partial x^{5}} \Biggr] \Biggr],\quad r=2,3,4, \ldots. \end{aligned}$$
(50)

Hence, using initial condition Eq. (45), we derive:

$$\begin{aligned} W_{0}={}&e_{0}- \frac{1680\xi p^{4}}{\beta_{1} (p x+q y\pm (\sqrt{-\frac {\beta_{2}p^{2}}{\beta_{3}}-q^{2}})z+\phi )^{4}}, \\ W_{1}={}&{-} \frac{6720\beta_{3}^{\frac{5}{2}}\xi e_{0} p^{5} t^{\gamma}}{\Gamma (\gamma+1) (\sqrt{\beta_{3}}(p x+q y+\phi)- (\sqrt{-\beta _{2}p^{2}-\beta_{3} q^{2}})z )^{5}}, \\ W_{2}={}& \biggl(\frac{1}{\delta}-1 \biggr)W_{1}- \frac{33\text{,}600 \beta_{1} \beta_{3}^{3}\xi e_{0}^{2} p^{6} t^{2\gamma }}{\Gamma(2\gamma+1) (\sqrt{\beta_{3}}(p x+q y+\phi)-( \sqrt {-\beta_{2}p^{2}-\beta_{3} q^{2}})z )^{6}}, \\ W_{3}={}& \biggl(\frac{1}{\delta}-1 \biggr)W_{2}- \frac{33\text{,}600\beta_{1}\beta_{3}^{3}\xi{e_{0}^{2}}p^{6}}{ (\sqrt {\beta_{3}}(p x+q y+\phi)-(\sqrt{-\beta_{2} p^{2}-\beta_{3} q^{2}})z )^{11}} \\ &{}\times \biggl( \frac{6\sqrt{\beta_{3}}\beta_{1} e_{0}pt^{3\gamma} (\sqrt {\beta_{3}}(px+qy+\phi)-(\sqrt{-\beta_{2} p^{2}-\beta_{3} q^{2}})z )^{4}}{\Gamma(3\gamma+1)} \\ &{}+ \frac{(1-\delta)t^{2\gamma} (\sqrt{\beta_{3}}(p x+q y+\phi)-(\sqrt{-\beta_{2}p^{2}-\beta_{3} q^{2}})z )^{5}}{\delta\Gamma(2\gamma+1)} \\ &{}+ \frac{13\text{,}440\beta_{3}^{\frac{5}{2}}\xi p^{5} t^{3\gamma }}{\Gamma(3\gamma+1)}- \frac{6720\beta_{3}^{\frac{5}{2}} \xi p^{5}\Gamma(2\gamma +1)t^{3\gamma}}{\Gamma(\gamma+1)^{2}\Gamma(3\gamma+1)} \biggr). \end{aligned}$$

Accordingly, we can obtain the remaining terms \(W_{r}, r=4, 5, 6, \ldots \) .

q-HATM Solution:

Implementing LT on Eq. (44) with Eq. (45), we obtain

$$\begin{aligned} &\mathscr{L}[W]-\frac{1}{s} \bigl(W(x,y,z,0) \bigr) \\ &\quad {}+ \frac{1}{s^{\gamma}} \mathscr{L} \biggl[\beta_{1}W\frac{\partial{W}}{\partial x}+ \beta_{2} \frac{\partial^{3} W}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W}{\partial x\,\partial y^{2}} + \beta_{3} \frac{\partial^{3} W}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W}{\partial x^{5}} \biggr]=0. \end{aligned}$$
(51)

The nonlinear operator \(\mathcal{N}(\phi)\), \(\phi=\phi(x,y,z,t;q)\), is given as

$$\begin{aligned} \mathcal{N}(\phi)={}&\mathscr{L}[\phi]-\frac{1}{s} \bigl(W(x,y,z,0) \bigr) \\ &{}+ \frac{1}{s^{\gamma}}\mathscr{L} \biggl[\beta_{1} \phi \frac{\partial{\phi}}{\partial x}+\beta_{2} \frac{\partial^{3} \phi}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} \phi}{\partial x\,\partial y^{2}} +\beta_{3} \frac{\partial^{3} \phi}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} \phi}{\partial x^{5}} \biggr]. \end{aligned}$$
(52)

Referring to Eq. (33) with \(\mathcal{H}=1\), the rth-order deformation equation is

$$\begin{aligned} \mathscr{L} \bigl[W_{r}-\Upsilon^{*}_{r} W_{r-1} \bigr]=\hbar \mathcal{R}_{r} (\vec{W}_{r-1} ), \end{aligned}$$
(53)

where

$$\begin{aligned} \mathcal{R}_{r} (\vec{W}_{r-1} )={}&\mathscr{L} [W_{r-1} ]- \biggl(1-\frac{\Upsilon^{*}_{r}}{n} \biggr)\frac{1}{s} \bigl\{ W(x,y,z,0) \bigr\} \\ &{}+ \frac{1}{s^{\gamma}}\mathscr{L} \Biggl[\beta_{1}\sum _{i=0}^{r-1}W_{i} \frac{\partial W_{r-i-1}}{\partial x}+\beta_{2} \frac{\partial^{3} W_{r-1}}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W_{r-1}}{\partial x\,\partial y^{2}} \\ &{} +\beta_{3} \frac{\partial^{3} W_{r-1}}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W_{r-1}}{\partial x^{5}} \Biggr]. \end{aligned}$$
(54)

An application of the inverse LT to Eq. (53) yields

$$\begin{aligned} W_{r}=\Upsilon^{*}_{r} W_{r-1}+\hbar \mathscr{L}^{-1} \bigl[ \mathcal{R}_{r} ( \vec{W}_{r-1} ) \bigr]. \end{aligned}$$
(55)

Solving Eqs. (55) using (47) and (54) for \(r=1,2,3,\ldots \) , we get:

$$\begin{aligned} W_{0}={}&e_{0}- \frac{1680\xi p^{4}}{\beta_{1} (p x+q y\pm (\sqrt{-\frac {\beta_{2}p^{2}}{\beta_{3}}-q^{2}})z+\phi )^{4}}, \\ W_{1}={}& \frac{6720\beta_{3}^{\frac{5}{2}}\xi e_{0} \hbar p^{5} t^{\gamma }}{\Gamma(\gamma+1) (\sqrt{\beta_{3}}(p x+q y+\phi)-( \sqrt {-\beta_{2}p^{2}-\beta_{3} q^{2}})z )^{5}}, \\ W_{2}={}&(n+\hbar)W_{1}- \frac{33\text{,}600 \beta_{1} \beta_{3}^{3}\xi e_{0}^{2} \hbar^{2} p^{6} t^{2\gamma}}{\Gamma(2\gamma+1) (\sqrt{\beta_{3}}(p x+q y+\phi)-( \sqrt{-\beta_{2}p^{2}-\beta_{3} q^{2}})z )^{6}}, \\ W_{3}={}&(n+\hbar)W_{2}+ \frac{33\text{,}600 \beta_{1}\beta_{3}^{3}\xi e_{0}^{2}\hbar ^{2}p^{6}}{ (\sqrt{\beta_{3}}(p x+qy+\phi)-z\sqrt{-\beta_{2} p^{2}-\beta_{3} q^{2}} )^{11}} \\ &{}\times \biggl( \frac{6\sqrt{\beta_{3}} \beta_{1} e_{0}\hbar{p}t^{3\gamma} (\sqrt{\beta_{3}}(p x+q y+\phi)-(\sqrt{-\beta_{2} p^{2}-\beta_{3} q^{2}})z )^{4}}{\Gamma(3\gamma+1)} \\ &{}- \frac{(n+\hbar)t^{2\gamma} (\sqrt{\beta_{3}}(p x+q y+\phi)-( \sqrt{-\beta_{2}p^{2}-\beta_{3} q^{2}})z )^{5}}{\Gamma(2\gamma+1)} \\ &{}+ \frac{13\text{,}440\beta_{3}^{\frac{5}{2}} \xi\hbar p^{5}t^{3\gamma }}{\Gamma(3 \gamma+1)}- \frac{6720\beta_{3}^{\frac{5}{2}} \xi\hbar p^{5}\Gamma(2\gamma+1) t^{3\gamma}}{\Gamma(\gamma+1)^{2}\Gamma(3 \gamma+1)} \biggr). \end{aligned}$$

Accordingly, we can derive the remaining terms.

