Investigating the Dynamics of Time-Fractional Drinfeld–Sokolov–Wilson System through Analytical Solutions

Abstract

This study addresses a nonlinear fractional Drinfeld–Sokolov–Wilson problem in dispersive water waves, which requires appropriate numerical techniques to obtain an approximative solution. The Adomian decomposition approach, the homotopy perturbation method, and Sumudu transform are combined to tackle the problem. The Caputo manner is used to describe fractional derivative, and He’s polynomials and Adomian polynomials are employed to address nonlinearity. By following these approaches, we obtain solutions in the form of convergent series. We verify and demonstrate the effectiveness of our suggested strategies by examining the assumed model in terms of fractional order. We use plots for various fractional orders to represent the physical behavior of the suggested technique solutions, and show a numerical simulation. The results demonstrate that the suggested algorithms are systematic, simple to use, effective, and accurate in analyzing the behavior of coupled nonlinear differential equations of fractional order in related scientific and engineering fields.

1. Introduction

Fractional partial differential equations (FPDEs) are a powerful mathematical tool used to model a wide range of complex physical phenomena. These equations involve partial derivatives of fractional order, which provide a means of describing the behavior of systems with long-range interactions or non-local effects. FPDEs have found applications in various fields, such as fluid mechanics, electromagnetism, finance, and biology, among others. Unlike classical partial differential equations, FPDEs have a non-local character, which makes their solutions more challenging to find. Nevertheless, the development of numerical methods, such as fractional finite differences, fractional calculus, and fractional Laplacians, has allowed researchers to study and simulate these equations in a more efficient way [1,2,3,4].

One of the advantages of using FPDEs is their ability to capture the memory effects and non-local interactions that are often observed in real-world systems. This feature makes FPDEs particularly useful in modeling complex systems, such as biological networks, financial markets, and porous media, where the interactions between the components can span over long distances [5,6,7]. In this context, understanding the properties and behavior of FPDEs has become a critical topic in applied mathematics and physics. Researchers continue to develop new techniques and methods to solve these equations and gain insights into the complex phenomena they describe [8,9,10,11,12,13,14,15,16].

Partial differential equations (PDEs) play a crucial role in describing many physical and mathematical phenomena. One important class of PDEs is the fractional partial differential equations (FPDEs), which involve derivatives of non-integer order. FPDEs have recently gained much attention due to their ability to accurately model anomalous diffusion, viscoelasticity, and other phenomena that cannot be captured by traditional integer-order PDEs [17,18,19]. FPDEs have a wide range of applications in various fields such as physics, engineering, finance, and biology. For example, in physics, FPDEs are used to describe the behavior of particles in fractal media, where the traditional diffusion equation fails to capture the true behavior of the particles [20,21]. In engineering, FPDEs are used to model the transport of fluids in porous media and to describe the viscoelastic properties of materials. In finance, FPDEs are used to model the dynamics of financial derivatives and to estimate the risk associated with financial investments. In biology, FPDEs are used to model the movement of molecules in biological systems and the spread of diseases [22,23,24,25,26].

The interaction of various long waves with various dispersion relations is illustrated by linked KdV equation systems which Hirota and Satsum convincingly establish in [27,28] and present as follows.

$$\begin{array}{cc}\hfill & {\phi }_{\Re }-\frac{1}{2}({\phi }_{\varrho \varrho \varrho }+6\phi {\phi }_{\varrho })=2b\zeta {\zeta }_{\varrho },\hfill \\ \hfill & {\zeta }_{\Re }+{\zeta }_{\varrho \varrho \varrho }+3\phi {\zeta }_{\varrho }=0.\hfill \end{array}$$

(1)

The expression for the forcing term in the KdV system is $2b\zeta {\zeta }_{\varrho }$, and according to Drinfeld and Sokolov, this equation is a specific case of the Kadomtsev–Petviashvili (KP) hierarchy with four reductions [29]. Moreover, in [30], Wilson established a relationship between Equation (1) and affine Lie algebras, showing how it can be obtained as a general Drinfeld and Sokolov construction. Wilson also associated this equation with the affine Lie algebra $C_2^{\left(1\right)}$. Later, Wilson introduced and provided an example of the remarkable equation known as the Drinfeld–Sokolov–Wilson (DSW) equation, which is given by:

$$\begin{array}{cc}\hfill & {\phi }_{\Re }+\kappa \zeta {\zeta }_{\varrho }=0,\hfill \\ \hfill & {\zeta }_{\Re }+\ell {\zeta }_{\varrho \varrho \varrho }+\tau \phi {\zeta }_{\varrho }+\delta {\phi }_{\varrho }\zeta =0.\hfill \end{array}$$

(2)

In the present framework, we consider a system of equations in which non-zero parameters $\kappa ,\ell ,\tau$ and $\delta$ represent the amplitude of wave modes. The functions $\phi (\varrho ,\Re )$ and $\phi (\varrho ,\Re )$ represent the amplitude of wave modes with respect to time ℜ and space $\mu$, respectively. This system is especially significant for modeling dispersive water waves when $\kappa =\delta =1$ and $\ell =\tau =2$. In this context, we consider the fractional DSW (FDSW) equation in the following form.

