Homotopy perturbation method-based soliton solutions of the time-fractional (2+1)-dimensional Wu–Zhang system describing long dispersive gravity water waves in the ocean

1 Introduction

The study of differential equations (DEs) is a pivotal topic as they capture most of the real-world phenomena, i.e., earthquakes [1, 2], natural gas consumption [3, 4], current flow [5], and cooking [6]. These equations can additionally be characterized into linear and non-linear differential equations. Many important and interesting phenomena like electrical circuits [7, 8], DNA sequencing [9, 10], disease modeling and analysis [11, 12], and food chain models [13, 14] are captured through differential equations. Since the order of a DE describes the nature and scope of the captured phenomena, it is therefore important for researchers to cater fractional-order derivatives for a more general study of the physical aspects of the considered phenomena. Fractional models allow better understanding of model dynamics and facilitate researchers to accurately predict changes in the physical systems. The chaos theory [15], nanotechnology [16], fluid flow [17], cosmology [18], and robotics [19] use differential equations for problem formulation. These equations also frequently appear in many branches of mathematics [20, 21], finance [22], economy [23], and biology [24].

The phrase “fractal” was first created in 1975 by mathematician Benoit Mandelbrot [25]. It is a geometric shape that exhibits the same level of non-regularity on all scales. Fractals are infinite patterns, which we frequently see in nature. Snowflakes, trees, mountains, clouds, and coastlines represent fractals as they are highly uneven at both large and small scales. Many important models including the diffusion model of red ink [26] and thin films [27], the vibration model for a concrete beam [28] and electronic devices [29], and the COVID-19 mathematical model [30] contain fractal geometry. The distinction between fractional and fractal is that the former is a statement of a fractional number, while the latter is a geometric figure that is similar at all scales.

The Wu–Zhang system [31] contains non-linear partial differential equations (PDEs) and deals with the motion of water waves in oceans. In 1996, three sets of model equations were first derived by Wu and Zhang and named the Wu–Zhang system of PDEs [31]. This system is used to customize several harbor and coastal designs. This non-linear (2 + 1)-dimensional fractional system describes shallow water dispersive long gravity waves in two horizontal directions, which are given as

$\begin{array}{cc}\hfill & \frac{{\partial }^{\zeta }\mathcal{U}}{\partial {\mathfrak{t}}^{\zeta }}+\mathcal{U}\frac{\partial \mathcal{U}}{\partial x}+\mathcal{V}\frac{\partial \mathcal{U}}{\partial y}+\frac{\partial \mathcal{W}}{\partial x}=0,\hfill \\ \hfill & \frac{{\partial }^{\zeta }\mathcal{V}}{\partial {\mathfrak{t}}^{\zeta }}+\mathcal{U}\frac{\partial \mathcal{V}}{\partial x}+\mathcal{V}\frac{\partial \mathcal{V}}{\partial y}+\frac{\partial \mathcal{W}}{\partial y}=0,\hfill \\ \hfill & \frac{{\partial }^{\zeta }\mathcal{W}}{\partial {\mathfrak{t}}^{\zeta }}+\frac{\partial \left(\mathcal{U}\mathcal{W}\right)}{\partial x}+\frac{\partial \left(\mathcal{V}\mathcal{W}\right)}{\partial y}+\frac{1}{3}\left(\frac{{\partial }^3\mathcal{U}}{\partial x^3}+\frac{{\partial }^3\mathcal{U}}{\partial x\partial y^2}+\frac{{\partial }^3\mathcal{V}}{\partial x^2\partial y}+\frac{{\partial }^3\mathcal{V}}{\partial y^3}\right)=0,\hfill \end{array}(1)$

where $\mathcal{U}$ and $\mathcal{V}$ represent the velocities at the surface of water in x and y directions, while $\mathcal{W}$ depicts the elevation of water waves. The aforementioned WZ system is a time fraction, while Wang and He [32] concluded that when time is fractional, space must also be fractional. This is called Wang–He’s spatiotemporal fractional relationship (for more details see [32]). Due to the substantial importance of WZ systems, many scholars have attempted to solve and analyze these systems through variety of methodologies like mVIM [33], ADM [34, 35], extended tanh and exp–function method [36], and dynamical analysis method [37]. Recently, for more generalized solutions and predictions, the WZ systems are also attempted fractionally by few of the scientists. Kaur and Gupta discussed dispersion analysis of the (2 + 1)-dimensional time-fractional WZ system [38]. Patel and Patel investigated the fractional-order WZ system analytically [39]. Different approaches of fractional derivatives can be utilized, such as Caputo [40], Atangana–Baleanu [41], Caputo–Fabrizio [42], and He’s fractional derivative [43].

In order to solve such highly non-linear fractional systems, many analytical and numerical methodologies are utilized by different researchers. Anjum et al. [44] applied Li–He’s modified homotopy perturbation approach to solve the microelectromechanical system. Baitiche et al. [45] used the monotone iterative method for fractional DEs with non-linearity at the boundary. Do et al. [46] extended Chebyshev wavelets to two-dimensional fractional DEs. Hashemi et al. [47] investigated multi-term FDEs using minimization techniques. Tian and Liu utilized the modified exp-function to fractional PDEs in [48]. Furthermore, to solve complex problems, the enhanced homotopy methods can be found in [49, 50]. In this study, a hybrid algorithm is proposed by mixing the classical homotopy perturbation method [51, 52] with the Laplace transform [53] along with different fractional derivatives (Atangana–Baleanu, Caputo–Fabrizio, and Caputo) for a highly non-linear time-fractional (2 + 1)-dimensional WZ system. In the rest of the paper, Section 2 contains preliminary definitions. Section 3 contains the proposed methodology for handling time-fractional (2 + 1)-dimensional WZ system, whereas proof of convergence and error analysis are given in Section 4. Solution and results and discussion are given in Sections 5 and 6, respectively, while a conclusion is given in Section 7.

2 Basic definitions

Definition 1: For a function $\mathcal{U}(t,x,y)$, the Caputo’s time-fractional derivative ${}^C\mathcal{D}_t^{\zeta }$ is [54]

${}^C\mathcal{D}_t^{\zeta }\mathcal{U}\left(t,x,y\right)=\frac{1}{\mathrm{\Gamma }\left(q-\zeta \right)}{\int }_0^t{\left(t-G\right)}^{q-\zeta -1}{\mathcal{U}}^{\left(q\right)}\left(G,x,y\right)dG,q-1<\zeta \le q.(2)$

Definition 2: According to [55], one can express the Laplace transform L of the function $\mathcal{U}(t,x,y)$ that has been subjected to the Caputo’s time-fractional derivative ${}^C\mathcal{D}_t^{\zeta }$.

