## 1 Introduction  
Dispersive shallow water waves are wave propagation phenomena that have received extensive attention in engineering due to their numerous applications in physics, engineering, mechanics, and mathematics. In the past few decades, several partial differential equations (PDEs), including the Camassa-Holm equation [1, 2], the Degasperis-Procesi equation [2, 3, 4], the Fornberg-Whitham equation [3, 5, 6, 7, 8, 9, 10, 11, 12], and the Schrodinger equation [13, 14, 15, 16, 17], have been presented to describe the traveling wave phenomenon and water waves of finite depths. These PDEs and their modified forms mathematically represent transient nonlinear PDEs that involve third-order time-space mixed partial derivatives or complex terms. Therefore, obtaining the numerical solutions of those PDEs is a significant issue for analyzing the traveling wave phenomenon. In previous studies, two types of methods were mainly used to obtain solutions for such equations: the approximate method, the spectral method [18, 19], and numerical simulation. Due to the complexity of these equations, finding exact solutions is generally challenging. Therefore, numerical methods play a crucial role in solving these equations. Traditional numerical approaches, relying on temporal and spatial discretization, face difficulties when dealing with high-order mixed partial derivatives. As a result, researchers have extensively employed approximate methods to derive numerical solutions, which offer convenient and effective alternatives to classical numerical treatments. For example, [PERSON] [2] used the extended tanh method to establish the new solitary wave solutionsof the modified Degasperis\(-\)Procesi and Camassa\(-\)Holm equations. [PERSON] and [PERSON] [10] applied the homotopy analysis method (HAM) to acquire the HAM results of the Fornberg\(-\)Whitham equation and compared them with [PERSON]'s decomposition method (ADM). [PERSON] et al. [5] utilized the modified variational iteration algorithm (MVIA) to obtain the approximate numerical solutions of the Fornberg\(-\)Whitham equation and its modified form. In addition to the above-mentioned methods, there are other approximate methods such as the method of phase portrait analysis [9], [PERSON]'s variational iteration method (VIM) [8], and the reproducing kernel Hilbert space method (RKHSM) [7]. Recently, due to the evolution of computer technology and numerical methods, novel numerical schemes have also been applied to solve such equations. These include the Godunov method [6], the cubic B-spline quasi-interpolation method [4], the hybrid radial basis function method [3], the linearized implicit pseudo-spectral method [18], and the Gegenbauer spectral method [19]. In the field of numerical discretization methods for PDEs, two main categories can be distinguished: mesh-based methods and meshless/free methods. Mesh-based methods include well-known techniques such as finite difference (FDM), finite volume, and finite element methods. These methods have been extensively developed and modified to solve engineering problems and analyze complex physical phenomena that are challenging to investigate through practical experiments. Despite its widespread application, mesh generation remains a challenging task. To overcome this issue, the concept of using numerical discretization methods without the need for a mesh has emerged as an important research direction. Meshless methods offer a solution by avoiding the requirement for mesh generation, making it easier to handle computational domains with complex shapes. In meshless methods, numerical discretization involves constructing interpolation functions on arbitrary distribution nodes to approximate the governing equation. Over the years, meshless methods have gained advantages such as simplified numerical procedures, easy implementation, flexibility, and the ability to construct hybrid numerical schemes tailored to specific research needs.