An Analytical Approach to Solve the Fractional Benney Equation Using the q-Homotopy Analysis Transform Method

Abstract

This paper introduces an analytical approach for solving the Benney equation using the q-homotopy analysis transform method. The Benney equation is a nonlinear partial differential equation that has applications in diverse areas of physics and engineering. The q-homotopy analysis transform method is a numerical technique that has been successfully employed to solve a broad range of nonlinear problems. By utilizing this method, we derive approximate analytical solutions for the Benney equation. The results demonstrate that this method is a powerful and effective tool for obtaining accurate solutions for the equation. The proposed method offers a valuable contribution to the existing literature on the behavior of the Benney equation and provides researchers with a useful tool for solving this equation in various applications.

1. Introduction

Fractional nonlinear partial differential equations (PDEs) are a type of mathematical model that describes the behavior of systems with complex dynamics. These equations involve fractional derivatives, which generalize the classical concept of differentiation to non-integer orders. Nonlinearity, on the other hand, refers to the property that the solutions of the equation are not proportional to the input, making it difficult to predict the behavior of the system [1,2,3,4].

Fractional derivatives appear naturally in many physical phenomena, such as diffusion, wave propagation, and viscoelasticity, which exhibit memory effects and long-range interactions. These effects are captured by fractional derivatives, which account for the history of the system and its nonlocal interactions. Nonlinearity, on the other hand, arises from the coupling of multiple variables and the nonlinear response of the system to external forces [5,6]. Examples of fractional nonlinear PDEs include the fractional diffusion equation, the fractional wave equation, the fractional Burgers equation, and the fractional KdV equation. These equations have been studied extensively in recent years, both analytically and numerically, to understand their properties and to develop efficient methods for solving them [7,8,9].

One of the main challenges in dealing with fractional nonlinear PDEs is their complexity and the lack of closed-form solutions. Analytical methods are limited to specific cases and often rely on approximation techniques or numerical simulations. Numerical methods, on the other hand, are more general but require high computational resources and careful validation to ensure accuracy and stability [10,11,12]. Despite these challenges, fractional nonlinear PDEs have numerous applications in science and engineering, such as in the modeling of transport phenomena, fluid dynamics, materials science, and biological systems. They provide a powerful tool for understanding the behavior of complex systems and for designing control strategies to optimize their performance [13,14,15].

Fractional partial differential equations (FPDEs) and symmetry are closely related concepts in mathematical physics. In particular, symmetry plays a crucial role in the study of fractional differential equations. Symmetry is the invariance of a mathematical object under certain transformations [16,17,18]. In the case of differential equations, it refers to the invariance of the equation under certain transformations of its solutions. Symmetry can be used to simplify the problem and to identify solutions that satisfy certain boundary conditions [19,20].

The investigation of nonlinear physical issues is of paramount importance in acquiring meaningful insights. Numerous scholars have employed diverse techniques to articulate nonlinear fractional differential equations (FDEs) in order to derive the necessary outcomes.

The Benney equation general form, as in [21], is given by

$${\psi }_{\omega }(\eta ,\omega )+{\left({\psi }^n(\eta ,\omega )\right)}_{\eta }+{\psi }_{\eta \eta }(\eta ,\omega )+\beta {\psi }_{\eta \eta \eta }(\eta ,\omega )+{\psi }_{\eta \eta \eta \eta }(\eta ,\omega )=0,$$

(1)

where $\beta$ is the positive constant called the characterizing dispersion and n is a positive integer.

The Benney equation has several applications in various practical domains such as solitons theory, dynamics, physics, and fluid mechanics. The long waves on a viscous fluid traveling down an inclined plane and the unstable drift waves are described by the Benney equation in plasma; to learn more about the Benney equation’s applications, see [21,22].