$\mathbf{L}\left\{{}^C\mathcal{D}_t^{\zeta }\mathcal{U}\left(t,x,y\right)\right\}=s^{\zeta }\mathbf{L}\left\{\mathcal{U}\left(t,x,y\right)\right\}-{\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}{\mathcal{U}}^{\left(p\right)}\left(0,x,y\right),q-1<\zeta \le q.(3)$

Definition 3: The Caputo–Fabrizio’s time-fractional derivative ${}^{CF}\mathcal{D}_t^{\zeta }$ of a function $\mathcal{U}(t,x,y)$ is [42]

${}^{CF}\mathcal{D}_t^{\zeta }\mathcal{U}\left(t,x,y\right)=\frac{1}{1-\zeta }{\int }_0^t{e}^{\frac{-\zeta \left(t-G\right)}{1-\zeta }}\frac{\partial \mathcal{U}\left(G,x,y\right)}{\partial G}dG,0<\zeta <1.(4)$

Definition 4: The Laplace transform L of the Caputo–Fabrizio’s time-fractional derivative ${}^{CF}\mathcal{D}_t^{\zeta }$ of a function $\mathcal{U}(t,x,y)$ is given as [56]

$\mathbf{L}\left\{{}^{CF}\mathcal{D}_t^{\zeta +q}\mathcal{U}\left(t,x,y\right)\right\}=\frac{s^{q+1}\mathbf{L}\left\{\mathcal{U}\left(t,x,y\right)\right\}-{\displaystyle \sum _{p=0}^q}{s}^{q-p}{\mathcal{U}}^{\left(p\right)}\left(0,x,y\right)}{s+\zeta \left(1-s\right)},0<\zeta \le 1.(5)$

Definition 5: A function $\mathcal{U}(t,x,y)$ in the sense of Atangana–Baleanu’s time-fractional derivative ${}^{AB}\mathcal{D}_t^{\zeta }$ is stated as [41]

${}^{AB}\mathcal{D}_t^{\zeta }\mathcal{U}\left(t,x,y\right)=\frac{\mathbb{K}\left(\zeta \right)}{1-\zeta }{\int }_0^t{E}_{\zeta }\left[-\frac{\zeta {\left(t-G\right)}^{\zeta }}{1-\zeta }\right]\frac{\partial \mathcal{U}\left(G,x,y\right)}{\partial G}dG,0<\zeta \le 1.(6)$

Here, $\mathbb{K}(\zeta )$ is a normalization function with properties $\mathbb{K}(0)$ = $\mathbb{K}(1)$ = 1.

Definition 6: The Laplace transform L connected with Atangana–Baleanu time-fractional derivative ${}^{AB}\mathcal{D}_t^{\zeta }$ of a function $\mathcal{U}(t,x,y)$ can be described as [57]

$\mathbf{L}\left\{{}^{AB}\mathcal{D}_t^{\zeta }\mathcal{U}\left(t,x,y\right)\right\}=AB\left(\zeta \right).\frac{s^{\zeta }\mathbf{L}\left\{\mathcal{U}\left(t,x,y\right)\right\}-s^{\zeta -1}\mathcal{U}\left(0,x,y\right)}{s^{\zeta }\left(1-\zeta \right)+\zeta },0\le \zeta \le 1.(7)$

Here, AB(ζ) is a normalization function.

Definition 7: He’s fractional derivative of a function $\mathcal{U}(t,x,y)$ can be defined by [43]

${\mathcal{D}}_t^{\zeta }\mathcal{U}\left(t,x,y\right)=\frac{1}{\mathrm{\Gamma }\left(q-\zeta \right)}\frac{d^q}{d{t}^q}{\int }_{t_0}^t{\left(G-t\right)}^{q-\zeta -1}\left[{\mathcal{U}}_0\left(G,x,y\right)-\mathcal{U}\left(G,x,y\right)\right]dG,q-1<\zeta \le q.(8)$

Definition 8: The core idea behind the two-scale dimension [58, 59], which commonly arises in the non-linear problem, is that while self-similarity is difficult to uncover in practical applications, fractal structures self-assemble on all scales. Creating models with the two-scale dimension allows for the successful description of various physical events.

Definition 9: A Banach space $\mathbb{B}$ is a normed space ‖. ∥, which is complete with respect to the metric derived from its norm.

3 Hybrid algorithm for (2 + 1)-dimensional time-fractional systems

Consider a (2 + 1)-dimensional, time-fractional system as

$\begin{array}{cc}\hfill & {\mathcal{D}}_t^{\zeta }\mathcal{A}1\left(t,x,y\right)+\mathcal{L}\left[\mathcal{A}r\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}r\left(t,x,y\right)\right]-\mathfrak{l}\left(t,x,y\right)=0,\hfill \\ \hfill & {\mathcal{D}}_t^{\zeta }\mathcal{A}2\left(t,x,y\right)+\mathcal{L}\left[\mathcal{A}r\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}r\left(t,x,y\right)\right]-\mathfrak{m}\left(t,x,y\right)=0,\hfill \\ \hfill & {\mathcal{D}}_t^{\zeta }\mathcal{A}3\left(t,x,y\right)+\mathcal{L}\left[\mathcal{A}r\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}r\left(t,x,y\right)\right]-\mathfrak{n}\left(t,x,y\right)=0,\hfill \\ \hfill & r=\mathrm{1,2,3},t>0,\hfill \\ \hfill & q-1<\zeta \le q,\hfill \end{array}(9)$

that has initial conditions

$\begin{array}{c}\hfill \mathcal{A}1\left(0,x,y\right)=\mathcal{J}1,\\ \hfill \mathcal{A}2\left(0,x,y\right)=\mathcal{J}2,\\ \hfill \mathcal{A}3\left(0,x,y\right)=\mathcal{J}3,\end{array}(10)$

where the unknown functions $\mathcal{A}1(t,x,y)$, $\mathcal{A}2(t,x,y)$, and $\mathcal{A}3(t,x,y)$ have time-fractional derivatives, and ${\mathcal{D}}_t^{\zeta }$, $\mathfrak{l}(t,x,y)$, $\mathfrak{m}(t,x,y)$, and $\mathfrak{n}(t,x,y)$ are some of its known functions. The symbols $\mathcal{N}$ and $\mathcal{L}$ represent non-linear and linear operators, respectively.

The procedure will start by applying the Laplace transform on (9), which gives

$\begin{array}{cc}\hfill & \mathcal{L}\left\{{\mathcal{D}}_t^{\zeta }\left[\mathcal{A}1\left(t,x,y\right)\right]\right\}+\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}r\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}r\left(t,x,y\right)\right]-\mathfrak{l}\left(t,x,y\right)\right\}=0,\hfill \\ \hfill & \mathcal{L}\left\{{\mathcal{D}}_t^{\zeta }\left[\mathcal{A}2\left(t,x,y\right)\right]\right\}+\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}r\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}r\left(t,x,y\right)\right]-\mathfrak{m}\left(t,x,y\right)\right\}=0,\hfill \\ \hfill & \mathcal{L}\left\{{\mathcal{D}}_t^{\zeta }\left[\mathcal{A}3\left(t,x,y\right)\right]\right\}+\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}r\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}r\left(t,x,y\right)\right]-\mathfrak{n}\left(t,x,y\right)\right\}=0.\hfill \end{array}(11)$

Now, by utilizing the basic definitions given in Section 2, we can find the Laplace transform of the fractional derivative. Definition (2) gives

$\begin{array}{c}\hfill \mathcal{L}\left[\mathcal{A}1\left(t,x,y\right)\right]-\left(\frac{1}{s^{\zeta }}\right){\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}\mathcal{A}{1}^{\left(p\right)}\left(0,x,y\right)+\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}r\left(t,x,y\right)\right]\right.\\ \hfill \left.+\mathcal{N}\left[\mathcal{A}r\left(t,x,y\right)\right]-\mathfrak{l}\left(t,x,y\right)\right\}=0,\\ \hfill \mathcal{L}\left[\mathcal{A}2\left(t,x,y\right)\right]-\left(\frac{1}{s^{\zeta }}\right){\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}\mathcal{A}{2}^{\left(p\right)}\left(0,x,y\right)+\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}r\left(t,x,y\right)\right]\right.\\ \hfill \left.+\mathcal{N}\left[\mathcal{A}r\left(t,x,y\right)\right]-\mathfrak{m}\left(t,x,y\right)\right\}=0,\\ \hfill \mathcal{L}\left[\mathcal{A}3\left(t,x,y\right)\right]-\left(\frac{1}{s^{\zeta }}\right){\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}\mathcal{A}{3}^{\left(p\right)}\left(0,x,y\right)+\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}r\left(t,x,y\right)\right]\right.\\ \hfill \left.+\mathcal{N}\left[\mathcal{A}r\left(t,x,y\right)\right]-\mathfrak{n}\left(t,x,y\right)\right\}=0.\end{array}(12)$

