Extension of the Optimal Auxiliary Function Method to Solve the System of a Fractional-Order Whitham–Broer–Kaup Equation

Abstract

In this paper, we introduce and implement the optimal auxiliary function method to solve a system of fractional-order Whitham–Broer–Kaup equations, a class of nonlinear partial differential equations with broad applications in mathematical physics. This method provides a systematic and efficient approach to finding accurate solutions for complex systems of fractional-order equations. We give a full analysis using tables and figures to demonstrate the reliability and accuracy of our approach. We confirm the effectiveness of our suggested method in solving the considered equations using numerical simulations and comparisons, emphasizing its potential for applications in a variety of scientific and engineering areas.

1. Introduction

An integral or derivative of arbitrary order is included in fractional calculus as a generalization of classical integral and differentiation [1]. It has become increasingly popular in a variety of fields, including physics, fluid mechanics, engineering, and entropy [2,3,4,5]. In some cases, fractional calculus is capable of accurately and appropriately describing some of the physical models and engineering processes. There is more scope for the use of Shannon entropy than fractional calculus-based entropies [6] due to its greater usefulness. Fractional entropy is a popular research field due to its board range of implementations [7]. As shown in [8], the fractional derivatives of the fluid–dynamic traffic model can eliminate the assumptions underlying continuum traffic flow models. The theory and implementation of fractional calculus have been extensively studied [9,10,11,12,13].

In the fields of engineering, applied sciences, and technology, partial differential equations with nonlinearities have numerous applications, including gravitation and dynamics [14,15,16]. In a wide variety of fields, including plasma physics, mathematical biology fluid dynamics, and solid-state physics, nonlinear PDEs are useful tools for modeling complex nonlinear dynamical phenomena [17]. A set of partial differential equations can be used to represent a variety of dynamical systems. There are well-known applications of partial differential equations in mathematics, such as the Poincare coefficient and the Calabi coefficient. Over the past few decades, nonlinear partial differential equations have been a major area of research. To examine the approximate results of nonlinear partial differential equations, a wide range of mathematical methods have been developed. According to Aminikhah and Biazar [18], the Homotopy Perturbation Method (HPM) is a powerful method for solving coupled models of Brusselator and Burger equations that obey a particular set of assumptions. For classically ordered partial differential equations [19,20,21,22,23,24,25,26,27], there are other methods for approximating the solutions. In the last few years, partial differential equations of fractional order have achieved significant advances [28,29,30,31,32,33], because they have been applied to various areas of applied science, including control theory, pattern reorganization, signal processing, identification of systems, and image analysis. The numerical solution of fractional-order differential equations is essential for many fields of science since they describe many physical phenomena more accurately than classical-order differential equations. Among the different variations available for nonlinear analysis, one of the most well-known is the Ritz method [34], and another is the variational iterations technique [35,36]. In our study of nonlinear equations that always substitute some ansatzes into the Lagrange functional, we were able to accurately capture the soliton solutions and their evolution by applying the variational approximation method. As a result, the variational parameters can be determined by solving the Euler–Lagrange equations [37,38]. A variety of advantages are available to users of variational-based methods in comparison to analytical or numerical methods [39,40]. In the first place, they provide a more comprehensive perspective on practical issues, as well as a better understanding of the nature of the solutions. Second, the proposed method is based on the idea that the solution domain of the PDE will be energy conserved throughout the entire domain of the solution, which requires much weaker local differentiabilities of variables than direct solutions to the PDE. As a final point, all possible trial functions were used to obtain the best solutions. According to Garner et al. [41], it is possible to solve the Korteweg–de Vries equation by multiple solitary solutions. Date et al. [42], have solved quasi-periodically the orthogonal Kadomtsev–Petviashvili equations. The variant Boussinesq equation is stated to have a single solution according to Wang et al. [43]. It was demonstrated by Fan and Zhang [44], that there is a shallow water solution to the Whitham–Broer–Kaup equation. There are multiple solites for Whitham–Broer–Kaup equations, as demonstrated by Naranmandula and Chen [45]. An exact solution class was discovered by Wang et al. [46], using the Whitham-Broer-Kaup equations. The Optimal Auxiliary Function Method (OAFM) can be used for the solution of a variety of different types of nonlinear differential equations. As a method for solving nonlinear differential equations, the OAFM was first introduced by Belendez and Hernandez [47]. To demonstrate the accuracy and dependability of the method, the authors solved a Duffing oscillator equation. The OAFM has been applied to solve a wide range of nonlinear physics and engineering problems since its invention. As part of their study of quantum mechanics [48], Belendez et al. (2010) [47] applied the OAFM to solve the nonlinear Schrodinger equation. OAFM provided accurate and precise results compared to numerical outcomes, according to the authors.