Example 2

Consider Eq. (44) with \(k=2\) given as

$$\begin{aligned} & \mathcal{D}_{t}^{\gamma}W+\beta_{1}W^{2} \frac{\partial{W}}{\partial x}+\beta_{2} \frac{\partial^{3} W}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} W}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W}{\partial x^{5}}=0, \\ &\quad 0< \gamma\leq{1}, t>0, \end{aligned}$$
(56)

with initial condition

$$\begin{aligned} W(x,y,z,0)=e_{0}+ \frac{6i\sqrt{10}\sqrt{\xi}p^{2}}{\sqrt{\beta_{1}} (p x+q y-\frac{\sqrt{-i\sqrt{10}\sqrt{\beta_{1}} \sqrt{\xi}e_{0} p^{2}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt{\beta_{3}}}z+\phi )^{2}}, \end{aligned}$$
(57)

where \(e_{0}, p, q\), and ϕ are arbitrary constants. The exact solution for \(\gamma=1\) is given by

$$\begin{aligned} W(x,y,z,t)=e_{0}+ \frac{6i\sqrt{10}\sqrt{\xi} p^{2}}{\sqrt{\beta_{1}} (p x+qy-\frac{\sqrt{-i\sqrt{10}\sqrt{\beta_{1}}\sqrt{\xi} e_{0} p^{2}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt{\beta_{3}}}z-\beta _{1} e_{0}^{2} p t+\phi )^{2}}. \end{aligned}$$
(58)

δ-HPTM Solution:

Application of δ-HPTM to Eq. (56) with Eq. (57) gives

$$\begin{aligned} \sum_{r=0}^{\infty}p^{r}W_{r}={}& \mathscr{L}^{-1} \biggl[\frac{1}{s} \bigl(W(x,y,z,0) \bigr) \biggr]-p \biggl(1-\frac{1}{\delta} \biggr) \Biggl(\sum _{r=0}^{ \infty}p^{r}W_{r}-W_{0} \Biggr) \\ &{}- p \Biggl\{ \mathscr{L}^{-1} \Biggl[ \frac{1}{s^{\gamma}}\mathscr{L} \Biggl[ \biggl(\beta_{2} \frac{\partial^{3} }{\partial x^{3}}+\beta_{3} \frac{\partial^{3} }{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} }{\partial x\,\partial z^{2}} \\ &{}+ \xi\frac{\partial^{5}}{\partial x^{5}} \biggr)\sum_{r=0}^{\infty }p^{r}W_{r}+ \beta_{1}\sum_{r=0}^{\infty}p^{r} \sum_{i=0}^{r}\sum _{j=0}^{i}W_{j}W_{i-j} \frac{\partial W_{r-i}}{\partial{x}} \Biggr] \Biggr] \Biggr\} . \end{aligned}$$
(59)

By equating the identical power terms of p in Eq. (59) we generate the sequence of δ-HPTM:

$$\begin{aligned} &p^{(0)}: W_{0}=W(x,y,z,0), \\ & p^{(1)}: W_{1}=-\mathscr{L}^{-1} \biggl[ \frac{1}{s^{\gamma}} \mathscr{L} \biggl[\beta_{1}W_{0}^{2} \frac{\partial{W_{0}}}{\partial x}+ \beta_{2}\frac{\partial^{3} W_{0}}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W_{0}}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} W_{0}}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W_{0}}{\partial x^{5}} \biggr] \biggr], \\ &p^{(2)}: W_{2}=- \biggl(1-\frac{1}{\delta} \biggr)W_{1}-\mathscr{L}^{-1} \biggl[\frac{1}{s^{\gamma}} \mathscr{L} \biggl[\beta_{1}W_{0}W_{0} \frac{\partial{W_{1}}}{\partial x}+\beta_{1}W_{0}W_{1} \frac{\partial{W_{0}}}{\partial x} \\ &\phantom{p^{(2)}: W_{2}=}{} + \beta_{1}W_{1}W_{0} \frac{\partial{W_{0}}}{\partial x}+\beta_{2} \frac{\partial^{3} W_{1}}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W_{1}}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} W_{1}}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W_{1}}{\partial x^{5}} \biggr] \biggr], \\ &\vdots \\ &p^{(r)}: W_{r}=- \biggl(1-\frac{1}{\delta} \biggr)W_{r-1}-\mathscr{L}^{-1} \Biggl[\frac{1}{s^{\gamma}} \mathscr{L} \Biggl[\beta_{1}\sum_{i=0}^{r-1} \sum_{j=0}^{i}W_{j}W_{i-j} \frac{\partial W_{(r-i-1)}}{\partial{x}} \\ &\phantom{p^{(r)}: W_{r}=}{} + \beta_{2} \frac{\partial^{3} W_{r-1}}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W_{r-1}}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} W_{r-1}}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W_{r-1}}{\partial x^{5}} \Biggr] \Biggr],\quad r=2,3,4, \ldots. \end{aligned}$$
(60)

Hence, using initial condition Eq. (57), we derive:

$$\begin{aligned} W_{0}={}&e_{0}+ \frac{6i\sqrt{10} \sqrt{\xi}p^{2}}{\sqrt{\beta_{1}} (p x+q y-\frac{\sqrt{-i\sqrt{10}\sqrt{\beta_{1}}\sqrt{\xi}e_{0} p^{2}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt{\beta_{3}}}z+\phi )^{2}}, \\ W_{1}={}& \frac{12i\sqrt{10}\sqrt{\beta_{1}} \sqrt{\xi} e_{0}^{2} p^{3} t^{\gamma}}{\Gamma(\gamma+1) (p x+q y-\frac{\sqrt{-\beta_{3} q^{2}+p^{2} (-\beta_{2}-i\sqrt{10}\sqrt{\beta_{1}}\sqrt{\xi }e_{0} )}}{\sqrt{\beta_{3}}}z+\phi )^{3}}, \\ W_{2}={}& \biggl(\frac{1}{\delta}-1 \biggr)W_{1}- \frac{36i\sqrt{10}\beta_{1}^{\frac{3}{2}}\sqrt{\xi}e_{0}^{4} p^{4} t^{2\gamma}}{\Gamma(2\gamma+1) (p x+q y-\frac{\sqrt {-\beta_{3} q^{2}+p^{2} (-\beta_{2}-i\sqrt{10}\sqrt{\beta _{1}} \sqrt{\xi}e_{0} )}}{\sqrt{\beta_{3}}}z+\phi )^{4}}, \\ W_{3}={}& \biggl(\frac{1}{\delta}-1 \biggr)W_{2}- \frac{36\beta_{1}^{\frac{3}{2}} \sqrt{\xi}e_{0}^{4} p^{4}}{ (p x+q y-\frac{\sqrt{-\beta_{3} q^{2}+p^{2} (-\beta_{2}-i\sqrt {10} \sqrt{\beta_{1}}\sqrt{\xi} e_{0} )}}{\sqrt{\beta _{3}}}z+\phi )^{9}}, \\ &{}\times \biggl( \frac{i \sqrt{10}(1-\delta) t^{2\gamma} (p x+q y-\frac{\sqrt {-\beta_{3} q^{2}+p^{2} (-\beta_{2}-i\sqrt{10} \sqrt{\beta _{1}}\sqrt{\xi} e_{0} )}}{\sqrt{\beta_{3}}}z+\phi )^{5}}{\delta\Gamma(2\gamma+1)} \\ &{}- \frac{4i\sqrt{10}\beta_{1} e_{0}^{2} p t^{3\gamma} (p x+q y-\frac{\sqrt{-\beta_{3} q^{2}+p^{2} (-\beta_{2}-i\sqrt{10} \sqrt{\beta_{1}}\sqrt{\xi} e_{0} )}}{\sqrt{\beta _{3}}}z+\phi )^{4}}{\Gamma(3\gamma+1)} \\ &{}+ \frac{240\sqrt{\beta_{1}} \sqrt{\xi}e_{0} p^{3} \Gamma(2 \gamma+1)t^{3\gamma} (p x+q y-\frac{\sqrt{-\beta_{3} q^{2}+p^{2} (-\beta_{2}-i\sqrt{10}\sqrt{\beta_{1}}\sqrt{\xi }e_{0} )}}{\sqrt{\beta_{3}}}z+\phi )^{2}}{\Gamma(\gamma +1)^{2} \Gamma(3\gamma+1)} \\ &{}- \frac{480\sqrt{\beta_{1}} \sqrt{\xi} e_{0} p^{3} t^{3 \gamma } (p x+q y-\frac{\sqrt{-\beta_{3} q^{2}+p^{2} (-\beta _{2}-i\sqrt{10}\sqrt{\beta_{1}}\sqrt{\xi}e_{0} )}}{\sqrt {\beta_{3}}}z+\phi )^{2}}{\Gamma(3\gamma+1)} \\ &{}+ \frac{1920i\sqrt{10}\xi p^{5} \Gamma(2\gamma+1)t^{3\gamma }}{\Gamma(\gamma+1)^{2}\Gamma(3\gamma+1)}- \frac{3840i\sqrt{10}\xi p^{5} t^{3\gamma}}{\Gamma(3\gamma+1)} \biggr). \end{aligned}$$