The homotopy of the system is

$\begin{array}{c}\hfill {\mathcal{H}}_1=\left(1-s\right)\left(\mathcal{L}\left\{\mathcal{A}1\left(t,x,y\right)\right\}-\mathcal{A}{1}_0\left(t,x,y\right)\right)+s(\mathcal{L}\left\{\mathcal{A}1\left(t,x,y\right)\right\}-\left(\frac{1}{s^{\zeta }}\right){\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}\\ \hfill \left.\mathcal{A}{1}^{\left(p\right)}\left(0,x,y\right)+\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}r\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}r\left(t,x,y\right)\right]-\mathfrak{l}\left(t,x,y\right)\right\}\right),\\ \hfill {\mathcal{H}}_2=\left(1-s\right)\left(\mathcal{L}\left\{\mathcal{A}2\left(t,x,y\right)\right\}-\mathcal{A}{2}_0\left(t,x,y\right)\right)+s(\mathcal{L}\left\{\mathcal{A}2\left(t,x,y\right)\right\}-\left(\frac{1}{s^{\zeta }}\right){\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}\\ \hfill \left.\mathcal{A}{2}^{\left(p\right)}\left(0,x,y\right)+\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}r\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}r\left(t,x,y\right)\right]-\mathfrak{m}\left(t,x,y\right)\right\}\right),\\ \hfill {\mathcal{H}}_3=\left(1-s\right)\left(\mathcal{L}\left\{\mathcal{A}3\left(t,x,y\right)\right\}-\mathcal{A}{3}_0\left(t,x,y\right)\right)+s(\mathcal{L}\left\{\mathcal{A}3\left(t,x,y\right)\right\}-\left(\frac{1}{s^{\zeta }}\right){\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}\\ \hfill \left.\mathcal{A}{3}^{\left(p\right)}\left(0,x,y\right)+\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}r\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}r\left(t,x,y\right)\right]-\mathfrak{n}\left(t,x,y\right)\right\}\right),\end{array}(13)$

where $\mathcal{A}{1}_0$, $\mathcal{A}{2}_0$, and $\mathcal{A}{3}_0$ are initial guesses. Expansion of $\mathcal{A}1(t,x,y)$, $\mathcal{A}2(t,x,y)$, and $\mathcal{A}3(t,x,y)$ in power series with respect to s leads to

$\begin{array}{c}\hfill \mathcal{A}1\left(t,x,y\right)=\mathcal{A}{1}_0\left(t,x,y\right)+s^1\mathcal{A}{1}_1\left(t,x,y\right)+s^2\mathcal{A}{1}_2\left(t,x,y\right)+\dots \\ \hfill \mathcal{A}2\left(t,x,y\right)=\mathcal{A}{2}_0\left(t,x,y\right)+s^1\mathcal{A}{2}_1\left(t,x,y\right)+s^2\mathcal{A}{2}_2\left(t,x,y\right)+\dots \\ \hfill \mathcal{A}3\left(t,x,y\right)=\mathcal{A}{3}_0\left(t,x,y\right)+s^1\mathcal{A}{3}_1\left(t,x,y\right)+s^2\mathcal{A}{3}_2\left(t,x,y\right)+\dots \end{array}(14)$

After substituting Eq. 14 in (13) and then comparing similar coefficients of s, we obtainAt s¹

$\begin{array}{c}\hfill \mathcal{L}\left\{\mathcal{A}{1}_1\left(t,x,y\right)\right\}+\mathcal{A}{1}_0\left(t,x,y\right)-\left(\frac{1}{s^{\zeta }}\right){\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}\mathcal{A}{1}^{\left(p\right)}\left(0,x,y\right)+\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}{r}_0\left(t,x,y\right)\right]\right.\\ \hfill \left.+\mathcal{N}\left[\mathcal{A}{r}_0\left(t,x,y\right)\right]-\mathfrak{l}\left(t,x,y\right)\right\}=0,\\ \hfill \mathcal{L}\left\{\mathcal{A}{2}_1\left(t,x,y\right)\right\}+\mathcal{A}{2}_0\left(t,x,y\right)-\left(\frac{1}{s^{\zeta }}\right){\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}\mathcal{A}{2}^{\left(p\right)}\left(0,x,y\right)+\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}{r}_0\left(t,x,y\right)\right]\right.\\ \hfill \left.+\mathcal{N}\left[\mathcal{A}{r}_0\left(t,x,y\right)\right]-\mathfrak{m}\left(t,x,y\right)\right\}=0,\\ \hfill \mathcal{L}\left\{\mathcal{A}{3}_1\left(t,x,y\right)\right\}+\mathcal{A}{3}_0\left(t,x,y\right)-\left(\frac{1}{s^{\zeta }}\right){\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}\mathcal{A}{3}^{\left(p\right)}\left(0,x,y\right)+\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}{r}_0\left(t,x,y\right)\right]\right.\\ \hfill \left.+\mathcal{N}\left[\mathcal{A}{r}_0\left(t,x,y\right)\right]-\mathfrak{n}\left(t,x,y\right)\right\}=0.\end{array}(15)$

The inverse Laplace transform leads to

$\begin{array}{c}\hfill \mathcal{A}{1}_1\left(t,x,y\right){+\mathcal{L}}^{-1}\left\{\mathcal{A}{1}_0\left(t,x,y\right)-\left(\frac{1}{s^{\zeta }}\right){\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}\mathcal{A}{1}^{\left(p\right)}\left(0,x,y\right)\right\}{+\mathcal{L}}^{-1}\left\{\left(\frac{1}{s^{\zeta }}\right)\right.\\ \hfill \mathcal{L}\{\mathcal{L}\left[\mathcal{A}{r}_0\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}{r}_0\left(t,x,y\right)\right]-\mathfrak{l}\left(t,x,y\right)\}=0,\\ \hfill \mathcal{A}{2}_1\left(t,x,y\right){+\mathcal{L}}^{-1}\left\{\mathcal{A}{2}_0\left(t,x,y\right)-\left(\frac{1}{s^{\zeta }}\right){\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}\mathcal{A}{2}^{\left(p\right)}\left(0,x,y\right)\right\}{+\mathcal{L}}^{-1}\left\{\left(\frac{1}{s^{\zeta }}\right)\right.\\ \hfill \mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}{r}_0\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}{r}_0\left(t,x,y\right)\right]-\mathfrak{m}\left(t,x,y\right)\right.\}=0,\\ \hfill \mathcal{A}{3}_1\left(t,x,y\right){+\mathcal{L}}^{-1}\left\{\mathcal{A}{3}_0\left(t,x,y\right)-\left(\frac{1}{s^{\zeta }}\right){\displaystyle \sum _{p=0}^{q-1}}{s}^{\zeta -p-1}\mathcal{A}{3}^{\left(p\right)}\left(0,x,y\right)\right\}{+\mathcal{L}}^{-1}\left\{\left(\frac{1}{s^{\zeta }}\right)\right.\\ \hfill \mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}{r}_0\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}{r}_0\left(t,x,y\right)\right]-\mathfrak{n}\left(t,x,y\right)\right.\}=0.\end{array}(16)$