We are particularly interested in the fractional-order Benney equation for our analysis:

$${}^{ABC}{D}_{\omega }^{\kappa }{\psi }_{\omega }(\eta ,\omega )+{\left({\psi }^3(\eta ,\omega )\right)}_{\eta }+{\psi }_{\eta \eta }(\eta ,\omega )+\beta {\psi }_{\eta \eta \eta }(\eta ,\omega )+{\psi }_{\eta \eta \eta \eta }(\eta ,\omega )=0.$$

(2)

There are several well-established techniques and algorithms that have gained prominence in providing analytical or numerical solutions for the fractional-order Benney problem in recent times. Akinlar et al. proposed a hybrid approach that effectively combined the advantages of both wavelets and fractional calculus to obtain an approximate solution to the Benney equation in fractional order [21]. Similarly, the extended homotopy perturbation approach was employed to investigate the Benney equation [23]. Furthermore, Kamal Shah et al. developed a novel technique known as LADM, which was well-suited for studying the Benney equation in the context of Caputo–Febrizio fractional derivatives [24]. These techniques have proven to be robust and effective in solving the fractional-order Benney problem, thereby contributing significantly to the field of fractional calculus.

Liao [25] introduced the homotopy analysis technique (HAM), in which, after choosing an auxiliary linear operator, an endless mapping was created from an initial hunch to a precise answer. The auxiliary parameter confirmed the convergence of the solution. Determining solutions to nonlinear issues representing real-world applications takes less time when semianalytical approaches are combined with an appropriate transform. The Laplace transform and the q-homotopy analysis transform technique (q-HATM) [26,27,28] have been combined. Its capacity to modify two potent computational approaches for investigating FDEs gives it an advantage. We may regulate the convergence area of the solution series in a sizable allowable domain by selecting the appropriate ℏ.

2. Preliminaries

The Laplace transform and several fundamental definitions of fractional derivatives are explained here.

Definition 1.

The fractional order Atangana–Baleanu (AB) derivative for a function $\psi \in {\mathbb{H}}^1(\mu ,ϵ)(ϵ>\mu ),\kappa \in [0,1]$ in the Riemann–Liouville sense is presented as follows

$${}_{\mu }^{ABC}{D}_{\omega }^{\kappa }\left(\psi \left(\omega \right)\right)=\frac{\mathcal{N}\left[\kappa \right]}{1-\kappa }\frac{d}{d\omega }{\int }_{\mu }^{\omega }{\psi }^{{}^{\prime }}\left(\rho \right)E_{\kappa }\left[\kappa \frac{{(\omega -\rho )}^{\kappa }}{\kappa -1}\right]d\rho .$$

(3)

Definition 2.

The AB fractional integral is defined by

$${}_{\mu }^{AB}{I}_{\omega }^{\kappa }\left(\psi \left(\omega \right)\right)=\frac{1-\kappa }{\mathcal{N}\left[\kappa \right]}\psi \left(\omega \right)+\frac{\kappa }{B\left[\kappa \right]\Gamma \left(\kappa \right)}{\int }_{\mu }^{\omega }\psi \left(\rho \right){(\omega -\rho )}^{\kappa -1}d\rho .$$

(4)

Definition 3.

The fractional-derivative Laplace transform (LT) is given by

$${\mathcal{L}}_{\omega }{[}_{\mu }^{AB}{D}_{\omega }^{\kappa }\left(\psi \left(\omega \right)\right)=\frac{\mathcal{N}\left[\kappa \right]}{1-\kappa }\frac{s^{\kappa }{\mathcal{L}}_{\omega }\left[\psi \left(\omega \right)\right]-s^{\kappa -1}\psi \left(0\right)}{s^{\kappa }+\kappa (1-\kappa )},0<\kappa \le 1.$$

(5)

3. Methodology

We provide the general methodology of the q-HATM [29,30,31,32] for the fractional order Benney equation

$${}^{ABC}{D}_{\omega }^{\kappa }\psi (\eta ,\omega )+{\left({\psi }^n(\eta ,\omega )\right)}_{\eta }+{\psi }_{\eta \eta }(\eta ,\omega )+\beta {\psi }_{\eta \eta \eta }(\eta ,\omega )+{\psi }_{\eta \eta \eta \eta }(\eta ,\omega )=0,$$

(6)

with the initial condition

$$\psi (\eta ,0)=f\left(\eta \right),$$

(7)

where ${}^{ABC}{D}_{\omega }^{\kappa }\psi (\eta ,\omega )$ is the AB derivative of $\psi (\eta ,\omega )$.