Following this procedure, we can obtain the remaining terms.

q-HATM Solution:

Implementing LT on Eq. (56) with Eq. (57), we obtain

$$\begin{aligned} & \mathscr{L}[W]-\frac{1}{s} \bigl(W(x,y,z,0) \bigr) \\ &\quad{}+ \frac{1}{s^{\gamma}} \mathscr{L} \biggl[\beta_{1}W^{2} \frac{\partial{W}}{\partial x}+\beta_{2} \frac{\partial^{3} W}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W}{\partial x\,\partial y^{2}} +\beta_{3} \frac{\partial^{3} W}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W}{\partial x^{5}} \biggr]=0. \end{aligned}$$
(61)

The nonlinear operator \(\mathcal{N}(\phi)\), \(\phi=\phi(x,y,z,t;q)\), is presented as

$$\begin{aligned} \mathcal{N}(\phi)={}&\mathscr{L}[\phi]-\frac{1}{s} \bigl(W(x,y,z,0) \bigr) \\ &{}+ \frac{1}{s^{\gamma}}\mathscr{L} \biggl[\beta_{1} \phi^{2} \frac{\partial{\phi}}{\partial x}+\beta_{2} \frac{\partial^{3} \phi}{\partial x^{3}}+ \beta_{3} \frac{\partial^{3} \phi}{\partial x\,\partial y^{2}} +\beta_{3} \frac{\partial^{3} \phi}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} \phi}{\partial x^{5}} \biggr]. \end{aligned}$$
(62)

Referring to Eq. (55), we have

$$\begin{aligned} \mathcal{R}_{r} (\vec{W}_{r-1} )={}&\mathscr{L} [W_{r-1} ]- \biggl(1-\frac{\Upsilon^{*}_{r}}{n} \biggr)\frac{1}{s} \bigl\{ W(x,y,z,0) \bigr\} \\ &{}+\frac{1}{s^{\gamma}}\mathscr{L} \Biggl[\beta_{1}\sum _{i=0}^{r-1} \sum _{j=0}^{i}W_{j}W_{i-j} \frac{\partial W_{r-i-1}}{\partial x} \\ &{}+ \beta_{2}\frac{\partial^{3} W_{r-1}}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W_{r-1}}{\partial x\,\partial y^{2}} +\beta_{3} \frac{\partial^{3} W_{r-1}}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W_{r-1}}{\partial x^{5}} \Biggr], \end{aligned}$$
(63)

where

$$\begin{aligned} W_{r}=\Upsilon^{*}_{r} W_{r-1}+\hbar \mathscr{L}^{-1} \bigl[ \mathcal{R}_{r} ( \vec{W}_{r-1} ) \bigr]. \end{aligned}$$
(64)

Solving Eqs. (64) using (57) and (63) for \(r=1,2,3,\ldots \) , we achieve the following:

$$\begin{aligned} W_{0}={}&e_{0}+ \frac{6i\sqrt{10} \sqrt{\xi}p^{2}}{\sqrt{\beta_{1}} (p x+q y-\frac{\sqrt{-i\sqrt{10}\sqrt{\beta_{1}}\sqrt{\xi}e_{0} p^{2}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt{\beta_{3}}}z+\phi )^{2}}, \\ W_{1}={}&{-} \frac{12i\sqrt{10}\sqrt{\beta_{1}} \sqrt{\xi}e_{0}^{2}\hbar p^{3}t^{\gamma}}{\Gamma(\gamma+1) (p x+q y-\frac{\sqrt{-\beta _{3} q^{2}+p^{2} (-\beta_{2}-i \sqrt{10}\sqrt{\beta_{1}}\sqrt {\xi} e_{0} )}}{\sqrt{\beta_{3}}}z+\phi )^{3}}, \\ W_{2}={}&(n+\hbar)W_{1}+ \frac{36i\sqrt{10}\beta_{1}^{3/2}\sqrt{\xi}e_{0}^{4}\hbar^{2} p^{4} t^{2\gamma}}{\Gamma(2\gamma+1) (p x+q y-\frac{\sqrt {-\beta_{3} q^{2}+p^{2} (-\beta_{2}-i\sqrt{10}\sqrt{\beta _{1}} \sqrt{\xi}e_{0} )}}{\sqrt{\beta_{3}}}z+\phi )^{4}}, \\ W_{3}={}&(n+\hbar)W_{2}+ \frac{36\beta_{1}^{\frac{3}{2}}\sqrt{\xi} e_{0}^{4}\hbar^{2} p^{4}}{ (p x+q y-\frac{\sqrt{-\beta_{3} q^{2}+p^{2} (-\beta_{2}-i\sqrt{10}\sqrt{\beta_{1}}\sqrt{\xi} e_{0} )}}{\sqrt{\beta_{3}}}z+\phi )^{9}} \\ &{}\times \biggl\{ \frac{i\sqrt{10}(n+\hbar)t^{2\gamma} (p x+q y-\frac{\sqrt {-\beta_{3} q^{2}+p^{2} (-\beta_{2}-i\sqrt{10}\sqrt{\beta _{1}} \sqrt{\xi}e_{0} )}}{\sqrt{\beta_{3}}}z+\phi )^{5}}{\Gamma(2\gamma+1)} \\ &{}- \frac{4i\sqrt{10}\beta_{1} e_{0}^{2}\hbar p t^{3\gamma} (p x+q y-\frac{\sqrt{-\beta_{3} q^{2}+p^{2} (-\beta_{2}-i\sqrt {10}\sqrt{\beta_{1}}\sqrt{\xi} e_{0} )}}{\sqrt{\beta _{3}}}z+\phi )^{4}}{\Gamma(3\gamma+1)} \\ &{}+ \frac{240\sqrt{\beta_{1}} \sqrt{\xi} e_{0}]\hbar p^{3}\Gamma (2\gamma+1) t^{3\gamma} (p x+q y-\frac{\sqrt{-\beta_{3} q^{2}+p^{2} (-\beta_{2}-i\sqrt{10}\sqrt{\beta_{1}} \sqrt{\xi }e_{0} )}}{\sqrt{\beta_{3}}}z+\phi )^{2}}{\Gamma(\gamma +1)^{2}\Gamma(3 \gamma+1)} \\ &{}- \frac{480\sqrt{\beta_{1}}\sqrt{\xi} e_{0}\hbar p^{3}t^{3\gamma} (p x+q y-\frac{\sqrt{-\beta_{3} q^{2}+p^{2} (-\beta_{2}-i\sqrt{10}\sqrt{\beta_{1}} \sqrt{\xi }e_{0} )}}{\sqrt{\beta_{3}}}z+\phi )^{2}}{\Gamma (3\gamma+1)} \\ &{}+ \frac{1920i\sqrt{10}\xi h p^{5} \Gamma(2\gamma+1)t^{3\gamma }}{\Gamma(\gamma+1)^{2}\Gamma(3\gamma+1)}- \frac{3840i\sqrt{10} \xi\hbar p^{5} t^{3\gamma}}{\Gamma(3\gamma+1)} \biggr\} . \end{aligned}$$

Accordingly, we can derive the remaining terms.