$$\begin{array}{cc}\hfill & D_{\Re }^{\sigma }\phi (\varrho ,\Re )+3\zeta (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )=0,\hfill \\ \hfill & D_{\Re }^{\sigma }\zeta (\varrho ,\Re )+2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )+2\phi (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )+{\phi }_{\varrho }(\varrho ,\Re )\zeta (\varrho ,\Re )=0,\hfill \end{array}$$

(3)

The use of the Caputo fractional operator allows us to define the fractional order $\sigma$ of the system, which incorporates memory effects and genetic consequences. This feature enables us to identify crucial physical characteristics of complex problems. The shallow water wave models were initially proposed by Drinfeld and Sokolov [31] and later refined by Wilson [30] to develop the classical DSW equations. The diffusive wave approximations of the shallow water equations (DSW) have been effectively used to model gravitational water flow dominated by shear stress, such as overland flows, dam breaks, flows through vegetation, and floods. These equations have significant importance in physics and engineering, and many researchers have made concerted efforts to find solutions for them. For example, Santillana [32] investigated the features of approximate DSW equation solutions using the Galerkin finite element method. Inc [33] employed the Adomian decomposition approach to obtain approximate doubly periodic wave solutions of the traditional DSW equations. Truncated Painlev’e and Mobius invariant forms were used [34] to find explicit solutions to the classical DSW equations with nonlocal symmetry. Zhao et al. [35] utilized the F-expansion technique to locate solutions to common DSW equations.

Although the precise solutions of fractional DSW equations are less frequently reported, Sahoo [36] derived double-periodic solutions using the Jacobi elliptical function approach. Misirli and Gurefe used the exp-function approach to study common DSW equations [37]. However, finding the exact solutions of the time-fractional DSW system is more challenging than for the classical DSW equations. Most studies focus on the classical DSW equations, and the research on fractional DSW equations is still in its nascent stage. Moreover, in addition to the approximate and periodic solutions, more solutions to the fractional DSW equations need to be determined [38,39,40].

In this study, we have used the Sumudu Transform Decomposition method (STDM) and the Homotopy Perturbation Transform method (HPTM) to derive an analytical solution for time-fractional coupled DSW equations. It is worth noting that these methods offer a superior performance, requiring less computational work while maintaining a high level of numerical precision. The paper is organized as follows: Section 2 provides fundamental definitions for fractional calculus, while Section 3 and Section 4 thoroughly explain the steps of the suggested methods. In Section 5, we apply the suggested methods to obtain approximate solutions for the time-fractional Drinfeld–Sokolov–Wilson system. Finally, the last section includes our observations and conclusions.

2. Basic Definitions

Definition 1.

A function $f\left(x\right)$ defined for $x>0$ belongs to the space $C_{\mu }$, where $\mu \in \mathbb{R}$, if there is a real number $p>\mu$ such that $f\left(x\right)=x^{p}h\left(x\right)$, where $h\left(x\right)\in C[0,\infty )$. Similarly, $f\left(x\right)$ is said to be in the space $C_{\mu }^m$, where $m\in \mathbb{N}$, if $f^{\left(m\right)}\in C_{\mu }$, meaning that the mth derivative of $f\left(x\right)$ belongs to $C_{\mu }$.

Definition 2.

The definition of the Riemann-Liouville fractional integral operator of order $\alpha \ge 0$ for a function $f\in C_{\mu },\mu \ge -1$, is as follows:

$$\begin{array}{c}{J}^{\alpha }f\left(x\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^x{(x-\Re )}^{\alpha -1}f(\Re )d\Re ,\alpha >0,x>0\\ J^{0}f\left(x\right)=f\left(x\right).\end{array}$$

Properties of the operator mentions only the following. For $f\in C_{\mu },\mu \ge -1,\alpha ,\beta \ge 0$, and $\gamma >-1$

$$\begin{array}{c}{J}^{\alpha }{J}^{\beta }f\left(x\right)=J^{\alpha +\beta }f\left(x\right),\\ J^{\alpha }{J}^{\beta }f\left(x\right)=J^{\beta }{J}^{\alpha }f\left(x\right),\\ J^{\alpha }{x}^{\gamma }=\frac{\mathsf{\Gamma }(\gamma +1)}{\mathsf{\Gamma }(\alpha +\gamma +1)}{x}^{\alpha +\gamma }.\end{array}$$

Lemma 1.

If $m-1<\alpha \le m,m\in \mathbb{N}$ and $f\in C_{\mu }^m$, and $\mu \ge -1$, then

$$\begin{array}{c}{D}^{\alpha }{J}^{\alpha }f\left(x\right)=f\left(x\right),\\ J^{\alpha }{D}_0^{\alpha }f\left(x\right)=f\left(x\right)-{\displaystyle \sum _{k=0}^{m-1}}{f}^{\left(k\right)}\left(0^{+}\right)\frac{x^k}{k!},x>0.\end{array}$$

Definition 3.