On using the LT on Equation (6), after simplification, we have

$$\mathcal{L}\left[\psi (\eta ,\omega )\right]-\frac{f\left(\eta \right)}{s}+\frac{1}{\mathcal{N}\left[\kappa \right]}\left(1-\kappa +\frac{\kappa }{s^{\kappa }}\right)\mathcal{L}\left[{\left({\psi }^n(\eta ,\omega )\right)}_{\eta }+{\psi }_{\eta \eta }(\eta ,\omega )+\beta {\psi }_{\eta \eta \eta }(\eta ,\omega )+{\psi }_{\eta \eta \eta \eta }(\eta ,\omega )\right]=0.$$

(8)

The nonlinear operator is defined as follows

$$\begin{array}{cc}\hfill N\left[\varphi \right(\eta ,\omega ;q\left)\right]=& \mathcal{L}\left[\varphi (\eta ,\omega ;q)\right]-\frac{f\left(\eta \right)}{s}+\frac{1}{\mathcal{N}\left[\kappa \right]}\left(1-\kappa +\frac{\kappa }{s^{\kappa }}\right)\mathcal{L}[{\left({\varphi }^n(\eta ,\omega ;q)\right)}_{\eta }+{\varphi }_{\eta \eta }(\eta ,\omega ;q)\hfill \\ \hfill & +\beta {\varphi }_{\eta \eta \eta }(\eta ,\omega ;q)+{\varphi }_{\eta \eta \eta \eta }(\eta ,\omega ;q)].\hfill \end{array}$$

(9)

Here, $\varphi (\eta ,\omega ;q)$ is the real-valued function with respect to $\eta$, $\omega$, and $q\in [0,\frac{1}{n}]$. Now, we define a homotopy as follows

$$(1-nq)\mathcal{L}[\varphi (\eta ,\omega ;q)-{\psi }_0(\eta ,\omega )]=\hslash qN\left[\varphi (\eta ,\omega ;q)\right],$$

(10)

where ℏ is an auxiliary parameter, $\mathcal{L}$ is the LT, and $q\in [0,\frac{1}{n}]$$(n\ge 1)$ is the embedding parameter. For $q=0$ and $q=\frac{1}{B}$, the following hold true

$$\varphi (\eta ,\omega ;0)={\psi }_0(\eta ,\omega ),\varphi (\eta ,\omega ;\frac{1}{n})=\psi (\eta ,\omega ).$$

(11)

Thus, by intensifying q from 0 to $\frac{1}{n}$, the solution $\varphi (\eta ,\omega ;q)$ varies from the initial guess ${\psi }_0(\eta ,\omega )$ to $\psi (\eta ,\omega )$. Defining $\varphi (\eta ,\omega ;q)$ with respect to q by using the Taylor theorem, we obtain

$$\varphi (\eta ,\omega ;q)={\psi }_0(\eta ,\omega )+\sum _{m=1}^{\infty }{\psi }_m(\eta ,\omega )q^m,$$

(12)

where

$${\psi }_m=\frac{1}{m!}\frac{{\partial }^m\varphi (\eta ,\omega ;q)}{\partial q^m}{|}_{q=0}.$$

(13)

The series (10) converges at $q=\frac{1}{n}$ for the proper choice of ${\psi }_0(\eta ,\xi ,\omega )$, n, and ℏ. Then,

$$\psi (\eta ,\omega )={\psi }_0(\eta ,\omega )+\sum _{m=1}^{\infty }{\psi }_m(\eta ,\omega ){\left(\frac{1}{n}\right)}^m.$$

(14)

Taking the derivative of Equation (10) with respect to the embedding parameter q, setting $q=0$, and dividing by $m!$, we obtain

$$\mathcal{L}[\psi (\eta ,\omega )-k_m{\psi }_{m-1}(\eta ,\omega )]=\hslash {\mathcal{R}}_m\left({\overrightarrow{\psi }}_{m-1}\right),$$

(15)

where $h\ne 0$ is an auxiliary parameter, and the vectors are defined as

$${\overrightarrow{\psi }}_m=[{\psi }_0(\eta ,\omega ),{\psi }_1(\eta ,\omega ),\cdots ,{\psi }_m(\eta ,\omega )].$$

(16)

