High-Performance Computational Method for an Extended Three-Coupled Korteweg–de Vries System

Abstract

This paper calculates numerical solutions of an extended three-coupled Korteweg–de Vries system by the q-homotopy analysis transformation method (q-HATM), which is a hybrid of the Laplace transform and the q-homotopy analysis method. Multiple investigations inspecting planetary oceans, optical cables, and cosmic plasma have employed the KdV model, significantly contributing to its development. The uniqueness, convergence, and maximum absolute truncation error of this algorithm are demonstrated. A numerical simulation has been performed to validate the accuracy and validity of the proposed approach. With high accuracy and few algorithmic processes, this algorithm supplies a series solution in the form of a recursive relation.

1. Introduction

In the past two decades, nonlinear science has begun to appear in modern mathematics, physics, engineering technology, and other important fields [1,2,3,4]. Presently, the primary topic of research in many categories, which include natural science and engineering technology, has shifted from linear problems to nonlinear problems. Such a form of complex problem necessitates extensive research and faces distinct difficulties. However, the study of nonlinear PDEs [5,6,7,8,9] has already started to encounter problems. In physics, fluid mechanics, communication technology, material science, dynamical systems, and biology, nonlinear coupled PDEs are commonly implemented. As a consequence, understanding how to solve nonlinear PDEs has both important theoretical and practical consequences.

Traditional numerical methods require more computer memory to calculate numerical solutions. This means that the semi-analytical technique and the Laplace transform eliminate the time-consuming drawbacks and need less CPU processing time while analyzing the numerical solutions of nonlinear phenomena in the actual world. A hybrid of the homotopy polynomials, the Laplace transform, and the q-HAM [10,11,12,13,14] is the q-HATM [15,16,17,18,19]. We analyze the uniqueness, convergence, and utmost absolute error [20,21] of q-HATM solutions. This algorithm has the advantages of both methods.

This article is going to investigate the extended three-coupled Korteweg–de Vries system [22,23,24],

$$\begin{array}{cc}\hfill & u_t={\beta }_1{u}_{xxx}+{\beta }_2{u}_{x}v+{\beta }_{2}u{v}_x,\hfill \\ \hfill & v_t={\beta }_1{\beta }_3{v}_{xxx}+2{\beta }_2{\beta }_{3}v{v}_x-{\beta }_2{w}_x,\hfill \\ \hfill & w_t={\beta }_1{w}_{xxx}+{\beta }_{2}v{w}_x-2{\beta }_{2}u{u}_x,\hfill \end{array}$$

(1)

with real differentiable functions associated with variables x and t including u, v, and w, while ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ are three non-zero real constants. In investigations of planetary oceans, optical fibers, and cosmic plasma, the KdV model has been extensively employed.

The main framework of this paper is as follows. The fundamental concepts of the sub-equation method [25,26,27] and the q-HATM are outlined in Section 2; the uniqueness and convergence of numerical solutions are also confirmed. The extended three-coupled Korteweg–de Vries system is discussed in Section 3 along with its analytical and numerical solutions. Section 4 shows the numerical results and comments. Section 5 concludes by providing some recommendations and outlining the key findings.

2. Analysis of the Presented Methodology

2.1. The Sub Equation Method

The Riccati equation is the foundation of the sub equation approach,

$${\psi }^{{}^{\prime }}\left(\eta \right)=\alpha +{\psi }^2\left(\eta \right).$$

(2)

When it comes to a particular PDE with two variables,

$$F(x,t,u_t,u_x,u_{xx},\dots )=0,$$

(3)

the travelling wave solution [25,26,27] is

$$\eta =bt+x,u(x,t)=U\left(\eta \right),$$

(4)

in this case, b is a constant that will be explained further.

Substituting Equation (4) into Equation (3), the following equation is established,

$$G(U\left(\eta \right),U^{{}^{\prime }}\left(\eta \right),\dots )=0.$$

(5)

Presumptive analytical solutions of Equation (5) are as follows:

$$U\left(\eta \right)=\sum _{i=0}^N{a}_i{\psi }^i\left(\eta \right),a_N\ne 0,$$

(6)

The principle of balance can be employed for estimating a positive integer N, and the coefficients $a_i(0\le i\le N)$ will be calculated afterward.

Equation (2) possesses five types of analytical solutions, which are listed below,

$$\psi \left(\eta \right)=\left\{\begin{array}{cc}-\sqrt{-\alpha }\tan \mathrm{h}\left(\sqrt{-\alpha }\eta \right),\hfill & \alpha <0,\hfill \\ -\sqrt{-\alpha }\cot \mathrm{h}\left(\sqrt{-\alpha }\eta \right),\hfill & \alpha <0,\hfill \\ -{\displaystyle \frac{1}{\eta }},\hfill & \alpha =0,\hfill \\ \sqrt{\alpha }\tan \left(\sqrt{\alpha }\eta \right),\hfill & \alpha >0,\hfill \\ -\sqrt{\alpha }\cot \left(\sqrt{\alpha }\eta \right),\hfill & \alpha >0.\hfill \end{array}\right.$$

(7)

With the help of the previously mentioned results, we can formulate equations for b and $a_i$$(i=1,2,\dots ,N)$ by juggling the coefficients of a polynomial describing $\psi \left(\eta \right)$. Then, we get the analytical solutions of Equation (3).