At s^k

$\begin{array}{c}\hfill \mathcal{L}\left\{\mathcal{A}{1}_k\left(t,x,y\right)\right\}+\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}{r}_{k-1}\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}{r}_{k-1}\left(t,x,y\right)\right]\right\}=0,\\ \hfill \mathcal{L}\left\{\mathcal{A}{2}_k\left(t,x,y\right)\right\}+\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}{r}_{k-1}\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}{r}_{k-1}\left(t,x,y\right)\right]\right\}=0,\\ \hfill \mathcal{L}\left\{\mathcal{A}{3}_k\left(t,x,y\right)\right\}+\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}{r}_{k-1}\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}{r}_{k-1}\left(t,x,y\right)\right]\right\}=0.\end{array}(17)$

Operating the inverse Laplace transform gives the following:

$\begin{array}{c}\hfill \mathcal{A}{1}_k\left(t,x,y\right){+\mathcal{L}}^{-1}\left\{\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}{r}_{k-1}\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}{r}_{k-1}\left(t,x,y\right)\right]\right\}\right\}=0,\\ \hfill \mathcal{A}{2}_k\left(t,x,y\right){+\mathcal{L}}^{-1}\left\{\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}{r}_{k-1}\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}{r}_{k-1}\left(t,x,y\right)\right]\right\}\right\}=0,\\ \hfill \mathcal{A}{3}_k\left(t,x,y\right){+\mathcal{L}}^{-1}\left\{\left(\frac{1}{s^{\zeta }}\right)\mathcal{L}\left\{\mathcal{L}\left[\mathcal{A}{r}_{k-1}\left(t,x,y\right)\right]+\mathcal{N}\left[\mathcal{A}{r}_{k-1}\left(t,x,y\right)\right]\right\}\right\}=0.\end{array}(18)$

The approximate solution of the given general time-fractional, (2 + 1)-dimensional PDE system is

$\begin{array}{cc}\hfill \tilde{\mathcal{A}}1& =\mathcal{A}{1}_0\left(t,x,y\right)+\mathcal{A}{1}_1\left(t,x,y\right)+\mathcal{A}{1}_2\left(t,x,y\right)+\mathcal{A}{1}_3\left(t,x,y\right)+\cdots ,\hfill \\ \hfill \tilde{\mathcal{A}}2& =\mathcal{A}{2}_0\left(t,x,y\right)+\mathcal{A}{2}_1\left(t,x,y\right)+\mathcal{A}{2}_2\left(t,x,y\right)+\mathcal{A}{1}_3\left(t,x,y\right)+\cdots ,\hfill \\ \hfill \tilde{\mathcal{A}}3& =\mathcal{A}{3}_0\left(t,x,y\right)+\mathcal{A}{3}_1\left(t,x,y\right)+\mathcal{A}{3}_2\left(t,x,y\right)+\mathcal{A}{3}_3\left(t,x,y\right)+\cdots .\hfill \end{array}(19)$

Residual errors of the system are

$\begin{array}{cc}\hfill \mathfrak{R}\mathfrak{e}\mathfrak{s}1& ={\mathcal{D}}_t^{\zeta }\left[\tilde{\mathcal{A}}1\right]+\mathcal{L}\left[\tilde{\mathcal{A}}r\right]+\mathcal{N}\left[\tilde{\mathcal{A}}r\right]-\mathfrak{l}\left(t,x,y\right),\hfill \\ \hfill \mathfrak{R}\mathfrak{e}\mathfrak{s}2& ={\mathcal{D}}_t^{\zeta }\left[\tilde{\mathcal{A}}2\right]+\mathcal{L}\left[\tilde{\mathcal{A}}r\right]+\mathcal{N}\left[\tilde{\mathcal{A}}r\right]-\mathfrak{m}\left(t,x,y\right),\hfill \\ \hfill \mathfrak{R}\mathfrak{e}\mathfrak{s}3& ={\mathcal{D}}_t^{\zeta }\left[\tilde{\mathcal{A}}3\right]+\mathcal{L}\left[\tilde{\mathcal{A}}r\right]+\mathcal{N}\left[\tilde{\mathcal{A}}r\right]-\mathfrak{n}\left(t,x,y\right).\hfill \end{array}(20)$

The same procedure can be extended to a system that comprises more than three equations.

4 Convergence and error analysis of the hybrid algorithm for (2 + 1)-dimensional fractional systems

4.1 Convergence

Theorem 1: If a Banach space has $\mathcal{A}{r}_n(t,x,y)$ and $\mathcal{A}r(t,x,y)$ defined in it for r = 1, 2, 3, then, the series solution of a fractional (2 + 1)-D system in Eq. 19 converges to the solution of (9) for a constant μ ϵ (0,1).

Proof: Let us define the sequence of partial sums of Eq. 19 as Qr_n. To demonstrate that Qr_n(t, x, y) forms a Cauchy sequence in the Banach space, we can proceed by using

$\begin{array}{cc}\hfill \Vert Q{r}_{n+1}\left(t,x,y\right)-Q{r}_n\left(t,x,y\right)\Vert & =\Vert \mathcal{A}{r}_{n+1}\left(t,x,y\right)\Vert \hfill \\ \hfill & \le \mu \Vert \mathcal{A}{r}_n\left(t,x,y\right)\Vert \hfill \\ \hfill & \le {\mu }^2\Vert \mathcal{A}{r}_{n-1}\left(t,x,y\right)\Vert \hfill \\ \hfill & \le \dots \le {\mu }^{n+1}\Vert \mathcal{A}{r}_0\left(t,x,y\right)\Vert .\hfill \end{array}(21)$

If Qr_n and Qr_m are partial sums with n ≥ m and n, m ϵ N, then utilization of triangle inequality gives

$\begin{array}{c}\hfill \Vert Q{r}_n-Q{r}_m\Vert =\Vert \left(Q{r}_n\left(t,x,y\right)-Q{r}_{n-1}\left(t,x,y\right)\right)+\left(Q{r}_{n-1}\left(t,x,y\right)-Q{r}_{n-2}\left(t,x,y\right)\right)\\ \hfill +\cdots +\left(Q{r}_{m+1}\left(t,x,y\right)-Q{r}_m\left(t,x,y\right)\right)\Vert \\ \hfill \le \Vert Q{r}_n\left(t,x,y\right)-Q{r}_{n-1}\left(t,x,y\right)\Vert +\Vert Q{r}_{n-1}\left(t,x,y\right)-Q{r}_{n-2}\left(t,x,y\right)\Vert \\ \hfill +\cdots +\Vert Q{r}_{m+1}\left(t,x,y\right)-Q{r}_m\left(t,x,y\right)\Vert .\end{array}(22)$