The development of this article is represented in five sections. In Section 2 and Section 3, we discuss some basic definitions and methodology of OAFM. The problem’s solution is represented in Section 4. Section 5 show the results and discussion. The conclusion of the work is written in Section 5.

2. Preliminaries

A few definitions and conclusions regarding Caputo fractional derivatives are discussed in this section.

Definition 1.

The Reiman–Liouville fractional integral operator is given as

$$J_{ϵ}^{\nu }\omega (f,ϵ)=\frac{1}{\Gamma \left(\nu \right)}{\int }_0^{ϵ}{(ϵ-r)}^{m-1}\omega (f,r)dr.$$

(1)

Definition 2.

Using the Caputo formula, the fractional derivative of $\omega (f,ϵ)$ is given by

$${}^C{D}_{ϵ}^{\nu }\omega (f,ϵ)=\frac{1}{\Gamma (m-\nu )}{\int }_0^{ϵ}{(ϵ-r)}^{m-\nu -1}\omega (f,r)dr,m-1<\nu \le m,ϵ>0.$$

(2)

Lemma 1.

For $m-1<\nu \le m$, $p>-1$, $ϵ\ge -1$, and $\lambda \in R$, we have:

1. 

$D_{ϵ}^{\nu }{ϵ}^p=\frac{\Gamma (\nu +1)}{\Gamma (p-\nu +1)}{ϵ}^{p-\nu }$.

2. 

$D_{ϵ}^{\nu }\lambda =0$.

3. 

$D_{ϵ}^{\nu }{I}_{ϵ}^{\nu }\omega (f,ϵ)=\omega (f,ϵ)$.

4. 

$I_{ϵ}^{\nu }=\omega (f,ϵ)-{\sum }_{i=0}^{n-1}{\partial }^i\omega (f,0)\frac{{ϵ}^i}{i!}$.

3. The Basic Concept of the Optimal Auxiliary Function Method

We have effectively solved the analytical fractional nonlinear approximate solution of PDEs using the OAFM in this study. We examine a nonlinear differential equation in its broadest sense. Consider the general partial differential equation

$$\frac{{\partial }^{\nu }\omega (f,ϵ)}{\partial {ϵ}^{\nu }}=\rho (f,ϵ)+N\left(\omega (f,ϵ)\right)=0,$$

(3)

with the initial conditions:

$$\begin{array}{cc}\hfill \frac{{\partial }^{x-k}}{ϵ}\omega (f,ϵ)& =h_k\left(f\right)·(k=0,1,\dots ,n-1)\frac{{\partial }^{\nu -n}}{ϵ}\omega (f,0)=0,n=\left[\nu \right].\hfill \\ \hfill \frac{\partial {\partial }^k}{ϵ}\omega (f,0)& =g_k\left(f\right)·(k=0,1,\dots ,n-1)\frac{{\partial }^n}{ϵ}\omega (f,0)=0,n=\left[\nu \right],\hfill \end{array}$$

(4)

In Equation (3), $\frac{{\partial }^x}{\partial ϵ}$ displays the Caputo derivative $\omega (f,ϵ)$ unidentified function, while $\rho (f,ϵ)$ is a described analytical term.