2.2. Fundamental Plan of the q-HATM

The following introduces [15,16,17,18,19] the basic idea of the q-HATM for nonlinear PDEs, assuming that the PDEs have the following form,

$$u_t(x,t)+Nu(x,t)+Ru(x,t)=f(x,t),$$

(8)

where N is Lipschitz continuous and a nonlinear differential operator. The linear differential operator R is bounded, meaning that given a number $\mu$, we acquire $\parallel Ru\parallel$≤$\mu \parallel u\parallel$. The source term is $f(x,t)$. Applying the Laplace transform to Equation (8), we have

$$s\mathcal{L}\left[u\right(x,t\left)\right]+\mathcal{L}\left[Nu\right(x,t\left)\right]+\mathcal{L}\left[Ru\right(x,t\left)\right]=u(x,0)+\mathcal{L}\left[f\right(x,t\left)\right],$$

(9)

The simplification of Equation (9) gives

$$\mathcal{L}\left[u(x,t)\right]+\frac{1}{s}\mathcal{L}[Nu(x,t)+Ru(x,t)-f(x,t)]-\frac{1}{s}u(x,0)=0.$$

(10)

The following is the nonlinear operator,

$$\mathcal{B}[\Theta (x,t;q)]=\mathcal{L}[\Theta (x,t;q)]+\frac{1}{s}\mathcal{L}[N\Theta (x,t;q)+R\Theta (x,t;q)-f(x,t)]-\frac{1}{s}\Theta (x,t;q)\left(0^{+}\right),$$

(11)

where the real function of q, x, and t is $\Theta (x,t;q)$ and $q\in [0,\frac{1}{n}]$$(n\ge 1)$ is an embedding parameter. The q-HATM supplies the following non-zero auxiliary function,

$$(1-nq)\mathcal{L}[\Theta (x,t;q)-u_0(x,t)]=\hslash qH(x,t)\mathcal{B}[\Theta (x,t;q)],$$

(12)

where the Laplace transform is designated by $\mathcal{L}$, the initial guess of $u(x,t)$ is represented as $u_0(x,t)$, and the auxiliary parameter $\hslash \ne 0$. In the following, we present the conclusions for $q=\frac{1}{n}$ and $q=0$, respectively,

$$\Theta (x,t;\frac{1}{n})=u(x,t),\Theta (x,t;0)=u_0(x,t).$$

(13)

The solutions $\Theta (x,t;q)$ converge to solutions $u(x,t)$ from the initial guess $u_0(x,t)$ as the embedding parameter q increases from 0 to $\frac{1}{n}$. Employing Taylor’s theorem [28] about q, the series expansion of function $\Theta (x,t;q)$ is given,

$$\Theta (x,t;q)=u_0(x,t)+\sum _{l=1}^{\infty }{u}_l(x,t)q^l,$$

(14)

where

$$u_l(x,t)=\frac{1}{l!}\frac{{\partial }^l\Theta (x,t;q)}{\partial q^l}{|}_{q=0}.$$

(15)

If Equation (14) converges at $q=\frac{1}{n}$ and the initial guess $u_0$, asymptotic parameter n, and auxiliary parameter ℏ values are set appropriately,

$$u(x,t)=\underset{q\to \frac{1}{n}}{lim}\Theta (x,t;q)=\sum _{l=0}^{\infty }{u}_l(x,t){(\frac{1}{n})}^l.$$

(16)

Dividing Equation (12) by $l!$ after differentiating it l times concerning q. Finalize by taking $q=0$ and becoming,

$$\mathcal{L}[u_l(x,t)-k_l{u}_{l-1}(x,t)]=\hslash H(x,t){\mathcal{R}}_l\left({\overrightarrow{u}}_{l-1}\right),$$

(17)

here

$${\overrightarrow{u}}_l=\left\{u_0,u_1\dots ,u_l\right\}.$$

(18)

The inverse Laplace transform acts on Equation (17), which becomes

$$u_l(x,t)=k_l{u}_{l-1}(x,t)+\hslash H(x,t){\mathcal{L}}^{-1}[{\mathcal{R}}_l({\overrightarrow{u}}_{l-1})],$$

(19)

where

$${\mathcal{R}}_l\left({\overrightarrow{u}}_{l-1}\right)=\frac{1}{(l-1)!}\frac{{\partial }^{l-1}\mathcal{B}(\Theta (x,t;q))}{\partial q^{l-1}}{|}_{q=0},k_l=\left\{\begin{array}{c}0,l\le 1,\hfill \\ n,l>1.\hfill \end{array}\right.$$

(20)

To acquire the iteration terms of $u_l(x,t)$, our suggested technique employs homotopy polynomials to provide a unique correction function. The numerical solutions for q-HATM series are

$$u^{\left[M\right]}(x,t)=\sum _{l=0}^M{u}_l(x,t){(\frac{1}{n})}^l.$$

(21)

2.3. Convergence Analysis of the q-HATM

The recommended approach has been employed during this study to investigate the convergence of Equation (8).

Theorem 1

(Uniqueness theorem). For any α in $(0,1)$, where $\gamma =\hslash (\lambda +\mu )T+n+\hslash$, the solution of Equation (8) generated employing the q-HATM is unique.

Proof. 