Let us consider a function $f\left(x\right)$ of n variables, denoted as $x_i$ for $i=1,\dots ,n$, and assume that it belongs to the class C on a domain $D\subseteq {\mathbb{R}}^n$.

$$a{\partial }_{\underline{\mathrm{x}}}^{\alpha }f={\left.\frac{1}{\mathsf{\Gamma }(m-\alpha )}{\int }_a^{x_i}{\left(x_i-t\right)}^{m-\alpha -1}{\partial }_{x_i}^{m}f\left(x_j\right)\right|}_{x_j=\Re }d\Re ,$$

We denote the standard partial derivative of integer order m with respect to $x_i$ as ${\partial }_{x_i}^m$.

Definition 4.

The definition of the Sumudu transform for the Caputo fractional derivative can be stated as follows:

$$\begin{array}{cc}\hfill S\left[D_{\Re }^{\alpha }f(\Re )\right]=& u^{-\alpha }S\left[f(\Re )\right]-\sum _{k=0}^{m-1}{u}^{-\alpha +k}{f}^{\left(k\right)}\left(0^{+}\right)(m-1<\alpha \le m).\hfill \end{array}$$

3. Fundamental Idea of HPTM

In this part, we take the arbitrary order differential equation to show the basic solution procedure.

$$D_{\lambda }^{\alpha }\phi (\varrho ,\Re )+{\mathcal{P}}_1\phi (\varrho ,\Re )+{\mathcal{Q}}_1\phi (\varrho ,\Re )=00<\alpha \le 1,$$

(4)

subject to the initial guess

$$\phi (\varrho ,0)=\xi \left(\varrho \right).$$

Here, $D_{\Re }^{\alpha }$ is the Caputo fractional derivative, ${\mathcal{P}}_1$ is representing the linear operator, and ${\mathcal{Q}}_1$ is depicting the nonlinear differential operator.

On using ST operator

$$S\left[D_{\Re }^{\alpha }\phi (\varrho ,\Re )\right]+S[{\mathcal{P}}_1\phi (\varrho ,\Re )+{\mathcal{Q}}_1\phi (\varrho ,\Re )]=0,$$

(5)

$$\frac{1}{u^{\alpha }}\{\mathcal{R}\left(u\right)-u\phi \left(0\right)\}+S[{\mathcal{P}}_1\phi (\varrho ,\Re )+{\mathcal{Q}}_1\phi (\varrho ,\Re )]=0.$$

(6)

We have after simplification

$$\mathcal{R}\left(\phi \right)=u\phi \left(0\right)-u^{\alpha }S[{\mathcal{P}}_1\phi (\varrho ,\Re )+{\mathcal{Q}}_1\phi (\varrho ,\Re )].$$

(7)

By operating inverse ST

$$\phi (\varrho ,\Re )=\phi \left(0\right)-S^{-1}\left[u^{\alpha }S[{\mathcal{P}}_1\phi (\varrho ,\Re )+{\mathcal{Q}}_1\phi (\varrho ,\Re )]\right],$$

(8)

Now, in terms of HPM,

$$\phi (\varrho ,\Re )=\sum _{k=0}^{\infty }{p}^k{\phi }_k(\varrho ,\Re ),$$

(9)

we express the nonlinear operator as

$${\mathcal{Q}}_1\phi (\varrho ,\Re )=\sum _{k=0}^{\infty }{p}^k{H}_k\left(\phi \right),$$

(10)

with He’s polynomials $H_k$ as

$$\begin{array}{cc}\hfill & H_k({\phi }_0,{\phi }_1,...,{\phi }_k)=\frac{1}{k!}\frac{{\partial }^k}{\partial p^k}{\left[{\mathcal{Q}}_1\left(\sum _{i=0}^{\infty }{p}^i{\phi }_i\right)\right]}_{p=0}.\hfill \\ \hfill & k=0,1,2,3\cdots \hfill \end{array}$$

(11)

Substituting (9) and (10) in Equation (8)

$$\sum _{k=0}^{\infty }{p}^k{\phi }_k(\varrho ,\Re )=\xi \left(\varrho \right)-p\times \left(S^{-1}\left[u^{\alpha }S\{{\mathcal{P}}_1\sum _{k=0}^{\infty }{p}^k{\phi }_k(\varrho ,\Re )+\sum _{k=0}^{\infty }{p}^k{H}_k\left(\phi \right)\}\right]\right).$$

(12)