On applying the inverse LT on Equation (15), one obtains

$${\psi }_m(\eta ,\omega )=k_m{\psi }_{m-1}(\eta ,\omega )+\hslash {\mathcal{L}}^{-1}\left[{\mathcal{R}}_m\left({\overrightarrow{\psi }}_{m-1}\right)\right],$$

(17)

$${\mathcal{R}}_m\left({\overrightarrow{\psi }}_{m-1}\right)={\left.\frac{1}{(m-1)!}\frac{{\partial }^{m-1}N\left[\varphi (x,t;q)\right]}{\partial q^{m-1}}\right|}_{q=0},$$

where

$$\begin{array}{cc}\hfill {\Re }_m\left({\overrightarrow{\psi }}_{m-1}\right)=& \mathcal{L}\left[{\psi }_{m-1}(\eta ,\omega )\right]-\left(1-\frac{k_m}{n}\right)\left(\frac{f\left(\eta \right)}{s}\right)+\frac{1}{\mathcal{N}\left[\kappa \right]}\left(1-\kappa +\frac{\kappa }{s^{\kappa }}\right)\mathcal{L}[{\left({\psi }^n(\eta ,\omega )\right)}_{\eta }\hfill \\ \hfill & +{\psi }_{\eta \eta }(\eta ,\omega )+\beta {\psi }_{\eta \eta \eta }(\eta ,\omega )+{\psi }_{\eta \eta \eta \eta }(\eta ,\omega )],\hfill \end{array}$$

(18)

and

$$k_m=\left\{\begin{array}{cc}0,\hfill & m\le 1,\hfill \\ n,\hfill & m>1.\hfill \end{array}\right.$$

(19)

Using Equations (17) and (18), one can obtain the series of ${\psi }_m(\eta ,\omega )$. Lastly, the series q-HATM solution is defined as

$$\psi (\eta ,\omega )=\sum _{m=0}^{\infty }{\psi }_m(\eta ,\omega ).$$

(20)

Theorem 1.

If $\exists$ a constant $0<\epsilon <1$ in such a way that $∥u_{m+1}(x,t)∥\le \epsilon ∥u_m(x,t)∥$ for each value of m, and if the truncated series ${\sum }_{m=0}^r{u}_m(x,t){\left(\frac{1}{n}\right)}^m$ is an approximate solution $u(x,t)$, then the maximum absolute truncated error is determined by [26]

$$∥u(x,t)-\sum _{m=0}^r{u}_m(x,t){\left(\frac{1}{n}\right)}^m∥\le \frac{{\epsilon }^{r+1}}{n^r(n-\epsilon )}∥u_0(x,t)∥.$$

4. Numerical Problems

Problem 1.

Consider the fractional order Benney equation with the ABR derivative given by

$$\begin{array}{cc}\hfill {}^{ABR}& D_{\omega }^{\kappa }\psi (\eta ,\omega )+3{\psi }^2(\eta ,\omega ){\psi }_{\eta }(\eta ,\omega )+{\psi }_{\eta \eta }(\eta ,\omega )+\beta {\psi }_{\eta \eta \eta }(\eta ,\omega )+{\psi }_{\eta \eta \eta \eta }(\eta ,\omega )=0,0<\kappa \le 1,\hfill \\ \hfill & \psi (\eta ,0)={\eta }^2.\hfill \end{array}$$

(21)

Using the Laplace transform on Equation (21) and the initial condition, we obtain

$$\mathcal{L}\left[\psi (\eta ,\omega )\right]=\frac{{\eta }^2}{s}-\frac{1}{\mathcal{N}\left[\kappa \right]}\left((1-\kappa +\frac{\kappa }{s^{\kappa }})\right)\mathcal{L}\left[3{\psi }^2(\eta ,\omega ){\psi }_{\eta }(\eta ,\omega )+{\psi }_{\eta \eta }(\eta ,\omega )+\beta {\psi }_{\eta \eta \eta }(\eta ,\omega )+{\psi }_{\eta \eta \eta \eta }(\eta ,\omega )\right].$$

(22)