The solution to Equation (8) is expressed as

$$u(x,t)=\sum _{l=0}^{\infty }{u}_l(x,t){(\frac{1}{n})}^l,$$

(22)

where

$$u_l(x,t)=\hslash {\mathcal{L}}^{-1}[\frac{1}{s}\mathcal{L}[Nu(x,t)+Ru(x,u)]]+(k_l+\hslash )u_{l-1}(x,t)-(1-\frac{k_l}{n})\frac{1}{s}[u(x,0)+f(x,t)].$$

(23)

Assuming that the two solutions to Equation (8) are u and $u^{▵}$, the above equation yields the following result:

$$|u-u^{▵}|=|\hslash {\mathcal{L}}^{-1}[\frac{1}{s}\mathcal{L}[N(u-u^{▵})+R(u-u^{▵})]]+(n+\hslash )(u-u^{▵})|.$$

(24)

It follows from the Laplace convolution theorem [29] that we possess

$$\begin{array}{cc}\hfill |u-u^{▵}|\le & \hslash {\int }_0^t\left(\right|N(u-u^{▵})|+|R(u-u^{▵})\left|\right)(t-\tau )d\tau +(n+\hslash )|u-u^{▵}|\hfill \\ \hfill \le & \hslash {\int }_0^t(\lambda +\mu )|u-u^{▵}|(t-\tau )d\tau +(n+\hslash )|u-u^{▵}|.\hfill \end{array}$$

(25)

Further, making use of the integral mean value theorem [30],

$$\begin{array}{cc}\hfill |u-u^{▵}|\le & \hslash (\lambda +\mu )|u-u^{▵}|T+(n+\hslash )|u-u^{▵}|\hfill \\ \hfill \le & \gamma |u-u^{▵}|,\hfill \end{array}$$

(26)

where $\gamma =\hslash (\lambda +\mu )T+n+\hslash$. This means that,

$$|u-u^{▵}|\le \gamma |u-u^{▵}|\Rightarrow (\gamma -1)|u-u^{▵}|\ge 0,$$

(27)

as $0<\gamma <1$; then, $u=u^{▵}$. As a result, Equation (8) has a unique solution. □

Theorem 2

(Convergence theorem). Considering that $Q:X\to X$ is a nonlinear mapping, alongside X, in this case, being a Banach space, this allows for

$$\parallel Q\left(u\right)-Q\left(w\right)\parallel \le \gamma \parallel u-w\parallel ,\forall u,w\in X.$$

(28)

In line with Banach’s theory [31] of fixed points, Q has a fixed point. The sequence generated via the q-HATM similarly converges to the fixed point of Q for $u_0$ and $w_0$, which has been selected at random among X, and

$$\parallel u_l-u_n\parallel \le \frac{{\gamma }^n}{1-\gamma }\parallel u_1-u_0\parallel ,\forall u_0,w_0\in X.$$

(29)

Proof. 

The norm is denoted by $\parallel g\left(t\right)\parallel$=max${}_{t\in J}$$\left|g\right(t\left)\right|$. The Banach space $\left(C\right[J],\parallel .\parallel )$ includes continuous functions performed on J. As previously mentioned, the sequence $\left\{u_n\right\}$ is a Cauchy-like sequence in the Banach space X,

$$\begin{array}{cc}\hfill & \parallel u_l-u_n\parallel =ma{x}_{t\in J}|u_l-u_n|\hfill \\ \hfill & =ma{x}_{t\in J}|\hslash {\mathcal{L}}^{-1}[\frac{1}{s}\mathcal{L}[N(u_{l-1}-u_{n-1})+R(u_{l-1}-u_{n-1})]]+(n+\hslash )(u_{l-1}-u_{n-1})|\hfill \\ \hfill & \le ma{x}_{t\in J}[\hslash {\mathcal{L}}^{-1}[\frac{1}{s}\mathcal{L}\left[\right|N(u_{l-1}-u_{n-1})|+\left|R(u_{l-1}-u_{n-1})\left|\right]\right]+(n+\hslash )|u_{l-1}-u_{n-1}\left|\right].\hfill \end{array}$$

(30)

The Laplace transform’s convolution theorem allows the following:

$$\begin{array}{cc}\hfill \parallel u_l-u_n\parallel \le & ma{x}_{t\in J}[\hslash {\int }_0^t\left[\right|N(u_{l-1}-u_{n-1})|+|R(u_{l-1}-u_{n-1})\left|\right](t-\tau )d\tau +(n+\hslash )|u_{l-1}-u_{n-1}\left|\right]\hfill \\ \hfill \le & ma{x}_{t\in J}[\hslash {\int }_0^t[[(\lambda +\mu )|u_{l-1}-u_{n-1}\left|\right](t-\tau )d\tau +(n+\hslash )|u_{l-1}-u_{n-1}\left|\right].\hfill \end{array}$$

(31)

Following that, implementing the integral mean value theorem,

$$\begin{array}{cc}\hfill \parallel u_l-u_n\parallel \le & ma{x}_{t\in J}[\hslash (\lambda +\mu )|u_{l-1}-u_{n-1}|T+(n+\hslash )|u_{l-1}-u_{n-1}\left|\right]\hfill \\ \hfill \le & \gamma \parallel u_{l-1}-u_{n-1}\parallel \iff \gamma =\hslash (\lambda +\mu )T+n+\hslash .\hfill \end{array}$$

(32)

If $l=n+1$, then the situation is as follows:

$$\parallel u_{n+1}-u_n\parallel \le \gamma \parallel u_n-u_{n-1}\parallel \le {\gamma }^2\parallel u_{n-1}-u_{n-2}\parallel \le \dots \le {\gamma }^n\parallel u_1-u_0\parallel .$$

(33)

Employing the trigonometric inequality produces

$$\begin{array}{cc}\hfill \parallel u_l-u_n\parallel =& \parallel u_{n+1}-u_n+u_{n+2}-u_{n+1}+\dots +u_l-u_{l-1}\parallel \hfill \\ \hfill \le & \parallel u_{n+1}-u_n\parallel +\parallel u_{n+2}-u_{n+1}\parallel +\dots +\parallel u_l-u_{l-1}\parallel \hfill \\ \hfill \le & [{\gamma }^n+{\gamma }^{n+1}+\dots +{\gamma }^{l-1}]\parallel u_1-u_0\parallel \hfill \\ \hfill \le & {\gamma }^n\times [1+\gamma +\dots +{\gamma }^{l-n-1}]\parallel u_1-u_0\parallel \hfill \\ \hfill \le & {\gamma }^n\times \frac{1-{\gamma }^{l-n-1}}{1-\gamma }\parallel u_1-u_0\parallel .\hfill \end{array}$$

(34)

Since $0<\gamma <1$, then $1-{\gamma }^{l-n-1}<1$, and we have

$$\parallel u_l-u_n\parallel \le \frac{{\gamma }^n}{1-\gamma }\parallel u_1-u_0\parallel .$$

(35)

However, $\parallel u_1-u_0\parallel <\infty$, so as $l\to \infty$, then $\parallel u_l-u_n\parallel \to 0$. Given that all Cauchy sequences are convergent, the sequence $\left\{u_n\right\}$ in the Banach space $C\left[J\right]$ is a Cauchy sequence. □

Theorem 3.