Equating in terms of identical powers p, we get

$$\begin{array}{cc}\hfill & p^0:{\phi }_0(\varrho ,\Re )=\xi \left(\varrho \right),\hfill \\ \hfill & p^1:{\phi }_1(\varrho ,\Re )=-S^{-1}\left[u^{\alpha }S({\mathcal{P}}_1{\phi }_0(\varrho ,\Re )+H_0\left(\phi \right))\right],\hfill \\ \hfill & p^2:{\phi }_2(\varrho ,\Re )=-S^{-1}\left[u^{\alpha }S({\mathcal{P}}_1{\phi }_1(\varrho ,\Re )+H_1\left(\phi \right))\right],\hfill \\ \hfill & ⋮\hfill \\ \hfill & p^k:{\phi }_k(\varrho ,\Re )=-S^{-1}\left[u^{\alpha }S({\mathcal{P}}_1{\phi }_{k-1}(\varrho ,\Re )+H_{k-1}\left(\phi \right))\right],\hfill \\ \hfill k>0,k\in N.\end{array}$$

(13)

Finally, we obtain the analytical solution by the truncated series as

$$\phi (\varrho ,\Re )=\underset{M\to \infty }{lim}\sum _{k=1}^M{\phi }_k(\varrho ,\Re ).$$

(14)

4. Fundamental Idea of STDM

In this part, we take the arbitrary order differential equation to show the basic solution procedure.

$$D_{\Re }^{\alpha }\phi (\varrho ,\Re )={\mathcal{P}}_1(\varrho ,\Re )+{\mathcal{Q}}_1(\varrho ,\Re )+{\mathcal{R}}_1(\varrho ,\Re ),0<\alpha \le 1,$$

(15)

subject to the initial guess

$$\phi (\varrho ,0)=\xi \left(\varrho \right).$$

Here, $D_{\Re }^{\alpha }=\frac{{\partial }^{\alpha }}{\partial {\Re }^{\alpha }}$ is the Caputo fractional derivative, ${\mathcal{P}}_1$ is representing the linear operator, and ${\mathcal{Q}}_1$ is depicting the nonlinear differential operator.

On using ST operator,

$$S\left[D_{\Re }^{\alpha }\phi (\varrho ,\Re )\right]=S[{\mathcal{P}}_1(\varrho ,\Re )+{\mathcal{Q}}_1(\varrho ,\Re )+{\mathcal{R}}_1(\varrho ,\Re )].$$

(16)

We have, after simplification,

$$\begin{array}{cc}\hfill & \frac{1}{u^{\alpha }}\{\mathcal{R}\left(u\right)-u\phi \left(0\right)\}=S[{\mathcal{P}}_1(\varrho ,\Re )+{\mathcal{Q}}_1(\varrho ,\Re )+{\mathcal{R}}_1(\varrho ,\Re )],\hfill \\ \hfill & \mathcal{R}\left(\phi \right)=u\phi \left(0\right)+u^{\alpha }S[{\mathcal{P}}_1(\varrho ,\Re )+{\mathcal{Q}}_1(\varrho ,\Re )+{\mathcal{R}}_1(\varrho ,\Re )].\hfill \end{array}$$

(17)

By operating inverse ST,

$$\phi (\varrho ,\Re )=\phi \left(0\right)+S^{-1}[u^{\alpha }S[{\mathcal{P}}_1(\varrho ,\Re )+{\mathcal{Q}}_1(\varrho ,\Re )+{\mathcal{R}}_1(\varrho ,\Re )].$$

(18)

Now, in terms of STDM,

$$\phi (\varrho ,\Re )=\sum _{m=0}^{\infty }{\phi }_m(\varrho ,\Re ).$$

(19)

We express the nonlinear operator as

$${\mathcal{Q}}_1(\varrho ,\Re )=\sum _{m=0}^{\infty }{\mathcal{A}}_m,$$

(20)

with Adomian polynomials ${\mathcal{A}}_m$ as

$${\mathcal{A}}_m=\frac{1}{m!}{\left[\frac{{\partial }^m}{\partial {\delta }^m}\left\{{\mathcal{Q}}_1\left(\sum _{k=0}^{\infty }{\delta }^k{\varrho }_k,\sum _{k=0}^{\infty }{\delta }^k{\Re }_k\right)\right\}\right]}_{\delta =0}.$$

(21)

Substituting (19) and (20) in Equation (18), we have

$$\sum _{m=0}^{\infty }{\phi }_m(\varrho ,\Re )=\phi \left(0\right)+S^{-1}\left[u^{\alpha }S\left\{{\mathcal{R}}_1(\varrho ,\Re )\right\}\right]+S^{-1}{u}^{\alpha }\left[S\left\{{\mathcal{P}}_1(\sum _{m=0}^{\infty }{\varrho }_m,\sum _{m=0}^{\infty }{\Re }_m)+\sum _{m=0}^{\infty }{\mathcal{A}}_m\right\}\right].$$

(22)

The following terms are obtained.

$${\phi }_0(\varrho ,\Re )=\phi \left(0\right)+S^{-1}\left[u^{\alpha }S\left\{{\mathcal{R}}_1(\varrho ,\Re )\right\}\right],$$

(23)

$${\phi }_1(\varrho ,\Re )=S^{-1}\left[u^{\alpha }S\{{\mathcal{P}}_1({\varrho }_0,{\Re }_0)+{\mathcal{A}}_0\}\right],$$

Thus the analytical solution by the truncated series for $m\ge 1$ is as

$${\phi }_{m+1}(\varrho ,\Re )=S^{-1}\left[u^{\alpha }S\{{\mathcal{P}}_1({\varrho }_m,{\Re }_m)+{\mathcal{A}}_m\}\right].$$

5. Applications

We obtain the analytical solutions for fractional coupled DSW equation by implementing the proposed techniques as follow.