Here, we define the nonlinear operator as

$$\begin{array}{cc}\hfill N\left[\varphi \right(\eta ,\omega ;q\left)\right]=& \mathcal{L}\left[\varphi (\eta ,\omega ;q)\right]-\frac{{\eta }^2}{s}+\frac{1}{\mathcal{N}\left[\kappa \right]}\left((1-\kappa +\frac{\kappa }{s^{\kappa }})\right)\mathcal{L}[3{\varphi }^2(\eta ,\omega ;q){\varphi }_{\eta }(\eta ,\omega ;q)+{\varphi }_{\eta \eta }(\eta ,\omega ;q)\hfill \\ \hfill & +\beta {\varphi }_{\eta \eta \eta }(\eta ,\omega ;q)+{\varphi }_{\eta \eta \eta \eta }(\eta ,\omega ;q)].\hfill \end{array}$$

(23)

The $m-th$ order deformation is given by

$$\mathcal{L}[{\psi }_m(\eta ,\omega )-k_m{\psi }_{m-1}(\eta ,\omega )]=\hslash {\mathcal{R}}_m\left({\overrightarrow{\psi }}_{m-1}\right),$$

(24)

where

$$\begin{array}{cc}\hfill {\mathcal{R}}_m\left({\overrightarrow{\psi }}_m\right)=& \mathcal{L}\left[\psi (\eta ,\omega )\right]-\frac{{\eta }^2}{s}+\frac{1}{\mathcal{N}\left[\kappa \right]}\left((1-\kappa +\frac{\kappa }{s^{\kappa }})\right)\mathcal{L}[3\sum _{i=0}^{m-1}\left(\sum _{j=0}^i{\psi }_j(\eta ,\omega ){\psi }_{i-j}(\eta ,\omega )\right)\frac{\partial }{\partial \eta }{\psi }_{m-i-1}(\eta ,\omega )\hfill \\ \hfill & +{\psi }_{\eta \eta }(\eta ,\omega )+\beta {\psi }_{\eta \eta \eta }(\eta ,\omega )+{\psi }_{\eta \eta \eta \eta }(\eta ,\omega )].\hfill \end{array}$$

(25)

Taking the inverse Laplace transform on Equation (24), we have

$${\psi }_m(\eta ,\omega )=k_m{\psi }_{m-1}(\eta ,\omega )+\hslash {\mathcal{L}}^{-1}\left({\mathcal{R}}_m\left({\overrightarrow{\psi }}_{m-1}\right)\right).$$

(26)

Solving the above equation, we find the following terms

$$\begin{array}{cc}\hfill {\psi }_0(\eta ,\omega )=& {\eta }^2,\hfill \\ \hfill {\psi }_1(\eta ,\omega )=& \frac{2\hslash \left(-1+3{\eta }^5(-1+\kappa )+\left(-\frac{(3{\eta }^5+1){\omega }^{\kappa }}{\Gamma (\kappa +1)}+1\right)\kappa \right)}{\mathcal{N}\left[\kappa \right]},\hfill \\ \hfill {\psi }_2(\eta ,\omega )=& \frac{2n\hslash \left(-1+3{\eta }^5(-1+\kappa )+\left(-\frac{(3{\eta }^5+1){\omega }^{\kappa }}{\Gamma (\kappa +1)}+1\right)\kappa \right)}{\mathcal{N}\left[\kappa \right]}+\hslash \left(\frac{1}{\Gamma {(1+\kappa )}^2\mathcal{N}{\left[\kappa \right]}^3}\right(24(\Gamma {(1+\kappa )}^2\hfill \\ \hfill & \times (-1+\kappa )\left(1+{\kappa }^2\left(1+\frac{2{\omega }^{2\kappa }}{\Gamma (2\kappa +1)}\right)-\kappa \left(\frac{3(-1+\kappa ){\omega }^{\kappa }}{\Gamma (\kappa +1)}+2\right)\right)+(-\frac{{\omega }^{3\kappa }\kappa }{\Gamma (3\kappa +1)}\hfill \\ \hfill & +\frac{{\omega }^{2\kappa }(-1+\kappa )}{\Gamma (2\kappa +1)})\Gamma \left(2\alpha +1\right){\kappa }^2){\left(3{\eta }^5+1\right)}^2{\hslash }^2\eta )+\frac{1}{\mathcal{N}{\left[\kappa \right]}^2}(2(\mathcal{N}\left[\kappa \right]\left(3{\eta }^5+1\right)(-1+\kappa )\hfill \\ \hfill & +3{(-1+\kappa )}^2{\eta }^2(21{\eta }^6+60\beta +22\eta )+(-720\eta +\frac{1}{\Gamma (\kappa +1)}(126{\eta }^8+360\beta {\eta }^2+132{\eta }^3\hfill \\ \hfill & -\mathcal{N}\left[\alpha \right](3{\eta }^5+1)+6(-(21{\eta }^7+60\beta \eta +22{\eta }^2+120)\kappa +120)\eta \left){\omega }^{\kappa }\right)\kappa +3\eta ({\kappa }^2\hfill \\ \hfill & \times \left(\frac{{\omega }^{2\kappa }}{\Gamma (2\kappa +1)}\left(21{\eta }^7+60\beta \eta +22{\eta }^2+120\right)+120\right)+120\left)\right)\hslash \left)\right),\hfill \\ \hfill & ⋮\hfill \end{array}$$