The utmost absolute truncation error will be calculated as follows if there is a real number $0<\gamma <1$ and $\parallel u_{l+1}\parallel \le \gamma \parallel u_l\parallel$ is satisfied:

$$\parallel u(x,t)-u^{\left[M\right]}(x,t)\parallel \le \frac{{\gamma }^{M+1}}{n^M(n-\gamma )}\parallel u_0(x,t)\parallel .$$

(36)

Proof. 

Given the values of $n(n\ge 1)$ and $\hslash (\hslash \ne 0)$, we have

$$\begin{array}{cc}\hfill \parallel u(x,t)-u^{\left[M\right]}(x,t)\parallel =& \parallel \sum _{l=0}^{\infty }{u}_l(x,t){(\frac{1}{n})}^l-\sum _{l=0}^M{u}_l(x,t){(\frac{1}{n})}^l\parallel \hfill \\ \hfill & \le \sum _{l=M+1}^{\infty }\parallel u_l(x,t)\parallel {(\frac{1}{n})}^l\hfill \\ \hfill & \le \sum _{l=M+1}^{\infty }{\gamma }^l\parallel u_0(x,t)\parallel {(\frac{1}{n})}^l\hfill \\ \hfill & \le \sum _{l=M+1}^{\infty }{(\frac{\gamma }{n})}^l\parallel u_0(x,t)\parallel \hfill \\ \hfill & \le {(\frac{\gamma }{n})}^{M+1}[1+\frac{\gamma }{n}+{(\frac{\gamma }{n})}^2+\dots ]\parallel u_0(x,t)\parallel \hfill \\ \hfill & \le {(\frac{\gamma }{n})}^{M+1}\frac{1}{1-\frac{\gamma }{n}}\parallel u_0(x,t)\parallel \hfill \\ \hfill & =\frac{{\gamma }^{M+1}}{n^M(n-\gamma )}\parallel u_0(x,t)\parallel .\hfill \end{array}$$

(37)

This proof of the theorem has now been finished. □

3. Applications of the Extended Three-Coupled Korteweg–de Vries System

3.1. Analytical Solutions for Equation (1)

We right now utilize the sub equation approach to discover the analytical solutions to Equation (1). As a consequence, we perform the wave transformation described below,

$$\eta =bt+x,u(x,t)=U\left(\eta \right),v(x,t)=V\left(\eta \right),w(x,t)=W\left(\eta \right).$$

(38)

Here are the simplified ODEs that result from inserting Equation (38) into Equation (1),

$$\begin{array}{cc}\hfill & b{U}^{{}^{\prime }}={\beta }_1{U}^{{}^{\prime \prime \prime }}+{\beta }_2{U}^{{}^{\prime }}V+{\beta }_{2}U{V}^{{}^{\prime }},\hfill \\ \hfill & b{V}^{{}^{\prime }}={\beta }_1{\beta }_3{V}^{{}^{\prime \prime \prime }}+2{\beta }_2{\beta }_{3}V{V}^{{}^{\prime }}-{\beta }_2{W}^{{}^{\prime }},\hfill \\ \hfill & b{W}^{{}^{\prime }}={\beta }_1{W}^{{}^{\prime \prime \prime }}+{\beta }_{2}V{W}^{{}^{\prime }}-2{\beta }_{2}U{U}^{{}^{\prime }}.\hfill \end{array}$$

(39)

The homogeneous balance between the nonlinear item and the highest-order derivative has to be obtained by Equation (39), which results in the following solutions,

$$\begin{array}{cc}\hfill & U\left(\eta \right)=a_0+a_1\psi \left(\eta \right)+a_2{\psi }^2\left(\eta \right),\hfill \\ \hfill & V\left(\eta \right)=b_0+b_1\psi \left(\eta \right)+b_2{\psi }^2\left(\eta \right),\hfill \\ \hfill & W\left(\eta \right)=c_0+c_1\psi \left(\eta \right)+c_2{\psi }^2\left(\eta \right).\hfill \end{array}$$

(40)

The following solutions are derived by computing the system of equations created by substituting Equation (40) into Equation (39),

$$\begin{array}{cc}\hfill & a_0=-\frac{a_2(-144\alpha {\beta }_1^3-18b_0{\beta }_1^2{\beta }_2+a_2^2{\beta }_2^3+144\alpha {\beta }_1^3{\beta }_3+36b_0{\beta }_1^2{\beta }_2{\beta }_3)}{108{\beta }_1^3},a_1=0,\hfill \\ \hfill & b_2=-\frac{6{\beta }_1}{{\beta }_2},c_2=\frac{a_2^2{\beta }_2}{3{\beta }_1},b_1=0,c_1=0,b=\frac{a_2^2{\beta }_2^3+144\alpha {\beta }_1^3{\beta }_3+36b_0{\beta }_1^2{\beta }_2{\beta }_3}{18{\beta }_1^2},\hfill \end{array}$$