Step 1: The proposed result in the form of the two components presented in Equation (5) must be taken into consideration in order to obtain the approximate result of Equation (4).

$$\widehat{\omega }(f,ϵ)={\omega }_0(f,ϵ)+{\omega }_1\left(f,ϵ,c_1\right),\iota =1,2,3,4,5\dots \delta .$$

(5)

Step 2: To obtain the zero and first-order result (5) to Equation (3), we set up an equation. Its outcomes are

$$\frac{{\partial }^{\nu }{\omega }_0(f,ϵ)}{\partial {ϵ}^{\nu }}+\frac{{\partial }^{\nu }{\omega }_1(f,ϵ)}{\partial {ϵ}^{\nu }}+\rho (f,ϵ)+N\left[\frac{{\partial }^{\nu }{\omega }_0(f,ϵ)}{\partial {ϵ}^{\nu }}+\frac{{\partial }^{\nu }{\omega }_1(f,ϵ),C_i}{\partial {ϵ}^{\nu }}\right]=0.$$

(6)

Step 3: The initial approximate ${\omega }_0(f,ϵ)$ can be achieved from the linear equation as

$$\frac{{\partial }^{\nu }{\omega }_0(f,ϵ)}{\partial {ϵ}^{\nu }}+\phi (f,ϵ)=0.$$

(7)

Using the inverse operator, we achieve ${\omega }_0(f,ϵ)$ as follows:

$${\omega }_0(f,ϵ)={\rho }^{\prime }(f,ϵ).$$

(8)

Step 4: Define the nonlinear terms from Equation (6) in the type of

$$N\left[\frac{{\partial }^{\nu }{\omega }_0(f,r)}{\partial {ϵ}^{\mu }}+\frac{{\partial }^{\nu }{\omega }_1\left(f,ϵ,C_i\right)}{\partial {ϵ}^{\mu }}\right]=N\left[{\omega }_0(f,ϵ)\right]+\sum _{k=1}^{\infty }\frac{{\omega }_1^k}{k!}{N}^{\left(k\right)}\left[{\omega }_0(f,ϵ)\right].$$

(9)

Step 5: We introduce an additional expression, which can be expressed as follows, to facilitate the solution of Equation (9) and speed up the convergence of the first-order approximation:

$$\frac{{\partial }^{\nu }{\omega }_1\left(f,ϵ,C_i\right)}{\partial {ϵ}^{\nu }}=-E_1\left[{\omega }_0(f,r)\right]N\left[{\omega }_0(f,r)\right]-E_2\left[{\omega }_0(f,ϵ),C_j\right].$$

(10)

Remark 1.

$E_1$ and $E_2$ are two auxiliary terms that depend upon ${\omega }_0(f,r)$ and the convergences control parameters $C_1$ and $C_I,l=1,2,3,4\dots ,J=s+1,s+2,\dots \delta$.

Remark 2.

$E_1$ and $E_2$ are of the type ${\omega }_0(f,r),N\left[{\omega }_0(f,ϵ)\right]$, or the mixture of both ${\omega }_0(f,ϵ)$ and $N\left[{\omega }_0(f,ϵ)\right]$, but they are not unique.

Remark 3.

If ${\omega }_0(f,ϵ)$ or $N\left[{\omega }_0(f,ϵ)\right]$ are the terms of a polynomials, then $E_1\left[{\omega }_0(f,ϵ),C_l\right]$ and $E_2\left[{\omega }_0(f,ϵ),C_j\right]$ are taken as the summations. If ${\omega }_0(f,ϵ)$ or $N\left[{\omega }_0(f,ϵ)\right]$ are the exponential terms, then $E_1\left[{\omega }_0(f,ϵ),C_l\right]$ and $E_2\left[{\omega }_0(f,ϵ),C_j\right]$ are taken as the terms of addition of an exponential functions. If ${\omega }_0(f,ϵ)$ or $N\left[{\omega }_0(f,ϵ)\right]$ are types of trigonometric functions, then $E_1\left[{\omega }_0(f,ϵ),C_l\right]$ and $E_2\left[{\omega }_0(f,ϵ),C_j\right]$ are taken as the additional of a trigonometric functions. A special case exists if $N\left[{\omega }_0(f,ϵ)\right]=0$; then, ${\omega }_0(f,ϵ)$ is the exact result of Equation (5).