Example 3

Consider Eq. (44) with \(k=4\) given as

$$\begin{aligned} &\mathcal{D}_{t}^{\gamma}W+\beta_{1}W^{4} \frac{\partial{W}}{\partial x}+\beta_{2} \frac{\partial^{3} W}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} W}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W}{\partial x^{5}}=0, \\ &\quad 0< \gamma\leq{1}, t>0, \end{aligned}$$
(65)

with initial condition

$$\begin{aligned} W(x,y,z,0)= \frac{2^{\frac{3}{4}}\sqrt[4]{-15}\sqrt[4]{\xi} p}{\sqrt[4]{\beta_{1}}} \tan \biggl(p x+q y- \frac{\sqrt{-20\xi p^{4}-\beta_{2}p^{2}-\beta_{3}q^{2}}}{\sqrt {\beta_{3}}}z \biggr), \end{aligned}$$
(66)

where p and q are arbitrary constants. The exact solution for \(\gamma=1\) is given by

$$\begin{aligned} W(x,y,z,t)= \frac{2^{\frac{3}{4}}\sqrt[4]{-15}\sqrt[4]{\xi}p}{\sqrt[4]{\beta_{1}}} \tan \biggl(px+qy- \frac{\sqrt{-20\xi{p^{4}}-\beta_{2}p^{2}-\beta_{3}q^{2}}}{\sqrt {\beta_{3}}}z+24 \xi{p^{5}}t \biggr). \end{aligned}$$
(67)

δ-HPTM Solution

Application of δ-HPTM to Eq. (65) with Eq. (66) gives

$$\begin{aligned} \sum_{r=0}^{\infty}p^{r}W_{r}={}& \mathscr{L}^{-1} \biggl[\frac{1}{s} \bigl(W(x,y,z,0) \bigr) \biggr]-p \biggl(1-\frac{1}{\delta} \biggr) \Biggl(\sum _{r=0}^{ \infty}p^{r}W_{r}-W_{0} \Biggr) \\ &{}- p \Biggl\{ \mathscr{L}^{-1} \Biggl[ \frac{1}{s^{\gamma}}\mathscr{L} \Biggl[ \biggl(\beta_{2} \frac{\partial^{3} }{\partial x^{3}}+\beta_{3} \frac{\partial^{3} }{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} }{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5}}{\partial x^{5}} \biggr) \\ &{}\times \sum_{r=0}^{\infty}p^{r}W_{r}+ \beta_{1}\sum_{r=0}^{ \infty}p^{r} \sum_{i=0}^{r}\sum _{j=0}^{i}\sum_{k=0}^{j} \sum_{l=0}^{k}W_{l}W_{k-l}W_{j-k}W_{i-j} \frac{\partial W_{r-i}}{\partial{x}} \Biggr] \Biggr] \Biggr\} . \end{aligned}$$
(68)

By equating the identical power terms of p in Eq. (68) we generate the sequence of δ-HPTM:

$$\begin{aligned} & p^{(0)}:W_{0}=W(x,y,z,0), \\ & p^{(1)}:W_{1}=-\mathscr{L}^{-1} \biggl[ \frac{1}{s^{\gamma}} \mathscr{L} \biggl[\beta_{1}W_{0}^{2} \frac{\partial{W_{0}}}{\partial x}+ \beta_{2}\frac{\partial^{3} W_{0}}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W_{0}}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} W_{0}}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W_{0}}{\partial x^{5}} \biggr] \biggr], \\ &p^{(2)}: W_{2}=- \biggl(1-\frac{1}{\delta} \biggr)W_{1}-\mathscr{L}^{-1} \biggl[\frac{1}{s^{\gamma}} \mathscr{L} \biggl[\beta_{1}W_{0}W_{0} \frac{\partial{W_{1}}}{\partial x}+\beta_{1}W_{0}W_{1} \frac{\partial{W_{0}}}{\partial x} \\ &\phantom{p^{(2)}: W_{2}=}{} +\beta_{1}W_{1}W_{0} \frac{\partial{W_{0}}}{\partial x}+\beta_{2} \frac{\partial^{3} W_{1}}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W_{1}}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} W_{1}}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W_{1}}{\partial x^{5}} \biggr] \biggr], \\ &\vdots \\ & p^{(r)}: W_{r}=- \biggl(1-\frac{1}{\delta} \biggr)W_{r-1}\\ &\phantom{p^{(r)}: W_{r}=}{}-\mathscr{L}^{-1} \Biggl[\frac{1}{s^{\gamma}} \mathscr{L} \Biggl[\beta_{1}\sum_{i=0}^{r-1} \sum_{j=0}^{i}\sum _{k=0}^{j}\sum_{l=0}^{k}W_{l}W_{k-l}W_{j-k}W_{i-j} \frac{\partial W_{(r-i-1)}}{\partial{x}} \\ &\phantom{p^{(r)}: W_{r}=}{} + \beta_{2} \frac{\partial^{3} W_{r-1}}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W_{r-1}}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} W_{r-1}}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W_{r-1}}{\partial x^{5}} \Biggr] \Biggr],\quad r=2,3,4, \ldots. \end{aligned}$$
(69)

Hence, using initial condition Eq. (66), we derive:

$$\begin{aligned} W_{0}= {}&\frac{ (2^{\frac{3}{4}}\sqrt[4]{-15}\sqrt[4]{\xi} p )}{\sqrt[4]{\beta_{1}}} \tan \biggl(px+qy- \frac{\sqrt{-20\xi p^{4}-\beta_{2}p^{2}-\beta_{3}q^{2}}}{\sqrt {\beta_{3}}}z \biggr), \\ W_{1}={}& \frac{24\sqrt[4]{-15}\xi^{\frac{5}{4}}2^{\frac{3}{4}}p^{6} t^{\gamma}}{\sqrt[4]{\beta_{1}}\Gamma(\gamma+1)} \sec^{2} \biggl(p x+q y- \frac{ \sqrt{-20\xi p^{4}-\beta_{2} p^{2}-\beta_{3}{q^{2}}}}{\sqrt {\beta_{3}}}z \biggr), \\ W_{2}={}& \biggl(\frac{1}{\delta}-1 \biggr)W_{1}+ \frac{1152 \sqrt[4]{-15} 2^{3/4} \xi^{9/4} p^{11} t^{2\gamma }}{\sqrt[4]{\beta_{1}}\Gamma(2 \gamma+1)} \\ &{}\times \biggl\{ \tan \biggl(p x+q y- \frac{\sqrt{-20\xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr)\\ &{}\times\sec^{2} \biggl(p x+q y- \frac{\sqrt{-20 \xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \biggr\} , \\ W_{3}={}& \biggl(\frac{1}{\delta}-1 \biggr)W_{2}- \frac{36\sqrt[4]{-15}2^{\frac{3}{4}}\xi^{\frac {9}{4}}p^{11}t^{2\gamma}}{\sqrt[4]{\beta_{1}}\delta\Gamma(\gamma +1)^{2}\Gamma(2\gamma+1)\Gamma(3\gamma+1)}\\ &{}\times \sec^{8} \biggl(p x+q y- \frac{\sqrt{-20 \xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}\times \biggl\{ 16\Gamma(\gamma+1)^{2} \biggl[2(\delta-1)\Gamma(3 \gamma+1)\sin \biggl(p x+q y- \frac{\sqrt{-20 \xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}\times \cos^{5} \biggl(px+qy- \frac{\sqrt{-20\xi{p^{4}}-\beta_{2}p^{2}-\beta_{3}q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}+ 3\delta\xi p^{5}\Gamma(2\gamma+1)t^{\gamma} \biggl(-569\cos \biggl(2p x+2q y- \frac{2\sqrt{-20\xi p^{4}-\beta_{2}{p^{2}}-\beta_{3}q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}+ 80\cos \biggl(4px+4qy- \frac{4\sqrt{-20 \xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}+ \cos \biggl(6 p x+6 q y- \frac{6\sqrt{-20 \xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr)+472 \biggr) \biggr] \\ &{}+ 3840\delta\xi p^{5}\Gamma(2 \gamma+1)^{2}t^{\gamma} \sin^{2} \biggl(p x+q y- \frac{\sqrt{-20\xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}\times \biggl(2\cos \biggl(2 p x+2 q y- \frac{2\sqrt{-20\xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr)-5 \biggr) \biggr\} . \end{aligned}$$