(41)

where $a_2$, $b_0$, and $c_0$ are arbitrary constants. In other words, we acquire three different travelling wave solutions for Equation (1): a soliton solution with $\alpha <0$,

$$\begin{array}{cc}\hfill & u(x,t)=-\frac{a_2(-144\alpha {\beta }_1^3-18b_0{\beta }_1^2{\beta }_2+a_2^2{\beta }_2^3+144\alpha {\beta }_1^3{\beta }_3+36b_0{\beta }_1^2{\beta }_2{\beta }_3)}{108{\beta }_1^3}\hfill \\ \hfill & -a_2\alpha {\tan \mathrm{h}}^2(\sqrt{-\alpha }(x+\frac{a_2^2{\beta }_2^3+144\alpha {\beta }_1^3{\beta }_3+36b_0{\beta }_1^2{\beta }_2{\beta }_3}{18{\beta }_1^2}t)),\hfill \\ \hfill & v(x,t)=b_0+\frac{6{\beta }_1}{{\beta }_2}\alpha {\tan \mathrm{h}}^2(\sqrt{-\alpha }(x+\frac{a_2^2{\beta }_2^3+144\alpha {\beta }_1^3{\beta }_3+36b_0{\beta }_1^2{\beta }_2{\beta }_3}{18{\beta }_1^2}t)),\hfill \\ \hfill & w(x,t)=c_0-\frac{a_2^2{\beta }_2}{3{\beta }_1}\alpha {\tan \mathrm{h}}^2(\sqrt{-\alpha }(x+\frac{a_2^2{\beta }_2^3+144\alpha {\beta }_1^3{\beta }_3+36b_0{\beta }_1^2{\beta }_2{\beta }_3}{18{\beta }_1^2}t)),\hfill \end{array}$$

(42)

a periodic solution with $\alpha >0$,

$$\begin{array}{cc}\hfill & u(x,t)=-\frac{a_2(-144\alpha {\beta }_1^3-18b_0{\beta }_1^2{\beta }_2+a_2^2{\beta }_2^3+144\alpha {\beta }_1^3{\beta }_3+36b_0{\beta }_1^2{\beta }_2{\beta }_3)}{108{\beta }_1^3}\hfill \\ \hfill & +a_2\alpha {\tan }^2(\sqrt{\alpha }(x+\frac{a_2^2{\beta }_2^3+144\alpha {\beta }_1^3{\beta }_3+36b_0{\beta }_1^2{\beta }_2{\beta }_3}{18{\beta }_1^2}t)),\hfill \\ \hfill & v(x,t)=b_0-\frac{6{\beta }_1}{{\beta }_2}\alpha {\tan }^2(\sqrt{\alpha }(x+\frac{a_2^2{\beta }_2^3+144\alpha {\beta }_1^3{\beta }_3+36b_0{\beta }_1^2{\beta }_2{\beta }_3}{18{\beta }_1^2}t)),\hfill \\ \hfill & w(x,t)=c_0+\frac{a_2^2{\beta }_2}{3{\beta }_1}\alpha {\tan }^2(\sqrt{\alpha }(x+\frac{a_2^2{\beta }_2^3+144\alpha {\beta }_1^3{\beta }_3+36b_0{\beta }_1^2{\beta }_2{\beta }_3}{18{\beta }_1^2}t)),\hfill \end{array}$$

(43)

and a rational solution with $\alpha =0$,

$$\begin{array}{cc}\hfill & u(x,t)=-\frac{a_2(-18b_0{\beta }_1^2{\beta }_2+a_2^2{\beta }_2^3+36b_0{\beta }_1^2{\beta }_2{\beta }_3)}{108{\beta }_1^3}+a_2{(\frac{18{\beta }_1^2}{18{\beta }_1^{2}x+a_2^2{\beta }_2^3+36b_0{\beta }_1^2{\beta }_2{\beta }_3})}^2,\hfill \\ \hfill & v(x,t)=b_0-\frac{6{\beta }_1}{{\beta }_2}{(\frac{18{\beta }_1^2}{18{\beta }_1^{2}x+a_2^2{\beta }_2^3+36b_0{\beta }_1^2{\beta }_2{\beta }_3})}^2,\hfill \\ \hfill & w(x,t)=c_0+\frac{a_2^2{\beta }_2}{3{\beta }_1}{(\frac{18{\beta }_1^2}{18{\beta }_1^{2}x+a_2^2{\beta }_2^3+36b_0{\beta }_1^2{\beta }_2{\beta }_3})}^2.\hfill \end{array}$$

(44)

3.2. Numerical Solutions for Equation (1)

Take Equation (1), for example,

$$\begin{array}{cc}\hfill & u_t={\beta }_1{u}_{xxx}+{\beta }_2{u}_{x}v+{\beta }_{2}u{v}_x,\hfill \\ \hfill & v_t={\beta }_1{\beta }_3{v}_{xxx}+2{\beta }_2{\beta }_{3}v{v}_x-{\beta }_2{w}_x,\hfill \\ \hfill & w_t={\beta }_1{w}_{xxx}+{\beta }_{2}v{w}_x-2{\beta }_{2}u{u}_x,\hfill \end{array}$$

(45)

with initial conditions

$$\begin{array}{cc}\hfill & u(x,0)=-\frac{a_2(-144\alpha {\beta }_1^3-18b_0{\beta }_1^2{\beta }_2+a_2^2{\beta }_2^3+144\alpha {\beta }_1^3{\beta }_3+36b_0{\beta }_1^2{\beta }_2{\beta }_3)}{108{\beta }_1^3}-a_2\alpha {\tan \mathrm{h}}^2\left(\sqrt{-\alpha }x\right),\hfill \\ \hfill & v(x,0)=b_0+\frac{6{\beta }_1}{{\beta }_2}\alpha {\tan \mathrm{h}}^2\left(\sqrt{-\alpha }x\right),w(x,0)=c_0-\frac{a_2^2{\beta }_2}{3{\beta }_1}\alpha {\tan \mathrm{h}}^2\left(\sqrt{-\alpha }x\right).\hfill \end{array}$$