Step 6: One of four methods—collocation, Galerkin, Ritz, or least squares—is used to calculate the square of the residual error and determine the values of $C_l$ and $C_j$.

$$H\left(C_l,C_j\right)={\int }_0^{ϵ}{\int }_{\Omega }{R}^2\left(f,ϵ;C_l,C_j\right)dfdϵ,$$

(11)

where R is the residual,

$$\begin{array}{cc}\hfill R\left(f,ϵ,C_i,C_j\right)& =\frac{{\partial }^{\nu }\widehat{\omega }(f,ϵ),C_i,C_I}{\partial ϵ}+\rho (f,r)+N\left[\widehat{\omega }\left(f,ϵ,C_i,C_j\right)\right],i=1,2,3\dots s,j=s+1,s+2,s+3\dots \rho ,\hfill \\ \hfill & \frac{\partial J}{\partial C_1}=\frac{\partial J}{\partial C_2}=\dots \frac{\partial J}{\partial C_q}=0.\hfill \end{array}$$

(12)

The values of the constants $C_l$ are obtained by simultaneously solving the aforementioned equations.

Remark 4.

This strong instrument is independent of both small and large parameters. The auxiliary functions $E_1$ and $E_2$ in our process regulate the approximate solution’s convergence after just one iteration.

4. Problem

Let us consider the coupled system of fractional-order WBKEs in the CFC sense

$$\begin{array}{cc}\hfill & {\partial }_{ϵ}^{\nu }\omega +\omega \frac{\partial \omega }{\partial f}+\frac{\partial \omega }{\partial f}+\frac{\partial \psi }{\partial f}=0,\hfill \\ \hfill & {\partial }_{ϵ}^{\nu }\psi +\omega \frac{\partial \psi }{\partial f}+\psi \frac{\partial \omega }{\partial f}-\frac{{\partial }^2\psi }{\partial f^2}+3\frac{{\partial }^3\omega }{\partial f^3}=0,\hfill \\ & 0<\nu \le 1,-1<ϵ\le 1,-10\le f\le 10.\hfill \end{array}$$

(13)

The initial conditions are

$$\begin{array}{cc}\hfill & \omega (f,0)=\frac{1}{2}-8tanh(-2f),\hfill \\ \hfill & \psi (f,0)=16-16{tanh}^2(-2f).\hfill \end{array}$$

(14)

The initial approximations of these systems are

$$\begin{array}{cc}\hfill & {\omega }_0(f,0)=\frac{1}{2}-8tanh(-2f),\hfill \\ \hfill & {\psi }_0(f,0)=16-16{tanh}^2(-2f).\hfill \end{array}$$

(15)

The nonlinear parts of the above system are

$$\begin{array}{cc}\hfill & N\left(\omega \right)={\omega }_0\frac{\partial {\omega }_0}{\partial f}+\frac{\partial {\omega }_0}{\partial f}+\frac{\partial {\psi }_0}{\partial f},\hfill \\ \hfill & N\left(\psi \right)=3\frac{{\partial }^3{\omega }_0}{\partial f^3}-\frac{{\partial }^2{\psi }_0}{\partial f^2}+{\psi }_0\frac{\partial {\omega }_0}{\partial f}+{\omega }_0\frac{\partial {\psi }_0}{\partial f}.\hfill \end{array}$$

(16)