Example

Consider the DSW equation having a fractional order as

$$\begin{array}{cc}\hfill & D_{\Re }^{\alpha }\phi (\varrho ,\Re )+3\zeta (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )=0,0<\alpha \le 1,\hfill \\ \hfill & D_{\Re }^{\alpha }\zeta (\varrho ,\Re )+2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )+2\phi (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )+{\phi }_{\varrho }(\varrho ,\Re )\zeta (\varrho ,\Re )=0,0<\alpha \le 1,\hfill \end{array}$$

(24)

subjected to the initial guesses,

$$\begin{array}{cc}\hfill & \phi (\varrho ,0)=3{sech}^2\left(\varrho \right),\hfill \\ \hfill & \zeta (\varrho ,0)=2sech\left(\varrho \right).\hfill \end{array}$$

By setting $\alpha =1$, we get an exact solution as

$$\begin{array}{cc}\hfill & \phi (\varrho ,\Re )=\frac{3\ell }{2}{sech}^2\left(\sqrt{\frac{\ell }{2}}(\varrho -\ell \Re )\right),\hfill \\ \hfill & \zeta (\varrho ,\Re )=\ell sech\sqrt{\frac{\ell }{2}}(\varrho -\ell \Re )).\hfill \end{array}$$

(25)

On using the ST operator,

$$\begin{array}{cc}\hfill & S\left\{\frac{{\partial }^{\alpha }\phi }{\partial {\Re }^{\alpha }}\right\}=S[-3\zeta (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )],\hfill \\ \hfill & S\left\{\frac{{\partial }^{\alpha }\zeta }{\partial {\Re }^{\alpha }}\right\}=S[-2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )-2\phi (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )-{\phi }_{\varrho }(\varrho ,\Re )\zeta (\varrho ,\Re )],\hfill \end{array}$$

We have, after simplification,

$$\begin{array}{cc}\hfill & \frac{1}{u^{\alpha }}\{\mathcal{R}\left(u\right)-u\phi \left(0\right)\}=S[-3\zeta (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )],\hfill \\ \hfill & \frac{1}{u^{\alpha }}\{\mathcal{R}\left(u\right)-u\zeta \left(0\right)\}=S[-2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )-2\phi (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )-{\phi }_{\varrho }(\varrho ,\Re )\zeta (\varrho ,\Re )],\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & \mathcal{R}\left(u\right)=u\phi \left(0\right)+u^{\alpha }S[-3\zeta (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )],\hfill \\ \hfill & \mathcal{R}\left(u\right)=u\zeta \left(0\right)+u^{\alpha }S[-2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )-2\phi (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )-{\phi }_{\varrho }(\varrho ,\Re )\zeta (\varrho ,\Re )].\hfill \end{array}$$

(27)

By operating inverse ST,

$$\begin{array}{cc}\hfill & \phi (\varrho ,\Re )=\phi \left(0\right)+S^{-1}{\displaystyle [}{u}^{\alpha }\left\{S\right(-3\zeta (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )\left)\right\}],\hfill \\ \hfill & \zeta (\varrho ,\Re )=\zeta \left(0\right)+S^{-1}\left[u^{\alpha }\right\{S(-2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )-2\phi (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )-{\phi }_{\varrho }(\varrho ,\Re )\zeta (\varrho ,\Re ))\left\}\right],\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & \phi (\varrho ,\Re )=3{sech}^2\left(\varrho \right)+S^{-1}\left[u^{\alpha }\right\{S(-3\zeta (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re ))\left\}\right],\hfill \\ \hfill & \zeta (\varrho ,\Re )=2sech\left(\varrho \right)+S^{-1}\left[u^{\alpha }\right\{S(-2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )-2\phi (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )-{\phi }_{\varrho }(\varrho ,\Re )\zeta (\varrho ,\Re ))\left\}\right].\hfill \end{array}$$

(29)