(46)

When the starting conditions from Equation (46) are coupled with the Laplace transform from Equation (45), we obtain

$$\begin{array}{cc}\hfill & \mathcal{L}\left[u(x,t)\right]=\frac{1}{s}\mathcal{L}[{\beta }_1\frac{{\partial }^{3}u}{\partial x^3}+{\beta }_2\frac{\partial u}{\partial x}v+{\beta }_{2}u\frac{\partial v}{\partial x}]+\frac{1}{s}u(x,0),\hfill \\ \hfill & \mathcal{L}\left[v(x,t)\right]=\frac{1}{s}\mathcal{L}[{\beta }_1{\beta }_3\frac{{\partial }^{3}v}{\partial x^3}+2{\beta }_2{\beta }_{3}v\frac{\partial v}{\partial x}-{\beta }_2\frac{\partial w}{\partial x}]+\frac{1}{s}v(x,0),\hfill \\ \hfill & \mathcal{L}\left[w(x,t)\right]=\frac{1}{s}\mathcal{L}[{\beta }_1\frac{{\partial }^{3}w}{\partial x^3}+{\beta }_{2}v\frac{\partial w}{\partial x}-2{\beta }_{2}u\frac{\partial u}{\partial x}]+\frac{1}{s}w(x,0).\hfill \end{array}$$

(47)

The following formula is used to create the zero-order deformation equation, where $i=1,2,3$, and ${\Theta }_i(x,t;q)={\Theta }_i$, respectively,

$$\begin{array}{cc}\hfill & (1-nq)\mathcal{L}[{\Theta }_1-u_0(x,t)]=\hslash qH(x,t){\mathcal{B}}_1({\Theta }_1,{\Theta }_2,{\Theta }_3),\hfill \\ \hfill & (1-nq)\mathcal{L}[{\Theta }_2-v_0(x,t)]=\hslash qH(x,t){\mathcal{B}}_2({\Theta }_1,{\Theta }_2,{\Theta }_3),\hfill \\ \hfill & (1-nq)\mathcal{L}[{\Theta }_3-w_0(x,t)]=\hslash qH(x,t){\mathcal{B}}_3({\Theta }_1,{\Theta }_2,{\Theta }_3).\hfill \end{array}$$

(48)

The nonlinear operators are:

$$\begin{array}{cc}\hfill & {{\mathcal{B}}_{\mathcal{1}}}_1({\Theta }_1,{\Theta }_2,{\Theta }_3)=\mathcal{L}\left[{\Theta }_1\right]-\frac{1}{s}u(x,0)-\frac{1}{s}\mathcal{L}[{\beta }_1\frac{{\partial }^3{\Theta }_1}{\partial x^3}+{\beta }_2\frac{\partial {\Theta }_1}{\partial x}{\Theta }_2+{\beta }_2{\Theta }_1\frac{\partial {\Theta }_2}{\partial x}],\hfill \\ \hfill & {{\mathcal{B}}_{\mathcal{2}}}_2({\Theta }_1,{\Theta }_2,{\Theta }_3)=\mathcal{L}[{\Theta }_2]-\frac{1}{s}v(x,0)-\frac{1}{s}\mathcal{L}[{\beta }_1{\beta }_3\frac{{\partial }^3{\Theta }_2}{\partial x^3}+2{\beta }_2{\beta }_3{\Theta }_2\frac{\partial {\Theta }_2}{\partial x}-{\beta }_2\frac{\partial {\Theta }_3}{\partial x}],\hfill \\ \hfill & {{\mathcal{B}}_{\mathcal{3}}}_3({\Theta }_1,{\Theta }_2,{\Theta }_3)=\mathcal{L}\left[{\Theta }_3\right]-\frac{1}{s}w(x,0)-\frac{1}{s}\mathcal{L}[{\Theta }_1\frac{{\partial }^3{\Theta }_3}{\partial x^3}+{\beta }_2{\Theta }_2\frac{\partial {\Theta }_3}{\partial x}-2{\beta }_2{\Theta }_1\frac{\partial {\Theta }_1}{\partial x}].\hfill \end{array}$$

(49)

The algorithm which has been proposed as follows has been employed to calculate the m-order deformation equation,

$$\begin{array}{cc}\hfill & \mathcal{L}[u_l(x,t)-k_l{u}_{l-1}(x,t)]=\hslash {\mathcal{R}}_{1,l}({\overrightarrow{u}}_{l-1},{\overrightarrow{v}}_{l-1},{\overrightarrow{w}}_{l-1}),\hfill \\ \hfill & \mathcal{L}[v_l(x,t)-k_l{v}_{l-1}(x,t)]=\hslash {\mathcal{R}}_{2,l}({\overrightarrow{u}}_{l-1},{\overrightarrow{v}}_{l-1},{\overrightarrow{w}}_{l-1}),\hfill \\ \hfill & \mathcal{L}[w_l(x,t)-k_l{w}_{l-1}(x,t)]=\hslash {\mathcal{R}}_{3,l}({\overrightarrow{u}}_{l-1},{\overrightarrow{v}}_{l-1},{\overrightarrow{w}}_{l-1}),\hfill \end{array}$$