From Eq. 21, we get

$\begin{array}{cc}\hfill \Vert Q{r}_n-Q{r}_m\Vert & \le {\mu }^n\Vert \mathcal{A}{r}_0\left(t,x,y\right)\Vert +{\mu }^{n-1}\Vert \mathcal{A}{r}_0\left(t,x,y\right)\Vert +\cdots +{\mu }^{m+1}\Vert \mathcal{A}{r}_0\left(t,x,y\right)\Vert \hfill \\ \hfill & \le \left({\mu }^n+{\mu }^{n-1}+\cdots +{\mu }^{m+1}\right)\Vert \mathcal{A}{r}_0\left(t,x,y\right)\Vert \hfill \\ \hfill & \le {\mu }^{m+1}\left({\mu }^{n-m-1}+{\mu }^{n-m-2}+\cdots +\mu +1\right)\Vert \mathcal{A}{r}_0\left(t,x,y\right)\Vert \hfill \\ \hfill & \le {\mu }^{m+1}\left(\frac{1-{\mu }^{n-m}}{1-\mu }\right)\Vert \mathcal{A}{r}_0\left(t,x,y\right)\Vert .\hfill \end{array}(23)$

Given 0 $<\mu <$ 1, hence, 1 − μ^n−m < 1. Thus, we have

$\begin{array}{c}\hfill \Vert Q{r}_n-Q{r}_m\Vert \le \frac{{\mu }^{m+1}}{1-\mu }\mathit{max}|\mathcal{A}{r}_0\left(t,x,y\right)|,\forall tϵ\left[0,T\right].\end{array}(24)$

Since $\mathcal{A}{r}_0$ is bounded, so

$\underset{n,m\to \infty }{\lim }\Vert Q{r}_n\left(t,x,y\right)-Q{r}_m\left(t,x,y\right)\Vert =0.(25)$

Thus, Qr_n(t, x, y) is a Cauchy sequence in the Banach space, and hence, the given statement is proved.

4.2 Error estimation

Theorem 1: One can determine the maximum absolute truncation error of the solution (19) for a fractional (2 + 1)-dimensional system (9) by using the following expression:

$\left|\mathcal{A}r\left(t,x,y\right)-{\displaystyle \sum _{j=0}^m}\mathcal{A}{r}_j\left(t,x,y\right)\right|\le \frac{{\mu }^{m+1}}{1-\mu }\Vert \mathcal{A}{r}_0\left(t,x,y\right)\Vert .(26)$

Proof: From Eq. 23, we have

$\Vert \mathcal{A}r\left(t,x,y\right)-Q{r}_m\Vert \le {\mu }^{m+1}\left(\frac{1-{\mu }^{n-m}}{1-\mu }\right)\Vert \mathcal{A}{r}_0\left(t,x,y\right)\Vert .(27)$

Since 0 $<\mu <$ 1, therefore, 1 − μ^n−m < 1. Thus, we have

$\left|\mathcal{A}r\left(t,x,y\right)-{\displaystyle \sum _{j=0}^m}\mathcal{A}{r}_j\left(t,x,y\right)\right|\le \frac{{\mu }^{m+1}}{1-\mu }\Vert \mathcal{A}{r}_0\left(t,x,y\right)\Vert .(28)$

5 Solution and analysis of the time-fractional Wu–Zhang system

Consider the following coupled time-fractional (2 + 1)-dimensional WZ system [39]:

$\begin{array}{cc}\hfill & \frac{{\partial }^{\zeta }\mathcal{U}}{\partial {\mathfrak{t}}^{\zeta }}+\mathcal{U}\frac{\partial \mathcal{U}}{\partial x}+\mathcal{V}\frac{\partial \mathcal{U}}{\partial y}+\frac{\partial \mathcal{W}}{\partial x}=0,\hfill \\ \hfill & \frac{{\partial }^{\zeta }\mathcal{V}}{\partial {\mathfrak{t}}^{\zeta }}+\mathcal{U}\frac{\partial \mathcal{V}}{\partial x}+\mathcal{V}\frac{\partial \mathcal{V}}{\partial y}+\frac{\partial \mathcal{W}}{\partial y}=0,\hfill \\ \hfill & \frac{{\partial }^{\zeta }\mathcal{W}}{\partial {\mathfrak{t}}^{\zeta }}+\frac{\partial \left(\mathcal{U}\mathcal{W}\right)}{\partial x}+\frac{\partial \left(\mathcal{V}\mathcal{W}\right)}{\partial y}+\frac{1}{3}\left(\frac{{\partial }^3\mathcal{U}}{\partial x^3}+\frac{{\partial }^3\mathcal{U}}{\partial x\partial y^2}+\frac{{\partial }^3\mathcal{V}}{\partial x^2\partial y}+\frac{{\partial }^3\mathcal{V}}{\partial y^3}\right)=0,0<\zeta \le 1\hfill \end{array}(29)$

that has the initial conditions

$\begin{array}{cc}\hfill \mathcal{U}\left(0,x,y\right)& =-\frac{d+ac}{b}+\frac{2\sqrt{3}}{3}b\tanh \left(bx+cy\right),\hfill \\ \hfill \mathcal{V}\left(0,x,y\right)& =a+\frac{2\sqrt{3}}{3}c\tanh \left(bx+cy\right),\hfill \\ \hfill \mathcal{W}\left(0,x,y\right)& =\frac{2}{3}\left(b^2+c^2\right){\text{sech}}^2\left(bx+cy\right),\hfill \end{array}(30)$

where $\mathcal{U}$ and $\mathcal{V}$ represent the velocity at the surface of water in the x and y directions, respectively, and $\mathcal{W}$ depicts the elevation of the water waves. a, b, c, and d are the non-zero arbitrary constants. The exact solution of (29) at ζ = 1 is

$\begin{array}{cc}\hfill \mathcal{U}\left(t,x,y\right)& =-\frac{d+ac}{b}+\frac{2\sqrt{3}}{3}b\tanh \left(bx+cy+dt\right),\hfill \\ \hfill \mathcal{V}\left(t,x,y\right)& =a+\frac{2\sqrt{3}}{3}c\tanh \left(bx+cy+dt\right),\hfill \\ \hfill \mathcal{W}\left(t,x,y\right)& =\frac{2}{3}\left(b^2+c^2\right){\text{sech}}^2\left(bx+cy+dt\right).\hfill \end{array}(31)$

Solution: The initial step of the He–Laplace procedure is the application of the Laplace transform on both sides of Eq. 29, which gives

$\begin{array}{cc}\hfill & \mathbf{L}\left\{\frac{{\partial }^{\zeta }\mathcal{U}}{\partial {\mathfrak{t}}^{\zeta }}\right\}+\mathbf{L}\left\{\mathcal{U}\frac{\partial \mathcal{U}}{\partial x}+\mathcal{V}\frac{\partial \mathcal{U}}{\partial y}+\frac{\partial \mathcal{W}}{\partial x}\right\}=0,\hfill \\ \hfill & \mathbf{L}\left\{\frac{{\partial }^{\zeta }\mathcal{V}}{\partial {\mathfrak{t}}^{\zeta }}\right\}+\mathbf{L}\left\{\mathcal{U}\frac{\partial \mathcal{V}}{\partial x}+\mathcal{V}\frac{\partial \mathcal{V}}{\partial y}+\frac{\partial \mathcal{W}}{\partial y}\right\}=0,\hfill \\ \hfill & \mathbf{L}\left\{\frac{{\partial }^{\zeta }\mathcal{W}}{\partial {\mathfrak{t}}^{\zeta }}\right\}+\mathbf{L}\left\{\frac{\partial \left(\mathcal{U}\mathcal{W}\right)}{\partial x}+\frac{\partial \left(\mathcal{V}\mathcal{W}\right)}{\partial y}+\frac{1}{3}\left(\frac{{\partial }^3\mathcal{U}}{\partial x^3}+\frac{{\partial }^3\mathcal{U}}{\partial x\partial y^2}+\frac{{\partial }^3\mathcal{V}}{\partial x^2\partial y}+\frac{{\partial }^3\mathcal{V}}{\partial y^3}\right)\right\}=0.\hfill \end{array}(32)$