(27)

In Figure 1, two-dimensional plots of approximate solutions at different fractional orders for the problem in 1. In Figure 2, three-dimensional plots of approximate solutions at different fractional orders for the problem 1. In

Table 1. The numerical values of approximate solutions at various fractional values for the problem 1 with $\beta =1$.

$\mathit{\omega }$	$\mathit{\eta }$	$\mathit{\kappa }=0.5$	$\mathit{\kappa }=0.7$	$\mathit{\kappa }=0.9$	$\mathit{\kappa }=1$
1	0	2.128379167	2.140766367	2.071557441	2
	0.1	85.31665749	75.96945949	54.8749427	40.6761086
	0.2	177.0418155	157.3943186	113.1421651	83.3731548
	0.3	278.2481880	247.2507879	177.4677326	130.5258330
	0.4	390.2002534	346.6438118	248.6191438	182.6863557
	0.5	515.0072901	457.3916866	327.8236424	240.7187498
	0.6	656.8171393	583.0369874	417.4220704	306.2414199
	0.7	824.6185442	731.2090832	522.3800063	382.6390723
	0.8	1038.759922	919.0860269	653.7294243	477.335391
	0.9	1345.435884	1185.467730	836.0769337	606.689905
	1	1846.975581	1615.918535	1123.085975	806.000000

, The numerical values of approximate solutions at various fractional values for the problem 1 with $\beta =1$.

Problem 2.

Consider the fractional order Benney equation with the ABR derivative given by

$$\begin{array}{cc}\hfill {}^{ABR}& D_{\omega }^{\kappa }\psi (\eta ,\omega )+3{\psi }^2(\eta ,\omega ){\psi }_{\eta }(\eta ,\omega )+{\psi }_{\eta \eta }(\eta ,\omega )+\beta {\psi }_{\eta \eta \eta }(\eta ,\omega )+{\psi }_{\eta \eta \eta \eta }(\eta ,\omega )=0,0<\kappa \le 1,\hfill \\ \hfill & \psi (\eta ,0)=cos\eta .\hfill \end{array}$$

(28)

Using the Laplace transform on Equation (28) and the initial condition, we obtain

$$\mathcal{L}\left[\psi (\eta ,\omega )\right]=\frac{cos\eta }{s}-\frac{1}{\mathcal{N}\left[\kappa \right]}\left((1-\kappa +\frac{\kappa }{s^{\kappa }})\right)\mathcal{L}\left[3{\psi }^2(\eta ,\omega ){\psi }_{\eta }(\eta ,\omega )+{\psi }_{\eta \eta }(\eta ,\omega )+\beta {\psi }_{\eta \eta \eta }(\eta ,\omega )+{\psi }_{\eta \eta \eta \eta }(\eta ,\omega )\right].$$

(29)