By following this procedure we can obtain other terms.

q-HATM Solution:

Implementing LT on Eq. (65) with Eq. (66), we obtain

$$\begin{aligned} & \mathscr{L}[W]-\frac{1}{s} \bigl(W(x,y,z,0) \bigr) \\ &\quad{}+ \frac{1}{s^{\gamma}} \mathscr{L} \biggl[\beta_{1}W^{4} \frac{\partial{W}}{\partial x}+\beta_{2} \frac{\partial^{3} W}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W}{\partial x\,\partial y^{2}} +\beta_{3} \frac{\partial^{3} W}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W}{\partial x^{5}} \biggr]=0. \end{aligned}$$
(70)

The nonlinear operator \(\mathcal{N}(\phi)\), \(\phi=\phi(x,y,z,t;q)\), is given as

$$\begin{aligned} \mathcal{N}(\phi)={}&\mathscr{L}[\phi]-\frac{1}{s} \bigl(W(x,y,z,0) \bigr) \\ &{}+ \frac{1}{s^{\gamma}}\mathscr{L} \biggl[\beta_{1} \phi^{4} \frac{\partial{\phi}}{\partial x}+\beta_{2} \frac{\partial^{3} \phi}{\partial x^{3}}+ \beta_{3} \frac{\partial^{3} \phi}{\partial x\,\partial y^{2}} +\beta_{3} \frac{\partial^{3} \phi}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} \phi}{\partial x^{5}} \biggr]. \end{aligned}$$
(71)

Referring to Eq. (55), we have

$$\begin{aligned} \mathcal{R}_{r} (\vec{W}_{r-1} )={}&\mathscr{L} [W_{r-1} ]- \biggl(1-\frac{\Upsilon^{*}_{r}}{n} \biggr)\frac{1}{s} \bigl\{ W(x,y,z,0) \bigr\} \\ &{}+ \frac{1}{s^{\gamma}}\mathscr{L} \Biggl[\beta_{1}\sum _{i=0}^{r-1} \sum _{j=0}^{i}\sum_{k=0}^{j} \sum_{l=0}^{k}W_{l}W_{k-l}W_{j-k}W_{i-j} \frac{\partial W_{(r-i-1)}}{\partial{x}} \\ &{}+ \beta_{2}\frac{\partial^{3} W_{r-1}}{\partial x^{3}}+\beta_{3} \frac{\partial^{3} W_{r-1}}{\partial x\,\partial y^{2}}+\beta_{3} \frac{\partial^{3} W_{r-1}}{\partial x\,\partial z^{2}}+\xi \frac{\partial^{5} W_{r-1}}{\partial x^{5}} \Biggr], \end{aligned}$$
(72)

where

$$\begin{aligned} W_{r}=\Upsilon^{*}_{r} W_{r-1}+\hbar \mathscr{L}^{-1} \bigl[ \mathcal{R}_{r} ( \vec{W}_{r-1} ) \bigr]. \end{aligned}$$
(73)

Solving Eqs. (73) using (66) and (72) for \(r=1,2,3,\ldots \) , we get:

$$\begin{aligned} W_{0}={}& \frac{ (2^{\frac{3}{4}}\sqrt[4]{-15}\sqrt[4]{\xi} p )}{\sqrt[4]{\beta_{1}}} \tan \biggl(px+qy- \frac{\sqrt{-20\xi p^{4}-\beta_{2}p^{2}-\beta_{3}q^{2}}}{\sqrt {\beta_{3}}}z \biggr), \\ W_{1}={}&{-} \frac{24\sqrt[4]{-15} 2^{3/4} \xi^{\frac{5}{4}}\hbar p^{6}t^{\gamma}\sec^{2} (p x+q y-\frac{z\sqrt{-20 \xi p^{4}-\beta_{2}p^{2}-\beta_{3}q^{2}}}{\sqrt{\beta_{3}}} )}{\sqrt[4]{\beta_{1}}\Gamma(\gamma+1)}, \\ W_{2}={}&(n+\hbar)W_{1}+ \frac{1152\sqrt[4]{-15} 2^{\frac{3}{4}}\xi^{\frac{9}{4}}\hbar^{2} p^{11}t^{2\gamma}}{\sqrt[4]{\beta_{1}}\Gamma(2\gamma+1)} \\ &{}\times \biggl\{ \tan \biggl(p x+q y- \frac{\sqrt{-20\xi{p^{4}}-\beta_{2}p^{2}-\beta_{3}q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}\times \sec^{2} \biggl(p x+q y- \frac{\sqrt{-20\xi p^{4}-\beta_{2} p^{2}-\beta_{3}q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \biggr\} , \\ W_{3}={}&(n+\hbar)W_{2}+ \frac{36\sqrt[4]{-15} 2^{\frac{3}{4}}\xi^{\frac{9}{4}}\hbar^{2} p^{11}t^{2\gamma}}{\sqrt[4]{\beta_{1}}\Gamma(\gamma+1)^{2}\Gamma (2\gamma+1) \Gamma(3\gamma+1)} \\ &{}\times\sec^{8} \biggl(px+\text{qy}- \frac{\sqrt{-20\xi p^{4}-\beta_{2} p^{2}-\beta_{3}q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}\times \biggl\{ 16\Gamma(\gamma+1)^{2} \biggl[2\Gamma(3\gamma+1) (n+ \hbar)\sin \biggl(p x+q y- \frac{\sqrt{-20\xi p^{4}-\beta_{2}p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}\times \cos^{5} \biggl(p x+q y- \frac{\sqrt{-20 \xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}+ 3\xi\hbar{p^{5}}\Gamma(2\gamma+1)t^{\gamma} \biggl(-569\cos \biggl(2 p x+2 q y- \frac{2\sqrt{-20\xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}+ 80\cos \biggl(4 p x+4 q y- \frac{4\sqrt{-20\xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}+ \cos \biggl(6 p x+6 q y- \frac{6\sqrt{-20\xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr)+472 \biggr) \biggr] \\ &{}+ 3840\xi\hbar{p^{5}}\Gamma(2\gamma+1)^{2}t^{\gamma} \sin^{2} \biggl(p x+q y- \frac{\sqrt{-20 \xi p^{4}-\beta_{2}p^{2}-\beta_{3}q^{2}}}{\sqrt {\beta_{3}}}z \biggr) \\ &{}\times \biggl(2\cos \biggl(2 p x+2 q y- \frac{2\sqrt{-20\xi p^{4}-\beta_{2} p^{2}-\beta_{3} q^{2}}}{\sqrt {\beta_{3}}}z \biggr)-5 \biggr) \biggr\} . \end{aligned}$$

Respectively, we can derive the remaining terms.

Numerical comparison

In this section, the δ-HPTM and q-HATM formulations are tested upon the generalized perturbed \((3+1)\)-dimensional Zakharov–Kuznetsov (gpZK) equation with Caputo fractional derivative. The δ-HPTM solution is presented as

$$\begin{aligned} \mathcal{W}_{(3)}(x,y,z,t;\delta)=\sum _{r=0}^{3} W_{r}\delta^{r}, \end{aligned}$$
(74)

and the q-HATM solution is presented as

$$\begin{aligned} \mathcal{W}_{(3)}{(x,t;n;\hbar)}=W_{0}+\sum _{r=1}^{3} W_{r} \biggl( \frac{1}{n} \biggr)^{r}. \end{aligned}$$
(75)

We observe that setting \(\delta=\frac{1}{n}\) in Eq. (74) yields

$$\begin{aligned} \mathcal{W}_{(3)} \biggl(x,y,z,t;\frac{1}{n} \biggr)=\sum_{r=0}^{3} W_{r} \biggl(\frac{1}{n} \biggr)^{r}, \end{aligned}$$
(76)

which is the solution of q-HATM. Thus we can conclude that this present modification (δ-HPTM) is more reliable and general. In Figs. 16, we present the response of the obtained solutions by the proposed methods with regard to the real and imaginary parts in terms of 2D and 3D plots. The 2D and 3D plots show the graphical comparison of the four-term approximation solutions obtain by δ-HPTM and q-HATM and their exact solutions. The 2D plots also present the effect and behavior of the distinct fractional orders on the solution profile. In addition, Figs. 14 exhibit different shapes of the exact and approximate soliton-like solutions, whereas Figs. 5, 6 represent the periodic wave solutions of the gpZK equation. The dynamics of the solution profile can obviously be noted and justify why gpZK should be examined to understand the effects in real-life applications.