Utilization of the Laplace transform on the Caputo’s time-fractional derivative (2) leads to

$\begin{array}{ll}\hfill \mathbf{L}\left\{\mathcal{U}\left(t,x,y\right)\right\}& -\frac{1}{s}\left(-\frac{d+ac}{b}+\frac{2\sqrt{3}}{3}b\tanh \left(bx+cy\right)\right)+\left(\frac{1}{s^{\zeta }}\right)\mathbf{L}\left\{\mathcal{U}\frac{\partial \mathcal{U}}{\partial x}+\mathcal{V}\frac{\partial \mathcal{U}}{\partial y}\right.\left.+\frac{\partial \mathcal{W}}{\partial x}\right\}=0,\mathbf{L}\left\{\mathcal{V}\left(t,x,y\right)\right\}\hfill \\ \hfill & -\frac{1}{s}\left(a+\frac{2\sqrt{3}}{3}c\tanh \left(bx+cy\right)\right)+\left(\frac{1}{s^{\zeta }}\right)\mathbf{L}\left\{\mathcal{U}\frac{\partial \mathcal{V}}{\partial x}+\mathcal{V}\frac{\partial \mathcal{V}}{\partial y}+\frac{\partial \mathcal{W}}{\partial y}\right\}=0,\mathbf{L}\left\{\mathcal{W}\left(t,x,y\right)\right\}\hfill \\ \hfill & -\frac{1}{s}\left(\frac{2}{3}\left(b^2+c^2\right){\text{sech}}^2\left(bx+cy\right)\right)+\left(\frac{1}{s^{\zeta }}\right)\times \mathbf{L}\left\{\frac{\partial \left(\mathcal{U}\mathcal{W}\right)}{\partial x}+\frac{\partial \left(\mathcal{V}\mathcal{W}\right)}{\partial y}+\frac{1}{3}\right.\left.\left(\frac{{\partial }^3\mathcal{U}}{\partial x^3}+\frac{{\partial }^3\mathcal{U}}{\partial x\partial y^2}+\frac{{\partial }^3\mathcal{V}}{\partial x^2\partial y}+\frac{{\partial }^3\mathcal{V}}{\partial y^3}\right)\right\}=0.\hfill \end{array}(33)$

We construct homotopies of the aforementioned system as

$\begin{array}{c}{H}_1=\left(1-p\right)\left(\mathbf{L}\left\{\mathcal{U}\left(t,x,y\right)\right\}-{\mathcal{U}}_0\left(t,x,y\right)\right)+p\left(\mathbf{L}\left\{\mathcal{U}\left(t,x,y\right)\right\}-\frac{1}{s}\left(-\frac{d+ac}{b}+\frac{2\sqrt{3}}{3}\right.\right.\hfill \\ \left.b\tanh \left(bx+cy\right))+\left(\frac{1}{s^{\zeta }}\right)\mathbf{L}\left\{\mathcal{U}\frac{\partial \mathcal{U}}{\partial x}+\mathcal{V}\frac{\partial \mathcal{U}}{\partial y}+\frac{\partial \mathcal{W}}{\partial x}\right\}\right),\hfill \\ H_2=\left(1-p\right)\left(\mathbf{L}\left\{\mathcal{V}\left(t,x,y\right)\right\}-{\mathcal{V}}_0\left(t,x,y\right)\right)+p\left(\mathbf{L}\left\{\mathcal{V}\left(t,x,y\right)\right\}-\frac{1}{s}\left(a+\frac{2\sqrt{3}}{3}c\right.\right.\hfill \\ \left.\tanh \left(bx+cy\right))+\left(\frac{1}{s^{\zeta }}\right)\mathbf{L}\left\{\mathcal{U}\frac{\partial \mathcal{V}}{\partial x}+\mathcal{V}\frac{\partial \mathcal{V}}{\partial y}+\frac{\partial \mathcal{W}}{\partial y}\right\}\right),\hfill \\ H_3=\left(1-p\right)\left(\mathbf{L}\left\{\mathcal{W}\left(t,x,y\right)\right\}-{\mathcal{W}}_0\left(t,x,y\right)\right)+p\left(\mathbf{L}\left\{\mathcal{W}\left(t,x,y\right)\right\}-\frac{1}{s}\left(\frac{2}{3}\left(b^2+c^2\right)\right.\right.\hfill \\ {\text{sech}}^2\left(bx+cy\right))+\left(\frac{1}{s^{\zeta }}\right)\mathbf{L}\left\{\frac{\partial \left(\mathcal{U}\mathcal{W}\right)}{\partial x}+\frac{\partial \left(\mathcal{V}\mathcal{W}\right)}{\partial y}+\frac{1}{3}\left(\frac{{\partial }^3\mathcal{U}}{\partial x^3}\right.\right.\left.\left.\left.+\frac{{\partial }^3\mathcal{U}}{\partial x\partial y^2}+\frac{{\partial }^3\mathcal{V}}{\partial x^2\partial y}+\frac{{\partial }^3\mathcal{V}}{\partial y^3}\right)\right\}\right),\hfill \end{array}(34)$

where ${\mathcal{U}}_0(t,x,y)$, ${\mathcal{V}}_0(t,x,y),$ and ${\mathcal{W}}_0(t,x,y)$ are the initial guesses.

$\begin{array}{cc}\hfill {\mathcal{U}}_0\left(t,x,y\right)& =-\frac{d+ac}{b}+\frac{2\sqrt{3}}{3}b\tanh \left(bx+cy\right),\hfill \\ \hfill {\mathcal{V}}_0\left(t,x,y\right)& =a+\frac{2\sqrt{3}}{3}c\tanh \left(bx+cy\right),\hfill \\ \hfill {\mathcal{W}}_0\left(t,x,y\right)& =\frac{2}{3}\left(b^2+c^2\right){\text{sech}}^2\left(bx+cy\right).\hfill \end{array}(35)$

In the next step, we will expand $\mathcal{U}(t,x,y)$, $\mathcal{V}(t,x,y),$ and $\mathcal{W}(t,x,y)$ in Taylor’s series form with respect to p as

$\begin{array}{cc}\hfill \mathcal{U}\left(t,x,y\right)& ={\displaystyle \sum _{m=1}^{\infty }}{p}^m{\mathcal{U}}_m,\hfill \\ \hfill \mathcal{V}\left(t,x,y\right)& ={\displaystyle \sum _{m=1}^{\infty }}{p}^m{\mathcal{V}}_m,\hfill \\ \hfill \mathcal{W}\left(t,x,y\right)& ={\displaystyle \sum _{m=1}^{\infty }}{p}^m{\mathcal{W}}_m.\hfill \end{array}(36)$

Substitution of Eq. 36 into Eq. 34 and then comparison of a similar coefficient with respect to p givesthe first-order problem