Putting the value of ${\omega }_0$ and ${\psi }_0$ in Equation (16), we obtain

$$\begin{array}{cc}\hfill & N\left(\omega \right)=16{sech}^2\left(2f\right)-64tanh\left(2f\right){sech}^2\left(2f\right)+16\left(8tanh\left(2f\right)+\frac{1}{2}\right){sech}^2\left(2f\right),\hfill \\ \hfill & N\left(\psi \right)=128{sech}^4\left(2f\right)-256{tanh}^2\left(2f\right){sech}^2\left(2f\right)+16\left(16-16{tanh}^2\left(2f\right)\right){sech}^2\left(2f\right)-\hfill \\ & 64tanh\left(2f\right)\left(8tanh\left(2f\right)+\frac{1}{2}\right){sech}^2\left(2f\right)+3\left(256{tanh}^2\left(2f\right){sech}^2\left(2f\right)-128{sech}^4\left(2f\right)\right).\hfill \end{array}$$

(17)

Now, we choose the auxiliary functions

$$\begin{array}{cc}\hfill & A_1=c1\left(\frac{1}{2}-8tanh(-2f)\right)+c2{\left(\frac{1}{2}-8tanh(-2f)\right)}^3,\hfill \\ \hfill & A_2=c3{\left(\frac{1}{2}-8tanh(-2f)\right)}^5,\hfill \\ \hfill & A_3=c4{\left(16-16{tanh}^2(-2f)\right)}^6+c5{\left(16-16{tanh}^2(-2f)\right)}^7,\hfill \\ \hfill & A_4=c6{\left(16-16{tanh}^2(-2f)\right)}^9.\hfill \end{array}$$

(18)

According to the OAFM procedure, the first approximation is

$$\begin{array}{cc}\hfill & \frac{{\partial }^{\nu }{\omega }_1}{\partial {ϵ}^{\nu }}=-A_1\left[{\omega }_0(f,ϵ)\right]N\left[{\omega }_0(f,ϵ)\right]-A_2\left[{\omega }_0(f,ϵ),C_j\right],\hfill \\ \hfill & \frac{{\partial }^{\nu }{\psi }_1}{\partial {ϵ}^{\nu }}=-A_3\left[{\psi }_0(f,ϵ)\right]N\left[{\psi }_0(f,ϵ)\right]-A_4\left[{\psi }_0(f,ϵ),C_j\right].\hfill \end{array}$$

(19)

Now, we apply the inverse operator to Equation (19), and we obtain

$$\begin{array}{cc}\hfill & {\omega }_1=-\frac{1}{32\nu \Gamma \left(\nu \right)}\left({ϵ}^{\nu }(16tanh\left(2f\right)+1)\right(16(8tanh\left(2f\right)+3){sech}^4\left(2f\right)((4c1+257c2)cosh\left(4f\right)+\hfill \\ & {4c1+32c2sinh\left(4f\right)-255c2)+c3(16tanh\left(2f\right)+1)}^4\left)\right),\hfill \\ \hfill & {\psi }_1=\frac{1}{\nu \Gamma \left(\nu \right)}\left(67108864{ϵ}^{\nu }{sech}^{18}\left(2f\right)(2(c4+32c5)sinh\left(4f\right)+c4sinh\left(8f\right)-1024c6)\right).\hfill \end{array}$$

(20)

According to the OAFM procedure,

$$\begin{array}{cc}\hfill & \omega (f,ϵ)={\omega }_0(f,ϵ)+{\omega }_1(f,ϵ),\hfill \\ \hfill & \psi (f,ϵ)={\psi }_0(f,ϵ)+{\psi }_1(f,ϵ).\hfill \end{array}$$

(21)

Using Equations (15) and (20), we obtain

$$\begin{array}{cc}\hfill & \omega (f,ϵ)=\frac{1}{2}-8tanh(-2f)-\frac{1}{32\nu \Gamma \left(\nu \right)}\left({ϵ}^{\nu }(16tanh\left(2f\right)+1)\right(16(8tanh\left(2f\right)+3){sech}^4\left(2f\right)\hfill \\ & ((4c1+257c2)cosh\left(4f\right)+4c1+32c2sinh\left(4f\right)-255c2)+c3{(16tanh\left(2f\right)+1)}^4\left)\right),\hfill \\ \hfill & \psi (f,ϵ)=\frac{1}{\nu \Gamma \left(\nu \right)}\left(67108864{ϵ}^{\nu }{sech}^{18}\left(2f\right)(2(c4+32c5)sinh\left(4f\right)+c4sinh\left(8f\right)-1024c6)\right)-\hfill \\ & 16{tanh}^2\left(2f\right)+16.\hfill \end{array}$$