(50)

where

$$\begin{array}{cc}\hfill & {\mathcal{R}}_{1,l}({\overrightarrow{u}}_{l-1},{\overrightarrow{v}}_{l-1},{\overrightarrow{w}}_{l-1})=\mathcal{L}\left[u_{l-1}(x,t)\right]-(1-\frac{k_l}{n})\frac{1}{s}u(x,0)\hfill \\ \hfill & -\frac{1}{s}\mathcal{L}[{\beta }_1\frac{{\partial }^3{u}_{l-1}}{\partial x^3}+{\beta }_2\sum _{k=0}^{l-1}{v}_k\frac{\partial u_{l-1-k}}{\partial x}+{\beta }_2\sum _{k=0}^{l-1}{u}_k\frac{\partial v_{l-1-k}}{\partial x}],\hfill \\ \hfill & {\mathcal{R}}_{2,l}({\overrightarrow{u}}_{l-1},{\overrightarrow{v}}_{l-1},{\overrightarrow{w}}_{l-1})=\mathcal{L}\left[v_{l-1}(x,t)\right]-(1-\frac{k_l}{n})\frac{1}{s}v(x,0)\hfill \\ \hfill & -\frac{1}{s}\mathcal{L}[{\beta }_1{\beta }_3\frac{{\partial }^3{v}_{l-1}}{\partial x^3}+2{\beta }_2{\beta }_3\sum _{k=0}^{l-1}{v}_k\frac{\partial v_{l-1-k}}{\partial x}-{\beta }_2\frac{\partial w_{l-1}}{\partial x}],\hfill \\ \hfill & {\mathcal{R}}_{3,l}({\overrightarrow{u}}_{l-1},{\overrightarrow{v}}_{l-1},{\overrightarrow{w}}_{l-1})=\mathcal{L}\left[w_{l-1}(x,t)\right]-(1-\frac{k_l}{n})\frac{1}{s}w(x,0)\hfill \\ \hfill & -\frac{1}{s}\mathcal{L}[{\beta }_1\frac{{\partial }^3{w}_{l-1}}{\partial x^3}+{\beta }_2\sum _{k=0}^{l-1}{v}_k\frac{\partial w_{l-1-k}}{\partial x}-2{\beta }_2\sum _{k=0}^{l-1}{u}_k\frac{\partial u_{l-1-k}}{\partial x}].\hfill \end{array}$$

(51)

In Equation (50), we can utilize the inverse Laplace transform to deduce

$$\begin{array}{cc}\hfill & u_l(x,t)=k_l{u}_{l-1}(x,t)+\hslash {\mathcal{L}}^{-1}{\mathcal{R}}_{1,l}({\overrightarrow{u}}_{l-1},{\overrightarrow{v}}_{l-1},{\overrightarrow{w}}_{l-1}),\hfill \\ \hfill & v_l(x,t)=k_l{v}_{l-1}(x,t)+\hslash {\mathcal{L}}^{-1}{\mathcal{R}}_{2,l}({\overrightarrow{u}}_{l-1},{\overrightarrow{v}}_{l-1},{\overrightarrow{w}}_{l-1}),\hfill \\ \hfill & w_l(x,t)=k_l{w}_{l-1}(x,t)+\hslash {\mathcal{L}}^{-1}{\mathcal{R}}_{3,l}({\overrightarrow{u}}_{l-1},{\overrightarrow{v}}_{l-1},{\overrightarrow{w}}_{l-1}).\hfill \end{array}$$

(52)

The following results have been achieved by dealing with the equations provided earlier,

$$\begin{array}{cc}\hfill & u_0(x,t)=-\frac{a_2(-144\alpha {\beta }_1^3-18b_0{\beta }_1^2{\beta }_2+a_2^2{\beta }_2^3+144\alpha {\beta }_1^3{\beta }_3+36b_0{\beta }_1^2{\beta }_2{\beta }_3)}{108{\beta }_1^3}-a_2\alpha {\tan \mathrm{h}}^2\left(\sqrt{-\alpha }x\right),\hfill \\ \hfill & v_0(x,t)=b_0+\frac{6{\beta }_1}{{\beta }_2}\alpha {\tan \mathrm{h}}^2\left(\sqrt{-\alpha }x\right),\hfill \\ \hfill & w_0(x,t)=c_0-\frac{a_2^2{\beta }_2}{3{\beta }_1}\alpha {\tan \mathrm{h}}^2\left(\sqrt{-\alpha }x\right),\hfill \\ \hfill & u_1(x,t)=\frac{\sqrt{-\alpha }\alpha a_{2}httanh\left(\sqrt{-\alpha }x\right){\sec \mathrm{h}}^2\left(\sqrt{-\alpha }x\right)\left(a_2^2{\beta }_2^3+36{\beta }_1^2{\beta }_3\left(4\alpha {\beta }_1+b_0{\beta }_2\right)\right)}{9{\beta }_1^2},\hfill \\ \hfill & v_1(x,t)=-\frac{2\sqrt{-\alpha }\alpha httanh\left(\sqrt{-\alpha }x\right){\sec \mathrm{h}}^2\left(\sqrt{-\alpha }x\right)\left(a_2^2{\beta }_2^3+36{\beta }_1^2{\beta }_3\left(4\alpha {\beta }_1+b_0{\beta }_2\right)\right)}{3{\beta }_1{\beta }_2},\hfill \\ \hfill & w_1(x,t)=-\frac{{\beta }_2\left({\beta }_2^3+36{\beta }_1^2\left({\beta }_2-4{\beta }_1\right){\beta }_3\right)httanh\left(x\right){\sec \mathrm{h}}^2\left(x\right)}{27{\beta }_1^3},\hfill \\ \hfill & ⋮\hfill \end{array}$$

(53)

Similar calculations can be carried out for the remaining iteration terms. Finally, the numerical solutions of Equation (45) are supplied as

$$\begin{array}{cc}\hfill & u^{\left[M\right]}(x,t)=u_0(x,t)+\sum _{l=1}^M{u}_l(x,t){(\frac{1}{n})}^l,\hfill \\ \hfill & v^{\left[M\right]}(x,t)=v_0(x,t)+\sum _{l=1}^M{v}_l(x,t){(\frac{1}{n})}^l,\hfill \\ \hfill & w^{\left[M\right]}(x,t)=w_0(x,t)+\sum _{l=1}^M{w}_l(x,t){(\frac{1}{n})}^l.\hfill \end{array}$$

(54)

As $N\to \infty$, the numerical solutions for Equation (1) correspondingly converge to the analytical solutions of Equation (45) for $n=1$ and $\hslash =-1$.