Now, in terms of HPM,

$$\begin{array}{cc}\hfill & \phi (\varrho ,\Re )={\phi }_0+{\phi }_{1}p+{\phi }_2{p}^2+\cdots ,\hfill \\ & \zeta (\varrho ,\Re )={\zeta }_0+{\zeta }_{1}p+{\zeta }_2{p}^2+\cdots ,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & \sum _{k=0}^{\infty }{p}^k{\phi }_k(\varrho ,\Re )=3{sech}^2\left(\varrho \right)-p\left(S^{-1}\left[u^{\alpha }S\left[2\left(\sum _{k=0}^{\infty }{p}^k{H}_k^1(\varrho ,\Re )\right)\right]\right]\right),\hfill \\ \hfill & \sum _{k=0}^{\infty }{p}^k{\zeta }_k(\varrho ,\Re )=2sech\left(\varrho \right)-p\left(S^{-1}\left[u^{\alpha }S\left[2{\left(\sum _{k=0}^{\infty }{p}^k{\zeta }_k(\varrho ,\Re )\right)}_{\varrho \varrho \varrho }+2\left(\sum _{k=0}^{\infty }{p}^k{H}_k^2(\varrho ,\Re )\right)+\left(\sum _{k=0}^{\infty }{p}^k{H}_k^3(\varrho ,\Re )\right)\right]\right]\right),\hfill \end{array}$$

(31)

we express the nonlinear first few terms as

$$\begin{array}{cc}\hfill & H_0^1\left(\varrho \right)={\zeta }_0{\zeta }_{0\varrho }\hfill \\ \hfill & H_1^1\left(\varrho \right)={\zeta }_1{\zeta }_{0\varrho }+{\zeta }_0{\zeta }_{1\varrho },\hfill \\ \hfill & H_2^1\left(\varrho \right)={\zeta }_2{\zeta }_{0\varrho }+{\zeta }_1{\zeta }_{1\varrho }+{\zeta }_0{\zeta }_{2\varrho },\hfill \\ \hfill & ⋮\hfill \\ \hfill & H_0^2\left(\varrho \right)={\phi }_0{\zeta }_{0\varrho },\hfill \\ \hfill & H_1^2\left(\varrho \right)={\phi }_1{\zeta }_{0\varrho }+{\phi }_0{\zeta }_{1\varrho },\hfill \\ \hfill & H_2^2\left(\varrho \right)={\phi }_2{\zeta }_{0\varrho }+{\phi }_1{\zeta }_{1\varrho }+{\phi }_0{\zeta }_{2\varrho },\hfill \\ \hfill & ⋮\hfill \\ \hfill & H_0^3\left(\varrho \right)={\phi }_{0\varrho }{\zeta }_0,\hfill \\ \hfill & H_1^3\left(\varrho \right)={\phi }_{1\varrho }{\zeta }_0+{\phi }_{0\varrho }{\zeta }_1,\hfill \\ \hfill & H_2^3\left(\varrho \right)={\phi }_{2\varrho }{\zeta }_0+{\phi }_{1\varrho }{\zeta }_1+{\phi }_{0\varrho }{\zeta }_2.\hfill \end{array}$$

Equating in terms of identical powers p, we get

$$\begin{array}{cc}\hfill & p^0:{\phi }_0(\varrho ,\Re )=3{sech}^2\left(\varrho \right),\hfill \\ \hfill & {\zeta }_0(\varrho ,\Re )=2sech\left(\varrho \right),\hfill \\ \hfill & p^1:{\phi }_1(\varrho ,\Re )=12{sech}^2\left(\varrho \right)tanh\left(\varrho \right)\frac{{\Re }^{\alpha }}{\mathsf{\Gamma }(\alpha +1)},\hfill \\ \hfill & {\zeta }_1(\varrho ,\Re )=4sech\left(\varrho \right)tanh\left(\varrho \right)\frac{{\Re }^{\alpha }}{\mathsf{\Gamma }(\alpha +1)},\hfill \\ \hfill & p^2:{\phi }_2(\varrho ,\Re )=24(-3{sech}^4\left(\varrho \right)+2{sech}^2\left(\varrho \right))\frac{{\Re }^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)},\hfill \\ \hfill & {\zeta }_2(\varrho ,\Re )=8(-2{sech}^3\left(\varrho \right)+sech\left(\varrho \right))\frac{{\Re }^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)},\hfill \\ \hfill & ⋮\hfill \end{array}$$

At $p\to 1$ we get

$$\begin{array}{cc}\hfill & \phi (\varrho ,\Re )={\phi }_0+{\phi }_1+{\phi }_2+{\phi }_3+\cdots \hfill \\ \hfill & =3{sech}^2\left(\varrho \right)+12{sech}^2\left(\varrho \right)tanh\left(\varrho \right)\frac{{\Re }^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+24(-3{sech}^4\left(\varrho \right)+2{sech}^2\left(\varrho \right))\frac{{\Re }^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}+\cdots \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \zeta (\varrho ,\Re )={\zeta }_0+{\zeta }_1+{\zeta }_2+{\zeta }_3+\cdots \hfill \\ \hfill & =2sech\left(\varrho \right)+4sech\left(\varrho \right)tanh\left(\varrho \right)\frac{{\Re }^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+8(-2{sech}^3\left(\varrho \right)+sech\left(\varrho \right))\frac{{\Re }^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}+\cdots \hfill \end{array}$$