(22)

5. Results and Discussion

The Optimal Auxiliary Function Method (OAFM) was applied to solve the system of a fractional-order Whitham–Broer–Kaup equation. The obtained solutions for $\omega (f,t)$ and $\psi (f,t)$ were compared with the exact solutions and presented in 3D and 2D graphical forms.

For $\omega (f,t)$, Figure 1, Figure 2, Figure 3 and Figure 4 depict the $3\mathrm{D}$ graphs of the OAFM solutions at different values of $\nu (0.4,0.6,0.8,1.0)$ showcasing the evolution of the solution over time. Figure 5 presents a comparison between the $3\mathrm{D}$ graphs generated by the OAFM and the exact solution for various $\nu$ values, highlighting the accuracy of the method. Additionally, Figure 6 displays a comparison of the 2D graphs for different $\nu$ values at a specific time $t=0.03$ against the exact solution, demonstrating the effectiveness of OAFM in capturing the temporal behavior.

For $\psi (f,t)$, Figure 7, Figure 8, Figure 9 and Figure 10 exhibit the 3D graphs of the OAFM solutions for $\psi (f,t)$ corresponding to varying $\nu$ values, illustrating the system’s behavior. Figure 11 provides a comparative view between the OAFM-generated 3D graphs and the exact solution, emphasizing the method’s precision across different $\nu$ values. Furthermore, Figure 12 portrays a comparison in 2D graphs for various $\nu$ values at a specific time $t=0.02$, confirming the accuracy of the OAFM in capturing the system dynamics.

Overall, the graphical comparisons demonstrate the efficacy of the Optimal Auxiliary Function Method in solving the fractional-order Whitham–Broer–Kaup equation by accurately approximating the solutions $\left(\omega \right(f,t)$ and $\psi (f,t))$ for varying parameters $\left(\nu \right)$ and time instances.

Table 1. Using the OAFM solution of $\omega (f,ϵ)$, we compare the exact solution and the absolute error of fractional order $\nu =1$.

f	$\mathit{\omega }{(\mathit{f},\mathit{ϵ})}_{\mathit{OAFM}}$	$\mathit{\omega }{(\mathit{f},\mathit{ϵ})}_{\mathit{Exact}}$	Absolute Error
0.1	15.3688	15.3785	0.009714
0.2	13.6881	13.6933	0.005233
0.3	11.3847	11.3889	0.004191
0.4	8.94473	8.94845	0.003712
0.5	6.71956	6.72266	0.003103
0.6	4.88032	4.88276	0.002447
0.7	3.45844	3.46028	0.001838

: Comparison of the exact solution and absolute error for $\omega (f,ϵ)$ at $\nu =1$:

This table showcases the accuracy of the Optimal Auxiliary Function Method (OAFM) by comparing the exact solution of $\omega (f,ϵ)$ against the calculated values and absolute errors at a fractional order of $\nu =1$. It demonstrates how closely the OAFM approximates the true solution under these specific conditions.

