## 2 The Mathematical Models of the Dispersive Shallow Water Waves  
The dispersive shallow water wave represents the behavior of the wave propagation, breaking wave, and dynamic system of the fluid. It is an important phenomenon in physics and engineering. In the past few decades, several researchers have proposed the \begin{table}
\begin{tabular}{l l} \hline \hline
**Abbreviations** & **Description** \\ \hline
1 st & first \\
1D & one-dimensional \\
2D & two-dimensional \\
2 nd & second \\
3 rd & third \\ ADM & [PERSON]’s decomposition method \\ Eq & equation \\ FDM-RK4 & finite difference method with the fourth-order \\  & Runge–Kutta method \\ GFDM & generalized finite difference method \\ HAM & homotopy analysis method \\ MAE & maximum absolute error \\ MVIA & modified variational iteration algorithm \\ PDE & partial differential equation \\ RBFCM & radial basis functions collocation method \\ RKHSM & reproducing kernel Hilbert space method \\ RMSE & root-mean-squared error \\ ST & space–time \\ ST-GFDM & space–time GFDM \\ VIM & He’s variational iteration method \\ _Superscript_ & \\ _ext_ & exact solution \\ _num_ & numerical solution \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The abbreviations and descriptions in this paper. mathematical model of the dispersive shallow water wave problem, and its general form of the governing equation [1; 2; 11] is described as,  
\[\frac{\partial u}{\partial t}-\frac{\partial^{3}u}{\partial x^{2}\partial t}+( \alpha+\beta u^{\omega})\frac{\partial u}{\partial x}=q\frac{\partial u}{ \partial x}\frac{\partial^{2}u}{\partial x^{2}}+u\frac{\partial^{3}u}{ \partial x^{3}},\;\;a\leq x\leq b,\;t\geq 0, \tag{1}\]  
where \(u\) is the wave speed; \(x\) and \(t\) are the spatial- and temporal-axes; \(\alpha\), \(\beta\), \(\omega\), and \(q\) are the constants. The initial conditions and boundary conditions are represented as follows:  
\[\left\{ \begin{array}{ll}u(x,0)=U_{0}(x)\\ u(a,t)=f_{1}(t)\\ u(b,t)=f_{2}(t)\end{array} .\right. \tag{2}\]  
By applying the constants \(a\), \(b\), \(\omega\), and \(q\), three different types of the dispersive shallow water wave equation are defined. The first one is the Fornberg-Whitham equation [5; 6; 7; 8] that is given with \(\alpha=1\), \(\beta=1\), \(\omega=1\), and \(q=3\) as,  
\[\frac{\partial u}{\partial t}-\frac{\partial^{3}u}{\partial x^{2}\partial t} +(1+u)\frac{\partial u}{\partial x}=3\frac{\partial u}{\partial x}\frac{ \partial^{2}u}{\partial x^{2}}+u\frac{\partial^{3}u}{\partial x^{3}},a\leq x \leq b,\;t\geq 0, \tag{3}\]  
and its modified form [5; 7; 10] is given (\(\alpha=1\), \(\beta=1\), \(k=2\), and \(q=3\)) as,  
\[\frac{\partial u}{\partial t}-\frac{\partial^{3}u}{\partial x^{2}\partial t} +\Big{(}1+u^{2}\Big{)}\frac{\partial u}{\partial x}=3\frac{\partial u}{ \partial x}\frac{\partial^{2}u}{\partial x^{2}}+u\frac{\partial^{3}u}{ \partial x^{3}}. \tag{4}\]  
The remaining two are similar: the Camassa-Holm equation [1; 2] and the Degasperis-Procesi equation [2; 3; 4], respectively. By giving \(\alpha=0\), \(\beta=3\), \(\omega=1\), and \(q=2\), the Camassa-Holm equation can be denoted as  
\[\frac{\partial u}{\partial t}-\frac{\partial^{3}u}{\partial x^{2}\partial t} +3u\frac{\partial u}{\partial x}=2\frac{\partial u}{\partial x}\frac{ \partial^{2}u}{\partial x^{2}}+u\frac{\partial^{3}u}{\partial x^{3}}, \tag{5}\]  
and the Degasperis-Procesi equation (\(\alpha=0\), \(\beta=4\), \(\omega=0\), and \(q=3\)) is described as  
\[\frac{\partial u}{\partial t}-\frac{\partial^{3}u}{\partial x^{2}\partial t} +4u\frac{\partial u}{\partial x}=3\frac{\partial u}{\partial x}\frac{ \partial^{2}u}{\partial x^{2}}+u\frac{\partial^{3}u}{\partial x^{3}}, \tag{6}\]  
As with the relationship between the Fornberg-Whitham equation and its modified form, the modified forms of the Camassa-Holm equation [2] and the Degasperis-Procesi equation [2; 3; 4] are represented by changing the nonlinear term \(u\frac{\partial u}{\partial x}\) into \(u^{2}\frac{\partial u}{\partial x}\).