Implementation of STDM

On using the ST operator,

$$\begin{array}{cc}\hfill & S\left\{\frac{{\partial }^{\alpha }\phi }{\partial {\Re }^{\alpha }}\right\}=S\left[-3\zeta (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )\right],\hfill \\ \hfill & S\left\{\frac{{\partial }^{\alpha }\zeta }{\partial {\Re }^{\alpha }}\right\}=S\left[-2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )-2\phi (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )-{\phi }_{\varrho }(\varrho ,\Re )\zeta (\varrho ,\Re )\right].\hfill \end{array}$$

We have, after simplification,

$$\begin{array}{cc}\hfill & \frac{1}{u^{\alpha }}\{\mathcal{R}\left(u\right)-u\phi \left(0\right)\}=S\left[-3\zeta (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )\right],\hfill \\ \hfill & \frac{1}{u^{\alpha }}\{\mathcal{R}\left(u\right)-u\zeta \left(0\right)\}=S\left[-2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )-2\phi (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )-{\phi }_{\varrho }(\varrho ,\Re )\zeta (\varrho ,\Re )\right].\hfill \end{array}$$

(32)

By operating inverse ST,

$$\begin{array}{cc}\hfill & \phi (\varrho ,\Re )=\phi \left(0\right)+S^{-1}\left[u^{\alpha }\left\{S\left(-3\zeta (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )\right)\right\}\right],\hfill \\ \hfill & \zeta (\varrho ,\Re )=\zeta \left(0\right)+S^{-1}\left[u^{\alpha }\left\{S\left(-2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )-2\phi (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )-{\phi }_{\varrho }(\varrho ,\Re )\zeta (\varrho ,\Re )\right)\right\}\right].\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & \phi (\varrho ,\Re )=cos\left(\varrho \right)+S^{-1}\left[u^{\alpha }\left\{S\left(-3\zeta (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )\right)\right\}\right],\hfill \\ \hfill & \zeta (\varrho ,\Re )=cos\left(\varrho \right)+S^{-1}\left[u^{\alpha }\left\{S\left(-2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )-2\phi (\varrho ,\Re ){\zeta }_{\varrho }(\varrho ,\Re )-{\phi }_{\varrho }(\varrho ,\Re )\zeta (\varrho ,\Re )\right)\right\}\right].\hfill \end{array}$$

(34)

Now, in terms of STDM,

$$\begin{array}{cc}\hfill & \phi (\varrho ,\Re )=\sum _{m=0}^{\infty }{\phi }_m(\varrho ,\Re ),and\hfill \\ \hfill & \zeta (\varrho ,\Re )=\sum _{m=0}^{\infty }{\zeta }_m(\varrho ,\Re ),\hfill \end{array}$$

where the nonlinear terms $\zeta {\zeta }_{\varrho }={\sum }_{m=0}^{\infty }{\mathcal{A}}_m$, $\phi {\zeta }_{\varrho }={\sum }_{m=0}^{\infty }{\mathcal{B}}_m$ and ${\phi }_{\varrho }\zeta ={\sum }_{m=0}^{\infty }{\mathcal{C}}_m$ are expressed in terms of Adomian polynomials as

$$\begin{array}{cc}\hfill & \sum _{m=0}^{\infty }{\phi }_m(\varrho ,\Re )=\phi (\varrho ,0)-S^{-1}\left[u^{\alpha }S\left[3\sum _{m=0}^{\infty }{\mathcal{A}}_m\right]\right],\hfill \\ \hfill & \sum _{m=0}^{\infty }{\zeta }_m(\varrho ,\Re )=\zeta (\varrho ,0)-S^{-1}\left[u^{\alpha }S\left[2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )+2\sum _{m=0}^{\infty }{\mathcal{B}}_m+\sum _{m=0}^{\infty }{\mathcal{C}}_m\right]\right],\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \sum _{m=0}^{\infty }{\phi }_m(\varrho ,\Re )=cos\left(\varrho \right)-S^{-1}\left[u^{\alpha }S\left[3\sum _{m=0}^{\infty }{\mathcal{A}}_m\right]\right],\hfill \\ \hfill & \sum _{m=0}^{\infty }{\zeta }_m(\varrho ,\Re )=sin\left(\varrho \right)-S^{-1}\left[u^{\alpha }S\left[2{\zeta }_{\varrho \varrho \varrho }(\varrho ,\Re )+2\sum _{m=0}^{\infty }{\mathcal{B}}_m+\sum _{m=0}^{\infty }{\mathcal{C}}_m\right]\right].\hfill \end{array}$$

(35)