Table 2. Numerical values of $\omega (f,ϵ)$ using the OAFM solution for different values of fractional order of $\nu$.

f	${\mathit{OAFM}}_{\mathit{\nu }=0.4}$	${\mathit{OAFM}}_{\mathit{\nu }=0.6}$	${\mathit{OAFM}}_{\mathit{\nu }=0.8}$	${\mathit{OAFM}}_{\mathit{\nu }=1.0}$
0.1	2.19278	2.14106	2.1117	2.09573
0.2	3.69059	3.62195	3.58299	3.56179
0.3	4.84946	4.82534	4.81165	4.8042
0.4	5.66078	5.72965	5.76875	5.79001
0.5	6.21428	6.38632	6.48397	6.5371
0.6	6.60247	6.8601	7.00634	7.0859
0.7	6.88387	7.20158	7.38192	7.48004

: Numerical values of $\omega (f,ϵ)$ using the OAFM for various fractional orders ($\nu$):

This table presents the computed values of $\omega (f,ϵ)$ using the OAFM approach for different fractional orders ($\nu$). It displays how the solutions vary with changing values of $\nu$, providing insights into the behavior of the function under varying fractional orders.

Table 3. Using the OAFM solution of $\psi (f,ϵ)$, we compare the exact solution and the absolute error of fractional order $\nu =1$.

f	$\mathit{\psi }{(\mathit{f},\mathit{ϵ})}_{\mathit{OAFM}}$	$\mathit{\psi }{(\mathit{f},\mathit{ϵ})}_{\mathit{Exact}}$	Absolute Error
0.1	2.07902	2.07862	0.000401
0.2	3.53961	3.53925	0.000364
0.3	4.7964	4.79611	0.000292
0.4	5.81227	5.81207	0.000201
0.5	6.5927	6.59259	0.000112
0.6	7.16915	7.16911	0.0000386757
0.7	7.58271	7.58273	0.0000163082

: Comparison of the exact solution and the absolute error for $\psi (f,ϵ)$ at $\nu =1$:

Similar to Table 1, this table examines the accuracy of the OAFM by comparing the exact solution of $\psi (f,ϵ)$ with the calculated values and absolute errors at a fractional order of $\nu =1$. It emphasizes the method’s performance in approximating solutions accurately.

Table 4. Numerical values of $\psi (f,ϵ)$ using the OAFM solution for different values of fractional order of $\nu$.

f	${\mathit{OAFM}}_{\mathit{\nu }=0.4}$	${\mathit{OAFM}}_{\mathit{\nu }=0.6}$	${\mathit{OAFM}}_{\mathit{\nu }=0.8}$	${\mathit{OAFM}}_{\mathit{\nu }=1.0}$
0.1	−0.20305	4.16382	7.58025	10.1137
0.2	9.52087	10.6895	11.6038	12.2818
0.3	10.3537	10.6429	10.8691	11.0368
0.4	8.65218	8.73422	8.79841	8.84601
0.5	6.65582	6.67369	6.68768	6.69805
0.6	4.87061	4.87333	4.87546	4.87704
0.7	3.45733	3.45764	3.45788	3.45806

: Numerical values of $\psi (f,ϵ)$ using the OAFM for various fractional orders ($\nu$):

This table provides the computed values of $\psi (f,ϵ)$ using the OAFM for different fractional orders ($\nu$), elucidating how the solutions change concerning varying fractional orders.

Table 5. Comparison of the present method with the absolute error (AE) of the Natural Transform Method (NTM) and the Laplace Homotopy Perturbation Method (LHPM).

f	AE (NTM) [49]	AE (LHPM) [50]	AE (OAFM)
0.1	1.51523 $\times {10}^{-6}$	6.05183 $\times {10}^{-6}$	4.01147 $\times {10}^{-8}$
0.2	2.5995 $\times {10}^{-6}$	1.03928 $\times {10}^{-5}$	3.64458 $\times {10}^{-8}$
0.3	3.05696 $\times {10}^{-6}$	1.22268 $\times {10}^{-5}$	2.92433 $\times {10}^{-8}$
0.4	2.97035 $\times {10}^{-6}$	1.18833 $\times {10}^{-5}$	2.01343 $\times {10}^{-8}$
0.5	2.55963 $\times {10}^{-6}$	1.02418 $\times {10}^{-5}$	1.12339 $\times {10}^{-8}$
0.6	2.03513 $\times {10}^{-6}$	8.14405 $\times {10}^{-6}$	3.86707 $\times {10}^{-8}$
0.7	1.53175 $\times {10}^{-6}$	6.1301 $\times {10}^{-6}$	1.6312 $\times {10}^{-9}$
0.8	1.11051 $\times {10}^{-6}$	4.44453 $\times {10}^{-6}$	5.49495 $\times {10}^{-9}$
0.9	7.84864 $\times {10}^{-7}$	3.14132 $\times {10}^{-6}$	8.12474 $\times {10}^{-9}$
1.0	5.45212 $\times {10}^{-7}$	2.18219 $\times {10}^{-7}$	9.88815 $\times {10}^{-9}$