Figure 1
figure1

The plots of the real part of δ-HPTM, q-HATM, and exact solution for Example 1

The selection of the auxiliary parameters δ in δ-HPTM and ħ in q-HATM are very crucial to guarantee fast convergence of the series solutions. For this reason, in Figs. 79, we have provided the so-called δ-curves and ħ-curves of the two proposed methods, which serve as a guide in our optimal selection of values in the present analysis. The horizontal line test is employed to attain the intervals containing optimal values. The comparative study for the case \(\gamma=1\) of the real and imaginary parts of the results obtained by δ-HPTM, q-HATM, and the exact solution as the benchmark are considered in Tables 16. From these tables and plots we can observe that the solutions obtained by the proposed methods are very accurate and in agreement with their respective exact solutions.

Table 1 The comparative study of \(\operatorname{Re}{[\mathcal {W}_{(3)}]}\) solutions of δ-HPTM, q-HATM, and exact solution for Example 1 at \(\beta_{1}=1, \beta_{2}=2, \beta_{3}=0.1, y=2, z=2, \xi=0.1, e_{0}=3, p=0.5, q=0.5, \phi=1\), and \(t=0.01\)
Table 2 The comparative study of \(\operatorname{Im}{[\mathcal {W}_{(3)}]}\) solutions of δ-HPTM, q-HATM, and exact solution for Example 1 at \(\beta_{1}=1, \beta_{2}=2, \beta_{3}=0.1, y=2, z=2, \xi=0.1, e_{0}=3, p=0.5, q=0.5, \phi=1\), and \(t=0.01\)
Table 3 The comparative study of \(\operatorname{Re}{[\mathcal {W}_{(3)}]}\) solutions of δ-HPTM, q-HATM, and exact solution for Example 2 at \(\beta_{1}=1, \beta_{2}=2, \beta_{3}=0.1, y=2, z=2, \xi=0.1, e_{0}=1, p=0.8, q=0.8, \phi=1\), and \(t=0.01\)
Table 4 The comparative study of \(\operatorname{Im}{[\mathcal {W}_{(3)}]}\) solutions of δ-HPTM, q-HATM, and exact solution for Example 2 at \(\beta_{1}=1, \beta_{2}=2, \beta_{3}=0.1, y=2, z=2, \xi=0.1, e_{0}=1, p=0.8, q=0.8, \phi=1\), and \(t=0.01\)
Table 5 The comparative study of \(\operatorname{Re}{[\mathcal {W}_{(3)}]}\) solutions of δ-HPTM, q-HATM, and exact solution for Example 3 at \(\beta_{1}=1, \beta_{2}=2, \beta_{3}=10, y=2, z=2, \xi=1, e_{0}=3, p=0.3, q=0.3, \phi=1\), and \(t=0.01\)
Table 6 The comparative study of \(\operatorname{Im}{[\mathcal {W}_{(3)}]}\) solutions of δ-HPTM, q-HATM, and exact solution for Example 3 at \(\beta_{1}=1, \beta_{2}=2, \beta_{3}=10, y=2, z=2, \xi=1, e_{0}=3, p=0.3, q=0.3\), and \(t=0.01\)

Remark 2

The parameter values used for Figs. 19 are as follows:

  • Figs. 1 and 2: \(\beta_{1}=1, \beta_{2}=2, \beta_{3}=0.1, y=z=2, \xi=0.1, e_{0}=3\), \(p=q=0.5\), \(\phi=1\), \(n=1\), \(\delta=1\), \(\hbar=-1\), and \(t=0.1\).

    Figure 2
    figure2

    The plots of the imaginary part of δ-HPTM, q-HATM, and exact solution for Example 1

  • Figs. 3 and 4: \(\beta_{1}=1, \beta_{2}=2, \beta_{3}=0.1, y=z=2, \xi=0.1, e_{0}=1, p=q=0.8, \phi=1\), \(n=1\), \(\delta=1, \hbar=-1\), and \(t=0.1\).

    Figure 3
    figure3

    The plots of the real part of δ-HPTM, q-HATM, and exact solution for Example 2

    Figure 4
    figure4

    The plots of the imaginary part of δ-HPTM, q-HATM, and exact solution for Example 2

  • Figs. 5 and 6: \(\beta_{1}=1, \beta_{2}=2, \beta_{3}=10, y=z=2, \xi=1, e_{0}=3, p=q=0.3\), \(n=1\), \(\delta=1\), \(\hbar=-1\), and \(t=0.5\).

    Figure 5
    figure5

    The plots of the real parts of δ-HPTM, q-HATM, and exact solution for Example 3

    Figure 6
    figure6

    The plots of the imaginary parts of δ-HPTM, q-HATM, and exact solution for Example 3

  • Fig. 7: \(\beta_{1}=1, \beta_{2}=2, \beta_{3}=0.1, \xi=0.1, e_{0}=3, p=0.5, q=0.5, \phi=1, z=y=2, n=1, x=1\), and \(t=0.01\).

    Figure 7
    figure7

    The curves plots of the real and imaginary parts of δ-HPTM and q-HATM solutions for Example 1

  • Fig. 8: \(\beta_{1}=1, \beta_{2}=2, \beta_{3}=0.1, \xi=0.1, e_{0}=1, p=0.8, q=0.8, \phi=1, z=y=2, n=1, x=1\), and \(t=0.01\).

    Figure 8
    figure8

    The curves plots of the real and imaginary parts of δ-HPTM and q-HATM solutions for Example 2

  • Fig. 9: \(\beta_{1}=1, \beta_{2}=2, \beta_{3}=10, \xi=1, e_{0}=3, p=0.3, q=0.3, z=y=2, n=1, x=1\), and \(t=0.5\).

    Figure 9
    figure9

    The curves plots of the real and imaginary parts of δ-HPTM and q-HATM solutions for Example 3

Conclusion

In this paper, we proposed a new modification of the homotopy perturbation method (HPM), called the δ-homotopy perturbation transform method (δ-HPTM), which consists of HPM, the Laplace transform method, and a control parameter δ for solving integer- and noninteger-order nonlinear problems. We effectively used the proposed method and q-HATM to obtain analytical approximate solutions of the generalized fractional perturbed \((3+1)\)-dimensional Zakharov–Kuznetsov equation. This equation characterizes nonlinear dust-ion-acoustic waves in the magnetized two-ion-temperature dusty plasmas. In comparison to the control parameters n and h in q-HATM, the control parameter δ in δ-HPTM also helps to adjust and control the convergence region of the series solutions and can overcome some limitations of HPM, HPTM, and He–Laplace method. The two methods present series solutions in the form of recurrence relation with high exactness and minimal computations. In reality, we consider HPM, HAM, HPTM, PHPM, and He–Laplace method as particular cases of δ-HPTM and more general when compared with q-HATM (see Eq. (76)). Finally, δ-HPTM can be considered as a good refinement of the existing numerical techniques and can be employed to study strongly nonlinear mathematical models describing natural phenomena.

Availability of data and materials

Not applicable.

References

  1. 1.

    Owusu-Mensah, I., Akinyemi, L., Oduro, B., Iyiola, O.S.: A fractional order approach to modeling and simulations of the novel COVID-19. Adv. Differ. Equ. 2020(1), 1 (2020). https://doi.org/10.1186/s13662-020-03141-7

    MathSciNet  Article  Google Scholar 

  2. 2.

    Kumar, S., Rashidi, M.M.: New analytical method for gas dynamic equation arising in shock fronts. Comput. Phys. Commun. 185, 1947–1954 (2014)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Kumar, D., Seadawy, A.R., Joardar, A.K.: Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology. Chin. J. Phys. 56(1), 75–85 (2018)

    Article  Google Scholar 

  4. 4.