The Adomian polynomials’ first few terms are given by

$$\begin{array}{cc}\hfill & {\mathcal{A}}_0={\zeta }_0{\zeta }_{0\varrho }\hfill \\ \hfill & {\mathcal{A}}_1={\zeta }_1{\zeta }_{0\varrho }+{\zeta }_0{\zeta }_{1\varrho },\hfill \\ \hfill & {\mathcal{A}}_2={\zeta }_2{\zeta }_{0\varrho }+{\zeta }_1{\zeta }_{1\varrho }+{\zeta }_0{\zeta }_{2\varrho },\hfill \\ \hfill & ⋮\hfill \\ \hfill & {\mathcal{B}}_0={\phi }_0{\zeta }_{0\varrho },\hfill \\ \hfill & {\mathcal{B}}_1={\phi }_1{\zeta }_{0\varrho }+{\phi }_0{\zeta }_{1\varrho },\hfill \\ \hfill & {\mathcal{B}}_2={\phi }_2{\zeta }_{0\varrho }+{\phi }_1{\zeta }_{1\varrho }+{\phi }_0{\zeta }_{2\varrho },\hfill \\ \hfill & ⋮\hfill \\ \hfill & {\mathcal{C}}_0={\phi }_{0\varrho }{\zeta }_0,\hfill \\ \hfill & {\mathcal{C}}_1={\phi }_{1\varrho }{\zeta }_0+{\phi }_{0\varrho }{\zeta }_1,\hfill \\ \hfill & {\mathcal{C}}_2={\phi }_{2\varrho }{\zeta }_0+{\phi }_{1\varrho }{\zeta }_1+{\phi }_{0\varrho }{\zeta }_2.\hfill \end{array}$$

The following terms are obtained:

$$\begin{array}{cc}\hfill & {\phi }_0(\varrho ,\Re )=3{sech}^2\left(\varrho \right),\hfill \\ \hfill & {\zeta }_0(\varrho ,\Re )=2sech\left(\varrho \right).\hfill \end{array}$$

For $m=0$,

$$\begin{array}{cc}\hfill & {\phi }_1(\varrho ,\Re )=12{sech}^2\left(\varrho \right)tanh\left(\varrho \right)\frac{{\Re }^{\alpha }}{\mathsf{\Gamma }(\alpha +1)},\hfill \\ \hfill & {\zeta }_1(\varrho ,\Re )=4sech\left(\varrho \right)tanh\left(\varrho \right)\frac{{\Re }^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}.\hfill \end{array}$$

For $m=1$,

$$\begin{array}{cc}\hfill & {\phi }_2(\varrho ,\Re )=24(-3{sech}^4\left(\varrho \right)+2{sech}^2\left(\varrho \right))\frac{{\Re }^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)},\hfill \\ \hfill & {\zeta }_2(\varrho ,\Re )=8(-2{sech}^3\left(\varrho \right)+sech\left(\varrho \right))\frac{{\Re }^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}.\hfill \end{array}$$

Thus, the analytical solution by the truncated series for $(m\ge 2)$ is calculated in the same manner.

$$\begin{array}{cc}\hfill & \phi (\varrho ,\Re )=\sum _{m=0}^{\infty }{\phi }_m(\varrho ,\Re )={\phi }_0(\varrho ,\Re )+{\phi }_1(\varrho ,\Re )+{\phi }_2(\varrho ,\Re )+\cdots \hfill \\ \hfill & \zeta (\varrho ,\Re )=\sum _{m=0}^{\infty }{\zeta }_m(\varrho ,\Re )={\zeta }_0(\varrho ,\Re )+{\zeta }_1(\varrho ,\Re )+\cdots \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \phi (\varrho ,\Re )=3{sech}^2\left(\varrho \right)+12{sech}^2\left(\varrho \right)tanh\left(\varrho \right)\frac{{\Re }^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+24(-3{sech}^4\left(\varrho \right)+2{sech}^2\left(\varrho \right))\frac{{\Re }^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}+\cdots \hfill \\ \hfill & \zeta (\varrho ,\Re )=2sech\left(\varrho \right)+4sech\left(\varrho \right)tanh\left(\varrho \right)\frac{{\Re }^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}+8(-2{sech}^3\left(\varrho \right)+sech\left(\varrho \right))\frac{{\Re }^{2\alpha }}{\mathsf{\Gamma }(2\alpha +1)}+\cdots \hfill \end{array}$$

In Figure 1, (a) show that the exact, (b) represent the analytical solutions and (c) the error plot of $\phi (\varrho ,\Re )$. In Figure 2, (a) show that the exact, (b) represent the analytical solutions and (c) the error plot of $\phi (\varrho ,\Re )$.

6. Conclusions

In conclusion, the nonlinear fractional Drinfeld–Sokolov–Wilson problem encountered in dispersive water waves requires a combination of appropriate numerical techniques, including the Adomian decomposition approach, the homotopy perturbation method, and Yang transform, to obtain an approximative solution. The use of the Caputo manner to describe the fractional derivative and the employment of He’s polynomials and Adomian polynomials help to address the nonlinearity of the problem. By following the steps of these approaches, we can obtain solutions in the form of a convergent series. Our investigation of the assumed model in terms of the fractional order demonstrates the effectiveness of these suggested strategies. We have represented the physical behavior of the suggested technique solutions through plots for various fractional orders and have provided a numerical simulation. The results obtained show that the suggested algorithms are extremely systematic, simple to use, effective, and accurate for analyzing the behavior of coupled nonlinear differential equations of fractional order in related scientific and engineering fields.