$\begin{array}{cc}\hfill & \mathbf{L}\left\{{\mathcal{U}}_1\left(t,x,y\right)\right\}+{\mathcal{U}}_0\left(t,x,y\right)-\frac{1}{s}\left(-\frac{d+ac}{b}+\frac{2\sqrt{3}}{3}b\tanh \left(bx+cy\right)\right)\hfill \\ \hfill & +\left(\frac{1}{s^{\zeta }}\right)\mathbf{L}\left\{{\mathcal{U}}_0\frac{\partial {\mathcal{U}}_0}{\partial x}+{\mathcal{V}}_0\frac{\partial {\mathcal{U}}_0}{\partial y}+\frac{\partial {\mathcal{W}}_0}{\partial x}\right\}=0,\hfill \\ \hfill & \mathbf{L}\left\{{\mathcal{V}}_1\left(t,x,y\right)\right\}-{\mathcal{V}}_0\left(t,x,y\right)-\frac{1}{s}\left(a+\frac{2\sqrt{3}}{3}c\tanh \left(bx+cy\right)\right)\hfill \\ \hfill & \left.+\left(\frac{1}{s^{\zeta }}\right)\mathbf{L}\left\{{\mathcal{U}}_0\frac{\partial {\mathcal{V}}_0}{\partial x}\right.+{\mathcal{V}}_0\frac{\partial {\mathcal{V}}_0}{\partial y}+\frac{\partial {\mathcal{W}}_0}{\partial y}\right\}=0,\hfill \\ \hfill & \mathbf{L}\left\{{\mathcal{W}}_1\left(t,x,y\right)\right\}-{\mathcal{W}}_0\left(t,x,y\right)-\frac{1}{s}\left(\frac{2}{3}\left(b^2+c^2\right){\text{sech}}^2\left(bx+cy\right)\right)\hfill \\ \hfill & \left.+\left(\frac{1}{s^{\zeta }}\right)\mathbf{L}\left\{\frac{\partial {\mathcal{U}}_0{\mathcal{W}}_0}{\partial x}\right.+\frac{\partial {\mathcal{V}}_0{\mathcal{W}}_0}{\partial y}+\frac{1}{3}\left(\frac{{\partial }^3{\mathcal{U}}_0}{\partial x^3}+\frac{{\partial }^3{\mathcal{U}}_0}{\partial x\partial y^2}+\frac{{\partial }^3{\mathcal{V}}_0}{\partial x^2\partial y}+\frac{{\partial }^3{\mathcal{V}}_0}{\partial y^3}\right)\right\}=0,\hfill \end{array}(37)$

with the condition

$\begin{array}{cc}\hfill {\mathcal{U}}_1\left(0,x,y\right)& =0,\hfill \\ \hfill {\mathcal{V}}_1\left(0,x,y\right)& =0,\hfill \\ \hfill {\mathcal{W}}_1\left(0,x,y\right)& =0.\hfill \end{array}(38)$

By operating the inverse Laplace transform, the solution at first order is

$\begin{array}{cc}\hfill {\mathcal{U}}_1\left(t,x,y\right)& =-\frac{4b{d}^2{t}^{2\zeta }\tanh \left(bx+cy\right){\text{sech}}^2\left(bx+cy\right)}{\sqrt{3}\mathrm{\Gamma }\left(2\zeta +1\right)},\hfill \\ \hfill {\mathcal{V}}_1\left(t,x,y\right)& =-\frac{4c{d}^2{t}^{2\zeta }\tanh \left(bx+cy\right){\text{sech}}^2\left(bx+cy\right)}{\sqrt{3}\mathrm{\Gamma }\left(2\zeta +1\right)},\hfill \\ \hfill {\mathcal{W}}_1\left(t,x,y\right)& =\frac{4\left(b^2+c^2\right)d^2{t}^{2\zeta }\left(\cosh \left(2\left(bx+cy\right)\right)-2\right){\text{sech}}^4\left(bx+cy\right)}{3\mathrm{\Gamma }\left(2\zeta +1\right)}.\hfill \end{array}(39)$

The second-order problem is

$\begin{array}{cc}\hfill & \mathbf{L}\left\{{\mathcal{U}}_2\left(t,x,y\right)\right\}+\left(\frac{1}{s^{\zeta }}\right)\mathbf{L}\left\{{\mathcal{U}}_1\frac{\partial {\mathcal{U}}_1}{\partial x}+{\mathcal{V}}_1\frac{\partial {\mathcal{U}}_1}{\partial y}+\frac{\partial {\mathcal{W}}_1}{\partial x}\right\}=0,\hfill \\ \hfill & \mathbf{L}\left\{{\mathcal{V}}_2\left(t,x,y\right)\right\}+\left(\frac{1}{s^{\zeta }}\right)\mathbf{L}\left\{{\mathcal{U}}_1\frac{\partial {\mathcal{V}}_1}{\partial x}+{\mathcal{V}}_1\frac{\partial {\mathcal{V}}_1}{\partial y}+\frac{\partial {\mathcal{W}}_1}{\partial y}\right\}=0,\hfill \\ \hfill & \mathbf{L}\left\{{\mathcal{W}}_2\left(t,x,y\right)\right\}+\left(\frac{1}{s^{\zeta }}\right)\mathbf{L}\left\{\frac{\partial {\mathcal{U}}_1{\mathcal{W}}_1}{\partial x}+\frac{\partial {\mathcal{V}}_1{\mathcal{W}}_1}{\partial y}+\frac{1}{3}\left(\frac{{\partial }^3{\mathcal{U}}_1}{\partial x^3}+\frac{{\partial }^3{\mathcal{U}}_1}{\partial x\partial y^2}+\frac{{\partial }^3{\mathcal{V}}_1}{\partial x^2\partial y}+\frac{{\partial }^3{\mathcal{V}}_1}{\partial y^3}\right)\right\}=0\hfill \end{array}(40)$

that has the condition

$\begin{array}{cc}\hfill {\mathcal{U}}_2\left(0,x,y\right)& =0,\hfill \\ \hfill {\mathcal{V}}_2\left(0,x,y\right)& =0,\hfill \\ \hfill {\mathcal{W}}_2\left(0,x,y\right)& =0.\hfill \end{array}(41)$

The inverse of the Laplace transform gives

$\begin{array}{cc}\hfill {\mathcal{U}}_2\left(t,x,y\right)& =\frac{2bd{t}^{\zeta }{\text{sech}}^2\left(bx+cy\right)}{\sqrt{3}\mathrm{\Gamma }\left(\zeta +1\right)},\hfill \\ \hfill {\mathcal{V}}_2\left(t,x,y\right)& =\frac{2cd{t}^{\zeta }{\text{sech}}^2\left(bx+cy\right)}{\sqrt{3}\mathrm{\Gamma }\left(\zeta +1\right)},\hfill \\ \hfill {\mathcal{W}}_2\left(t,x,y\right)& =-t^{\zeta }\left(\frac{4}{3}d{b}^2\tanh \left(bx+cy\right){\text{sech}}^2\left(bx+cy\right)+\frac{4}{3}{c}^2\right.\hfill \\ \hfill & \left.d\tanh \left(bx+cy\right){\text{sech}}^2\left(bx+cy\right)\right)/\mathrm{\Gamma }\left(\zeta +1\right).\hfill \end{array}(42)$

The same procedure is applied for higher-order problems. Thus, the approximate solution at the higher order of the (2 + 1)-dimensional Wu–Zhang system can be obtained by