Here, we define the nonlinear operator as

$$\begin{array}{cc}\hfill N\left[\varphi \right(\eta ,\omega ;q\left)\right]=& \mathcal{L}\left[\varphi (\eta ,\omega ;q)\right]-\frac{cos\eta }{s}+\frac{1}{\mathcal{N}\left[\kappa \right]}\left((1-\kappa +\frac{\kappa }{s^{\kappa }})\right)\mathcal{L}[3{\varphi }^2(\eta ,\omega ;q){\varphi }_{\eta }(\eta ,\omega ;q)+{\varphi }_{\eta \eta }(\eta ,\omega ;q)\hfill \\ \hfill & +\beta {\varphi }_{\eta \eta \eta }(\eta ,\omega ;q)+{\varphi }_{\eta \eta \eta \eta }(\eta ,\omega ;q)].\hfill \end{array}$$

(30)

The $m-th$ order deformation is given by

$$\mathcal{L}[{\psi }_m(\eta ,\omega )-k_m{\psi }_{m-1}(\eta ,\omega )]=\hslash {\mathcal{R}}_m\left({\overrightarrow{\psi }}_{m-1}\right),$$

(31)

where

$$\begin{array}{cc}\hfill {\mathcal{R}}_m\left({\overrightarrow{\psi }}_m\right)=& \mathcal{L}\left[\psi (\eta ,\omega )\right]-\frac{cos\eta }{s}+\frac{1}{\mathcal{N}\left[\kappa \right]}\left((1-\kappa +\frac{\kappa }{s^{\kappa }})\right)\mathcal{L}[3\sum _{i=0}^{m-1}\left(\sum _{j=0}^i{\psi }_j(\eta ,\omega ){\psi }_{i-j}(\eta ,\omega )\right)\frac{\partial }{\partial \eta }{\psi }_{m-i-1}(\eta ,\omega )\hfill \\ \hfill & +{\psi }_{\eta \eta }(\eta ,\omega )+\beta {\psi }_{\eta \eta \eta }(\eta ,\omega )+{\psi }_{\eta \eta \eta \eta }(\eta ,\omega )].\hfill \end{array}$$

(32)

Taking the inverse Laplace transform on Equation (31), we have

$${\psi }_m(\eta ,\omega )=k_m{\psi }_{m-1}(\eta ,\omega )+\hslash {\mathcal{L}}^{-1}\left({\mathcal{R}}_m\left({\overrightarrow{\psi }}_{m-1}\right)\right),$$

(33)

Solving the above equation, we find the following terms

$$\begin{array}{cc}\hfill {\psi }_0(\eta ,\omega )=& cos\eta ,\hfill \\ \hfill {\psi }_1(\eta ,\omega )=& \frac{\hslash \left(1+\kappa \left(-1+\frac{{\omega }^{\kappa }}{\Gamma (\kappa +1)}\right)\right)(3cos{\left(\eta \right)}^2-\beta )sin\left(\eta \right)}{\mathcal{N}\left[\kappa \right]},\hfill \\ \hfill {\psi }_2(\eta ,\omega )=& \frac{n\hslash \left(1+\kappa \left(-1+\frac{{\omega }^{\kappa }}{\Gamma (\kappa +1)}\right)\right)(3cos{\left(\eta \right)}^2-\beta )sin\left(\eta \right)}{\mathcal{N}\left[\kappa \right]}+\hslash (\frac{1}{\Gamma {(\kappa +1)}^2\mathcal{N}\left[\kappa \right]}\hfill \\ \hfill & (3{\left(3cos{\left(\eta \right)}^2-\beta \right)}^{2}sin{\left(\eta \right)}^3{\hslash }^2(\left(-1-{\kappa }^2\left(1+\frac{2{\omega }^{2\kappa }}{\Gamma (2\kappa +1)}\right)+\kappa \left(\frac{3(-1+\kappa ){\omega }^{\kappa }}{\Gamma (\kappa +1)}+2\right)\right)\hfill \\ \hfill & \times \Gamma {(\kappa +1)}^2(-1+\kappa )+{\kappa }^2\left(\frac{\kappa {\omega }^{3\kappa }}{\Gamma \left(\right)3\kappa +1}-\frac{{\omega }^{2\kappa (-1+\kappa )}}{\Gamma (2\kappa +1)}\right)\Gamma (2\kappa +1)\left)\right)\hfill \\ \hfill & +\frac{1}{\mathcal{N}{\left[\kappa \right]}^2}(\hslash (\frac{1}{\Gamma (2\kappa +1)}(-9cos{\left(\eta \right)}^5+24\beta cos{\left(\eta \right)}^3+54sin\left(\eta \right)(1-4cos{\left(\eta \right)}^2)\hfill \\ \hfill & +cos\left(\eta \right)(9sin{\left(\eta \right)}^2(3cos{\left(\eta \right)}^2-7\beta )-{\beta }^2)){\omega }^{2\kappa }{\kappa }^2+\frac{1}{\Gamma (\kappa +1)}\left(\kappa \right((54-54\kappa \hfill \\ \hfill & -\beta \mathcal{N}\left[\kappa \right]+3cos{\left(\eta \right)}^2(-126+\mathcal{N}\left[\kappa \right]+126\kappa ))sin\left(\eta \right)+2(-24\beta cos{\left(\eta \right)}^3+9cos{\left(\eta \right)}^5\hfill \\ \hfill & -27sin{\left(\eta \right)}^3+(9(-3cos{\left(\eta \right)}^2+7\beta )sin{\left(\eta \right)}^2+{\beta }^2)cos\left(\eta \right)\left)(-1+\kappa )\right){\omega }^{\kappa })\hfill \\ \hfill & +(-54+\beta \mathcal{N}\left[\kappa \right]+54\kappa +3(72-\mathcal{N}\left[\kappa \right]-72\kappa )cos{\left(\eta \right)}^2)sin\left(\eta \right)(-1+\kappa )\hfill \\ \hfill & +{(-1+\kappa )}^{2}cos\left(\eta \right)(24cos{\left(\eta \right)}^2\beta -9cos{\left(\eta \right)}^4+9sin{\left(\eta \right)}^2(3cos{\left(\eta \right)}^2-7\beta )-{\beta }^2)\left)\right)),\hfill \\ \hfill & ⋮\hfill \end{array}$$