4. Numerical Results and Simulation

With the q-HATM results in mind, the extended three-coupled Korteweg–de Vries system will be numerically simulated in this section. The comparisons between numerical solutions and analytical solutions generated by q-HATM with a five-term approximation are shown in Figure 1, Figure 2 and Figure 3. The absolute error function of Equation (1) is shown in Figure 4. The ℏ curves are shown in Figure 5, Figure 6 and Figure 7 for distinct n and x values. By altering the horizontal line in the ℏ curve, which symbolizes the convergence range of the extended coupled Korteweg–de Vries system, we may change the convergence range of the series solutions. Figure 5, Figure 6 and Figure 7 emphatically demonstrate that the numerical solutions converge in the intervals $(-2.1,0.1)$ and $(-3.8,0.2)$ for $n=1$ and $n=2$, respectively. As a result, we discovered that the range of the permitted ℏ convergence interval is wider when $n=3$. Figure 5, Figure 6 and Figure 7 demonstrate that when the n value increases, the convergence range also does. The comparisons between numerical solutions and analytical solutions are displayed in

Table 1. Numerical results for numerical and analytical solutions of $u(x,t)$ when $n=1$ and $\hslash =-1$.

t	x	Numerical	Exact	Absolute Error
0.1	−10	−0.25000001	−0.25000001	$1.99840\times {10}^{-15}$
	−5	−0.25016025	−0.25016025	$4.51094\times {10}^{-11}$
	0	−1.24610392	−1.24610390	$2.24716\times {10}^{-8}$
	5	−0.25020576	−0.25020576	$4.70228\times {10}^{-11}$
	10	−0.25000001	−0.25000000	$2.10942\times {10}^{-15}$
0.2	−10	−0.25000001	−0.25000001	$6.45040\times {10}^{-14}$
	−5	−0.25014142	−0.25014142	$1.41446\times {10}^{-9}$
	0	−1.23453776	−1.23453633	$1.42947\times {10}^{-6}$
	5	−0.25023315	−0.25023315	$1.53703\times {10}^{-9}$
	10	−0.25000001	−0.25000001	$6.99441\times {10}^{-14}$

,

Table 2. Numerical results for numerical and analytical solutions of $v(x,t)$ when $n=1$ and $\hslash =-1$.

t	x	Numerical	Exact	Absolute Error
0.1	−10	−0.99999999	−0.99999999	$4.02223\times {10}^{-15}$
	−5	−0.99967950	−0.99967950	$9.02187\times {10}^{-11}$
	0	0.99220785	0.99220780	$4.49431\times {10}^{-8}$
	5	−0.99958848	−0.99958848	$9.40457\times {10}^{-11}$
	10	−0.99999998	−0.99999998	$4.20917\times {10}^{-15}$
0.2	−10	−0.99999999	−0.99999999	$1.28923\times {10}^{-13}$
	−5	−0.99971716	−0.99971716	$2.82892\times {10}^{-9}$
	0	0.96907552	0.96907266	$2.85893\times {10}^{-6}$
	5	−0.99953370	−0.99953370	$3.07407\times {10}^{-9}$
	10	−0.99999998	−0.99999998	$1.39870\times {10}^{-13}$

and

Table 3. Numerical results for numerical and analytical solutions of $w(x,t)$ when $n=1$ and $\hslash =-1$.

t	x	Numerical	Exact	Absolute Error
0.1	−10	1.99999999	1.99999999	$2.01112\times {10}^{-15}$
	−5	1.99983975	1.99983975	$4.51093\times {10}^{-11}$
	0	1.00389608	1.00389610	$2.24716\times {10}^{-8}$
	5	1.99979424	1.99979424	$4.70228\times {10}^{-11}$
	10	1.99999999	1.99999999	$2.10459\times {10}^{-15}$
0.2	−10	1.99999999	1.99999999	$6.44650\times {10}^{-14}$
	−5	1.99985858	1.99985858	$1.41446\times {10}^{-9}$
	0	1.01546224	1.01546367	$1.42947\times {10}^{-6}$
	5	1.99976685	1.99976685	$1.53703\times {10}^{-9}$
	10	1.99999999	1.99999999	$6.99350\times {10}^{-14}$

. These charts show the accuracy of the results of the proposed method.

5. Conclusions

In the current study, the q-HATM and the Laplace transform have been employed to evaluate numerical solutions of an extended coupled Korteweg–de Vries system. The suggested technique has the benefit of not necessitating any discretization, linearization, or interference. The asymptotic parameter n and auxiliary parameter ℏ have been incorporated in the numerical solutions of the q-HATM, which provides us with a rapid means to modify the convergence speed and range within the found series solutions. The numerical outcomes show what a successful, accurate, and robust procedure iteration is for solving PDEs. Finally, we can prove that the proposed method is more systematic and accurate, and it can be used to study the complicated processes of nonlinear phenomena.