$\begin{array}{cc}\hfill \tilde{\mathcal{U}}& ={\displaystyle \sum _{m=0}^{\infty }}{\mathcal{U}}_m\left(t,x,y\right),\hfill \\ \hfill \tilde{\mathcal{V}}& ={\displaystyle \sum _{m=0}^{\infty }}{\mathcal{V}}_m\left(t,x,y\right),\hfill \\ \hfill \tilde{\mathcal{W}}& ={\displaystyle \sum _{m=0}^{\infty }}{\mathcal{W}}_m\left(t,x,y\right).\hfill \end{array}(43)$

By replacing the approximate solutions (43) in the given system (29), we obtain residual errors

$\begin{array}{cc}\hfill & R1=\frac{{\partial }^{\zeta }\tilde{\mathcal{U}}}{\partial {\mathfrak{t}}^{\zeta }}+\tilde{\mathcal{U}}\frac{\partial \tilde{\mathcal{U}}}{\partial x}+\tilde{\mathcal{V}}\frac{\partial \tilde{\mathcal{U}}}{\partial y}+\frac{\partial \tilde{\mathcal{W}}}{\partial x},\hfill \\ \hfill & R2=\frac{{\partial }^{\zeta }\tilde{\mathcal{V}}}{\partial {\mathfrak{t}}^{\zeta }}+\tilde{\mathcal{U}}\frac{\partial \tilde{\mathcal{V}}}{\partial x}+\tilde{\mathcal{V}}\frac{\partial \tilde{\mathcal{V}}}{\partial y}+\frac{\partial \tilde{\mathcal{W}}}{\partial y},\hfill \\ \hfill & R3=\frac{{\partial }^{\zeta }\tilde{\mathcal{W}}}{\partial {\mathfrak{t}}^{\zeta }}+\frac{\partial \tilde{\mathcal{U}}\tilde{\mathcal{W}}}{\partial x}+\frac{\partial \tilde{\mathcal{V}}\tilde{\mathcal{W}}}{\partial y}+\frac{1}{3}\left(\frac{{\partial }^3\tilde{\mathcal{U}}}{\partial x^3}+\frac{{\partial }^3\tilde{\mathcal{U}}}{\partial x\partial y^2}+\frac{{\partial }^3\tilde{\mathcal{V}}}{\partial x^2\partial y}+\frac{{\partial }^3\tilde{\mathcal{V}}}{\partial y^3}\right).\hfill \end{array}(44)$

6 Results and discussion

The objective of this study is to propose a new soliton solution of the non-linear time-fractional Wu–Zhang system. This (2 + 1)-dimensional system describes the phenomena of long dispersive waves. The current section is focused on the numerical and graphical results of the WZ system through a hybrid approach by using homotopy perturbation with the Laplace transform, which is known as the He–Laplace algorithm (method). Initially, solutions are captured through the He–Laplace algorithm, considering the fractional derivative in Caputo sense. The obtained results are then analyzed at both fractional and integral orders. Table 1 depicts the residual error at $\mathcal{U},\mathcal{V},\mathcal{W}$ along with overall system errors at various fractional parameter values. These errors clearly indicate the reliability of proposed methodology across the complete fractional domain. It is also observed that error is reduced when fractional parameter approaches one.

TABLE 1

TABLE 1. He–Laplace errors for different values of ζ, when a = d = 0.13, b = 0.11, c = 0.12, x = 3, and y = 6. Here, ${\mathcal{R}}_{\mathit{u}}$, ${\mathcal{R}}_{\mathit{v}}$, ${\mathcal{R}}_{\mathit{w}}$, and $\mathcal{R}$ represent residual errors of $\mathcal{U}$, $\mathcal{V}$, $\mathcal{W}$, and system errors, respectively.

Table 2 shows the comparison of results obtained through He–Laplace and other methods at the integer order that is ζ = 1. This numerical comparison indicates that He–Laplace surpasses other mentioned schemes in terms of accuracy. Figure 1 depicts the He–Laplace solution of the WZ system in 3D at the integer order. This graphical illustration confirms that in the WZ system, surface water velocities in x and y directions are very high, while elevation in water waves decreases with time. Error analysis at ζ = 0.4, 0.8, and 1 as 3D structures can be seen from Figure 2 for $\mathcal{U}$, $\mathcal{V},$ and $\mathcal{W},$ respectively. At ζ = 1, the errors are lesser than ζ = 0.8, and the same can be observed in case of ζ = 0.4.

TABLE 2

TABLE 2. Error comparison of the He–Laplace algorithm with other methods, when ζ = 1, a = b = 0.1, c = d = 0.01, t = 5, and y = 20.

FIGURE 1

FIGURE 1. Graphical illustration of the He–Laplace solution at ζ = 1, a = c = 2, b = d = 1, and t = 2.

FIGURE 2

FIGURE 2. Error analysis at different values of the fractional parameter ζ, when a = c = 0.2, b = d = 0.1, and t = 2.

The impact of the fractional parameter on the water surface is depicted in Figure 3. Research findings indicate that a rise in ζ results in a reduction of the water surface velocity, in both the x and y directions. However, water wave elevation $(\mathcal{W})$ shows inverse behavior in this case. Comparative analysis of different fractional derivative approaches (Atangana–Baleanu, Caputo–Fabrizio, and Caputo) on the solution profile can be seen in Figure 4. Analysis of this figure shows that water surface velocities are highest in the Atangana–Baleanu fractional approach as compared to Caputo and Caputo–Fabrizio fractional approaches. On the other hand, $\mathcal{W}$ depicts opposite behavior as compared to $\mathcal{U}$ and $\mathcal{V}$.

FIGURE 3

FIGURE 3. Effect of the fractional parameter ζ on the water surface level, when a = 0.8, c = 0.9, b = d = 0.7, y = 3, and x = 2.

FIGURE 4

FIGURE 4. Comparison of Caputo, Caputo–Fabrizio, and Atangana–Baleanu fractional derivative approaches on the solution profile, when a = 0.6, b = 0.8, c = 0.9, d = 0.7, y = 2, and x = 5.

7 Conclusion

In this article , a hybrid approach is proposed to solve and analyze the highly non-linear time-fractional (2 + 1)-dimensional WZ system, which is famous for capturing long dispersive waves. A hybrid approach in which homotopy perturbation is combined with the Laplace transform along with different fractional derivatives is proposed for the solution and analysis of the fractional WZ system. Efficiency of the obtained solution is checked over the entire fractional domain to show the validity and convergence of the proposed methodology. Error analysis is also performed in comparison with other well-known numerical methods, which confirms the efficiency of the proposed approach. Graphical analysis shows that water surface velocities increase, while surface elevation decreases, when fractional parameter increases. Also, it is noted that the Atangana–Baleanu approach uplifts water velocities in x and y directions more than Caputo and Caputo–Fabrizio approaches. Analysis of the results also concludes that the proposed method is a reliable technique, which can be extended to more complex fractional systems.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

Author contributions

Conceptualization: MQ. Data curation: EA. Formal analysis: ST. Validation: EA. Writing—original draft: MQ and SS. Writing—review editing: HA and SA. All authors contributed to the article and approved the submitted version.

Funding

This Project is funded by King Saud University, Riyadh, Saudi Arabia.

Acknowledgments

Research Supporting Project number (RSP2023R167), King Saud University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

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Abbreviations

WZ, Wu–Zhang; DEs, differential equations; PDEs, partial differential equations; FDEs, fractional differential equations; HPM, homotopy perturbation method; HLM, He–Laplace method; mVIM, modified variation iteration method; mADM, modified Adomian decomposition method; FRDTM, fractional reduced differential transform method.

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Nomenclature