    Baleanu, D., Wu, G.C., Zeng, S.D.: Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos Solitons Fractals 102, 99–105 (2017)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Ghanbari, B., Kumar, S., Kumar, R.: A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative. Chaos Solitons Fractals 133, 1–11 (2020). https://doi.org/10.1016/j.chaos.2020.109619

    MathSciNet  Article  Google Scholar 

  6. 6.

    Kumar, S., Kumar, R., Agarwal, R.P., Samet, B.: A study on fractional Lotka Volterra population model by using Haar wavelet and Adams Bashforth–Moulton methods. Math. Methods Appl. Sci. 43(8), 5564–5578 (2020). https://doi.org/10.1002/mma.6297

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kumar, S., Ghosh, S., Samet, B., Goufo, E.F.D.: An analysis for heat equations arises in diffusion process using new Yang–Abdel–Aty–Cattani fractional operator. Math. Methods Appl. Sci. 43(9), 6062–6080 (2020)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Nasrolahpour, H.: A note on fractional electrodynamics. Commun. Nonlinear Sci. Numer. Simul. 18, 2589–2593 (2013)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Hilfer, R., Anton, L.: Fractional master equations and fractal time random walks. Phys. Rev. E 51, R848–R851 (1995)

    Google Scholar 

  10. 10.

    Zhang, Y., Pu, Y.F., Hu, J.R., Zhou, J.L.: A class of fractional-order variational image in-painting models. Appl. Math. Inf. Sci. 6(2), 299–306 (2012)

    MathSciNet  Google Scholar 

  11. 11.

    Pu, Y.F.: Fractional differential analysis for texture of digital image. J. Algorithms Comput. Technol. 1(3), 357–380 (2007)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Baleanu, D., Guvenc, Z.B., Machado, J.T.: New Trends in Nanotechnology and Fractional Calculus Applications. Springer, Berlin (2010)

    Book  Google Scholar 

  13. 13.

    Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)

    Book  Google Scholar 

  14. 14.

    Salahshour, S., Ahmadian, A., Senu, N., Baleanu, D., Agarwal, P.: On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem. Entropy 17, 885–902 (2015)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Ruzhansky, M.V., Je Cho, Y., Agarwal, P., Area, I.: Advances in Real and Complex Analysis with Applications. Springer, Singapore (2017)

    Book  Google Scholar 

  16. 16.

    Jain, S., Agarwal, P., Kilicman, A.: Pathway fractional integral operator associated with 3m-parametric Mittag-Leffler functions. Int. J. Appl. Comput. Math. 4(5), 115 (2018)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Qureshi, S., Yusuf, A.: Mathematical modeling for the impacts of deforestation on wildlife species using Caputo differential operator. Chaos Solitons Fractals 126, 32–40 (2019)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Nigmatullina, R.R., Agarwal, P.: Direct evaluation of the desired correlations: verification on real data. Phys. A, Stat. Mech. Appl. 534, 121558 (2019)

    Article  Google Scholar 

  19. 19.

    Rekhviashvili, S., Pskhu, A., Agarwal, P., Jain, S.: Application of the fractional oscillator model to describe damped vibrations. Turk. J. Phys. 43(3), 236–242 (2019)

    Article  Google Scholar 

  20. 20.

    Qureshi, S., Yusuf, A.: Fractional derivatives applied to MSEIR problems: comparative study with real world data. Eur. Phys. J. Plus 134(4), 171 (2019)

    Article  Google Scholar 

  21. 21.

    Caputo, M.: Elasticita e Dissipazione. Zanichelli, Bologna (1969)

    Google Scholar 

  22. 22.

    Liao, S.J.: Homotopy analysis method: a new analytic method for nonlinear problems. Appl. Math. Mech. 19, 957–962 (1998)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    Google Scholar 

  24. 24.

    Miller, K.S., Ross, B.: An Introduction to Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    Google Scholar 

  25. 25.

    Zhang, J., Wei, Z., Yong, L., Xiao, Y.: Analytical solution for the time fractional BBM-Burger equation by using modified residual power series method. Complexity 2018, 1–11 (2018). https://doi.org/10.1155/2018/2891373

    Article  Google Scholar 

  26. 26.

    Alquran, M., Al-Khaled, K., Chattopadhyay, J.: Analytical solutions of fractional population diffusion model: residual power series. Nonlinear Stud. 22(1), 31–39 (2015)

    MathSciNet  Google Scholar 

  27. 27.

    Senol, M., Iyiola, O.S., Daei Kasmaei, H., Akinyemi, L.: Efficient analytical techniques for solving time-fractional nonlinear coupled Jaulent–Miodek system with energy-dependent Schrödinger potential. Adv. Differ. Equ. 2019, 462 (2019)

    Article  Google Scholar 

  28. 28.

    Senol, M.: Analytical and approximate solutions of \((2+1)\)-dimensional time-fractional Burgers–Kadomtsev–Petviashvili equation. Commun. Theor. Phys. 72, 055003 (2020)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Akinyemi, L., Iyiola, O.S.: Exact and approximate solutions of time-fractional models arising from physics via Shehu transform. Math. Methods Appl. Sci. 1–23 (2020). https://doi.org/10.1002/mma.6484

  30. 30.

    Khuri, S.A.: A Laplace decomposition algorithm applied to class of nonlinear differential equations. J. Math. Appl. 1(4), 141–155 (2001)

    MathSciNet  Article  Google Scholar 

  31. 31.

    El-Tawil, M.A., Huseen, S.N.: The q-homotopy analysis method (qHAM). Int. J. Appl. Math. Mech. 8, 51–75 (2012)

    Google Scholar 

  32. 32.

    El-Tawil, M.A., Huseen, S.N.: On convergence of the q-homotopy analysis method. Int. J. Contemp. Math. Sci. 8, 481–497 (2013)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Akinyemi, L., Iyiola, O.S., Akpan, U.: Iterative methods for solving fourth and sixth order time-fractional Cahn–Hillard equation. Math. Methods Appl. Sci. 43(7), 4050–4074 (2020). https://doi.org/10.1002/mma.6173

    MathSciNet  Article  Google Scholar 

  34. 34.

    Akinyemi, L.: q-homotopy analysis method for solving the seventh-order time-fractional Lax’s Korteweg–deVries and Sawada–Kotera equations. Comput. Appl. Math. 38, 1–22 (2019)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Adomian, G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic, Norwell (1994)

    Book  Google Scholar 

  36. 36.

    Keskin, Y., Oturanc, G.: Reduced differential transform method: a new approach to fractional partial differential equations. Nonlinear Sci. Lett. A, Math. Phys. Mech. 1, 61–72 (2010)

    Google Scholar 

  37. 37.

    Akinyemi, L.: A fractional analysis of Noyes–Field model for the nonlinear Belousov–Zhabotinsky reaction. Comput. Appl. Math. 39, 1–34 (2020). https://doi.org/10.1007/s40314-020-01212-9

    MathSciNet  Article  Google Scholar 

  38. 38.

    He, J.H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Odibat, Z.M., Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 7, 27–34 (2006)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Liao, S.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004)

    MathSciNet  Google Scholar 

  41. 41.

    Kazem, S., Abbasbandy, S., Kumar, S.: Fractional-order Legendre functions for solving fractional-order differential equations. Appl. Math. Model. 37, 5498–5510 (2013)

    MathSciNet  Article  Google Scholar 

  42. 42.

    Rashidi, M.M., Hosseini, A., Pop, I., Kumar, S., Freidoonimehr, N.: Comparative numerical study of single and two-phase models of nano-fluid heat transfer in wavy channel. Appl. Math. Mech. Engl. 35, 831–848 (2014)

    Article  Google Scholar 

  43. 43.

    Kumar, S.: A new analytical modeling for telegraph equation via Laplace transform. Appl. Math. Model. 38, 3154–3163 (2014)

    MathSciNet  Article  Google Scholar 

  44. 44.

    Kumar, S., Kumar, A., Baleanu, D.: Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves. Nonlinear Dyn. 85(2), 699–715 (2016)

    MathSciNet  Article  Google Scholar 

  45. 45.