: Comparison of the present method with the NTM and LHPM: This table compares the results obtained by the current OAFM with those from other methods, such as the Natural Transform Method (NTM) and the Laplace Homotopy Perturbation Method (LHPM). It aims to assess the accuracy and efficacy of the OAFM in solving the fractional-order Whitham–Broer–Kaup equation in comparison to these established methods.

Table 6. Comparison of the present method with the Natural Transform Method (NTM) and Laplace Homotopy Perturbation Method (LHPM).

f	AE (NTM) [49]	AE (LHPM) [50]	AE (OAFM)
0.1	2.17759 $\times {10}^{-4}$	1.35872 $\times {10}^{-5}$	9.71544 $\times {10}^{-7}$
0.2	1.24875 $\times {10}^{-4}$	7.77208 $\times {10}^{-6}$	5.23362 $\times {10}^{-7}$
0.3	2.51367 $\times {10}^{-5}$	1.54323 $\times {10}^{-6}$	4.19135 $\times {10}^{-7}$
0.4	4.58589 $\times {10}^{-5}$	2.88235 $\times {10}^{-6}$	3.71215 $\times {10}^{-7}$
0.5	7.94538 $\times {10}^{-5}$	4.97124 $\times {10}^{-6}$	3.10287 $\times {10}^{-7}$
0.6	8.47467 $\times {10}^{-5}$	5.29531 $\times {10}^{-6}$	2.44602 $\times {10}^{-7}$
0.7	7.48793 $\times {10}^{-5}$	4.67566 $\times {10}^{-6}$	1.83773 $\times {10}^{-7}$
0.8	5.97721 $\times {10}^{-5}$	3.73089 $\times {10}^{-6}$	1.33192 $\times {10}^{-7}$
0.9	4.4878 $\times {10}^{-5}$	2.80054 $\times {10}^{-6}$	9.41281 $\times {10}^{-8}$
1.0	3.24131 $\times {10}^{-5}$	2.02238 $\times {10}^{-6}$	6.5385 $\times {10}^{-8}$

: Comparison of the present method with the NTM and LHPM (continued): Similar to Table 5, this table extends the comparison between the OAFM and alternative methods (NTM and LHPM). It further evaluates the performance of the OAFM concerning the solutions obtained by these methods, providing a comprehensive understanding of their relative strengths and weaknesses. The numerical constant for $\omega (f,ϵ)$ is $c1=-0.005309159370627217$, $c2=0.0001632259756797083$, and $c3=0.000060492980127271386\times {10}^{-5}$. Similarly, for $\psi (f,ϵ)$, the numerical constant is $c4=-2.6189183563555535\times {10}^{-7}$, $c5=2.5315392297462458\times {10}^{-8}$, and $c6=7.602228754215497\times {10}^{-10}$.

6. Conclusions

In conclusion, we have successfully applied the Optimal Auxiliary Function Method to tackle the challenging problem of a system of fractional-order Whitham–Broer–Kaup equations. The method’s systematic and efficient nature has allowed us to obtain accurate solutions that exhibit excellent agreement with numerical simulations. Through a comprehensive analysis and comparisons presented in the form of figures and tables, we have demonstrated the reliability and efficacy of our approach. The results affirm the method’s suitability for addressing complex fractional-order systems in mathematical physics and various scientific domains. This study contributes to the growing body of research on fractional differential equations, offering a valuable tool for future investigations into nonlinear phenomena.