(34)

In Figure 3, Two-dimensional plots of approximate solutions at different fractional orders for the problem 2. In Figure 4, Three-dimensional plots of approximate solutions at different fractional orders for the problem 2. In

Table 2. Numerical values of approximate solutions at various fractional values for the problem 2 with $\beta =1$.

$\mathit{\omega }$	$\mathit{\eta }$	$\mathit{\kappa }=0.5$	$\mathit{\kappa }=0.7$	$\mathit{\kappa }=0.9$	q-HATM $\mathit{\kappa }=1$	LADM $\mathit{\kappa }=1$
1	0	15.89865417	14.25381518	10.52430944	8	8
	0.1	−1.832024984	−1.543389489	−0.8803186628	−0.4295903926	−0.4295903926
	0.2	−19.14110255	−16.95979593	−12.00254110	−8.646209940	−8.646209940
	0.3	−34.63295823	−30.75026556	−21.94082484	−15.98192520	−15.98192520
	0.4	−47.06736293	−41.81265793	−29.90352005	−21.85367303	−21.85367303
	0.5	−55.44182498	−49.26000190	−35.25928442	−25.79968055	−25.79968055
	0.6	−59.07713582	−52.49340697	−37.58497881	−27.51235592	−27.51235592
	0.7	−57.68973832	−51.26276939	−36.70405550	−26.86375081	−26.86375081
	0.8	−51.42898240	−45.69773137	−32.70565981	−23.91791002	−23.91791002
	0.9	−40.86344304	−36.29615175	−25.93720850	−18.92581335	−18.92581335
	1	−26.91555476	−23.86942018	−16.96996307	−12.30256052	−12.30256052

, Numerical values of approximate solutions at various fractional values for the problem 2 with $\beta =1$.

5. Conclusions

In conclusion, the q-homotopy analysis transform method is a highly promising approach for solving the Benney equation and other nonlinear differential equations. The method’s ability to break down complex equations into simpler parts, coupled with its accuracy and consistency with other numerical methods, make it a valuable tool in the field of mathematics. Moreover, the q-homotopy analysis transform method offers several advantages over traditional numerical methods, making it easier to implement and providing greater accuracy. Given its potential, this method is expected to play an increasingly important role in solving nonlinear differential equations in the future, and researchers are encouraged to explore its applications further.