    He, J.H.: Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng. 178(3–4), 257–262 (1999)

    MathSciNet  Article  Google Scholar 

  46. 46.

    He, J.H.: A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int. J. Non-Linear Mech. 35(1), 37–43 (2000)

    Article  Google Scholar 

  47. 47.

    He, J.H.: Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135, 73–79 (2003)

    MathSciNet  Google Scholar 

  48. 48.

    He, J.H.: Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlinear Sci. Numer. Simul. 6(2), 207–208 (2005)

    MathSciNet  Google Scholar 

  49. 49.

    He, J.H.: Some asymptotic methods for strongly nonlinear equations. Int. J. Mod. Phys. B 20(10), 1141–1199 (2006)

    MathSciNet  Article  Google Scholar 

  50. 50.

    He, J.H.: A short review on analytical methods for a fully fourth-order nonlinear integral boundary value problem with fractal derivatives. Int. J. Numer. Methods Heat Fluid Flow 30(11), 4933–4943 (2020)

    Article  Google Scholar 

  51. 51.

    Adamu, M.Y., Ogenyi, P.: Parameterized homotopy perturbation method. Nonlinear Sci. Lett. A, Math. Phys. Mech. 8(2), 240–243 (2017)

    Google Scholar 

  52. 52.

    Adamu, M.Y., Ogenyi, P.: New approach to parameterized homotopy perturbation method. Therm. Sci. 22, 1815–1870 (2018)

    Article  Google Scholar 

  53. 53.

    Nadeem, M., Li, F.Q.: He–Laplace method for nonlinear vibration systems and nonlinear wave equations. J. Low Freq. Noise Vib. Act. Control 38(3–4), 1060–1074 (2019)

    Article  Google Scholar 

  54. 54.

    Anjum, N., He, J.H.: Homotopy perturbation method for N/MEMS oscillators. Math. Methods Appl. Sci. 1–15 (2020). https://doi.org/10.1002/mma.6583

  55. 55.

    Khan, Y., Wu, Q.: Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput. Math. Appl. 61, 1963–1967 (2011). https://doi.org/10.1016/j.camwa.2010.08.022

    MathSciNet  Article  Google Scholar 

  56. 56.

    Madani, M., Fathizadeh, M.: Homotopy perturbation algorithm using Laplace transformation. Nonlinear Sci. Lett. A, Math. Phys. Mech. 1, 263–267 (2010)

    Google Scholar 

  57. 57.

    Zhen, H., Tian, B., Wang, Y., Sun, W., Liu, L.: Soliton solutions and chaotic motion of the extended Zakharov–Kuznetsov equations in a magnetized two-ion-temperature dusty plasma. Phys. Plasmas 21, 073709 (2014)

    Article  Google Scholar 

  58. 58.

    Seadawy, A.R., Lu, D.: Ion acoustic solitary wave solutions of three-dimensional nonlinear extended Zakharov–Kuznetsov dynamical equation in a magnetized two-ion-temperature dusty plasma. Results Phys. 6, 590–593 (2016)

    Article  Google Scholar 

  59. 59.

    Lu, D., Seadawy, A.R., Arshad, M., Wang, J.: New solitary wave solutions of \((3+1)\)-dimensional nonlinear extended Zakharov–Kuznetsov and modified KdV–Zakharov–Kuznetsov equations and their applications. Results Phys. 7, 899–909 (2017). https://doi.org/10.1016/j.rinp.2017.02.002

    Article  Google Scholar 

  60. 60.

    Liu, Z.M., Duan, W.S., He, G.J.: Effects of dust size distribution on dust acoustic waves in magnetized two-ion-temperature dusty plasmas. Phys. Plasmas 15, 083702 (2008)

    Article  Google Scholar 

  61. 61.

    Seadawy, A.R.: Stability analysis for Zakharov–Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput. Math. Appl. 67, 172–180 (2014)

    MathSciNet  Article  Google Scholar 

  62. 62.

    Kumar, S., Kumar, D.: Solitary wave solutions of \((3 + 1)\)-dimensional extended Zakharov–Kuznetsov equation by Lie symmetry approach. Comput. Math. Appl. 77, 2096–2113 (2019)

    MathSciNet  Article  Google Scholar 

  63. 63.

    Luchko, Y.F., Srivastava, H.M.: The exact solution of certain differential equations of fractional order by using operational calculus. Comput. Math. Appl. 29, 73–85 (1995)

    MathSciNet  Article  Google Scholar 

  64. 64.

    Dhaigude, C.D., Nikam, V.R.: Solution of fractional partial differential equations using iterative method. Fract. Calc. Appl. Anal. 15(4), 684–699 (2012)

    MathSciNet  Article  Google Scholar 

  65. 65.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)

    Book  Google Scholar 

  66. 66.

    Huseen, S.N., Akinyemi, L.: The δ-homotopy perturbation method. Under review in Hacet. J. Math. Stat.

  67. 37.

    Lang, S.: Real and Functional Analysis, 3rd edn. Springer, Berlin (1993)

    Book  Google Scholar 

  68. 68.

    Akinyemi, L., Huseen, S.N.: A powerful approach to study the new modified coupled Korteweg–de Vries system. Math. Comput. Simul. 177, 556–567 (2020). https://doi.org/10.1016/j.matcom.2020.05.021

    MathSciNet  Article  Google Scholar 

  69. 69.

    Akinyemi, L., Iyiola, O.S.: A reliable technique to study nonlinear time-fractional coupled Korteweg–de Vries equations. Adv. Differ. Equ. 2020(169), 1 (2020). https://doi.org/10.1186/s13662-020-02625-w

    MathSciNet  Article  Google Scholar 

  70. 70.

    Kumar, D., Agarwal, R.P., Singh, J.: A modified numerical scheme and convergence analysis for fractional model of Lienard’s equation. J. Comput. Appl. Math. 399, 405–413 (2018)

    MathSciNet  Article  Google Scholar 

  71. 71.

    Prakash, A., Veeresha, P., Prakasha, D.G., Goyal, M.: A homotopy technique for fractional order multi-dimensional telegraph equation via Laplace transform. Eur. Phys. J. Plus 134(19), 1–18 (2019)

    Google Scholar 

  72. 72.

    Singh, J., Kumar, D., Baleanu, D., Rathore, S.: An efficient numerical algorithm for the fractional Drinfeld–Sokolov–Wilson equation. Appl. Math. Comput. 335, 12–24 (2018)

    MathSciNet  Google Scholar 

  73. 73.

    Srivastava, H.M., Kumar, D., Singh, J.: An efficient analytical technique for fractional model of vibration equation. Appl. Math. Model. 45, 192–204 (2017)

    MathSciNet  Article  Google Scholar 

  74. 62.

    Kumara, D., Singha, J., Baleanu, D.: A new analysis for fractional model of regularized long-wave equation arising in ion acoustic plasma waves. Math. Methods Appl. Sci. 40, 5642–5653 (2017)

    MathSciNet  Article  Google Scholar 

  75. 75.

    Veeresha, P., Prakasha, D.G., Qurashi, M.A., Baleanu, D.: A reliable technique for fractional modified Boussinesq and approximate long wave equations. Adv. Differ. Equ. 2019(1), 1 (2019)

    MathSciNet  Article  Google Scholar 

  76. 76.

    Odibat, Z.M., Shawagfeh, N.T.: Generalized Taylor’s formula. Appl. Math. Comput. 186(1), 286–293 (2007)

    MathSciNet  Google Scholar 

  77. 77.

    Argyros, I.K.: Convergence and Applications of Newton-Type Iterations. Springer, New York (2008)

    Google Scholar 

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Acknowledgements

The authors thank the referees for a number of suggestions, which have improved many aspects of this paper.

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Akinyemi, L., Şenol, M. & Huseen, S.N. Modified homotopy methods for generalized fractional perturbed Zakharov–Kuznetsov equation in dusty plasma. Adv Differ Equ 2021, 45 (2021). https://doi.org/10.1186/s13662-020-03208-5

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Keywords

  • Laplace transform
  • δ-homotopy transform perturbation method
  • q-homotopy analysis transform method
  • Perturbed Zakharov–Kuznetsov equation
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