## 1 Introduction
At present, there are several meshless numerical methods, for example, the radial basis functions collocation method (RBFCM) [3], the virtual boundary meshless Galerkin method [20], the moving particle semi-implicit method [21], the radial point interpolation meshless method [22], the singular boundary method [23], the localized scheme based on boundary-type method [24; 25; 26], the generalized finite difference method (GFDM) [27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37], etc. The GFDM is a localized domain-type meshless collocation method based on the multivariate Taylor series expansion theory and the moving least-squares method. In 1980, the idea of FDM without mesh was proposed by [PERSON] and [PERSON] [27]. After [PERSON] et al. [28] proposed the explicit numerical process in 2001, numerical applications related to the GFDM have been proposed in the past two decades. By utilizing the GFDM, the approximate expression of the partial derivative terms can be represented by a linear combination of the weight coefficients and function values in a local support domain. This support domain only contains the goal node and its nearby nodes. Once the nodes are distributed in the computational domain, the governing equation can be discretized into a system of algebraic equations. In addition, the resultant matrix system is a sparse matrix due to the localized property of the GFDM. The above characteristics make the GFDM capable of applying to complex problems, such as theoretical analysis [29], shallow water equation [30; 31], porous media flow [32], wave propagation [33], elliptic interface problems [34], extended Fisher-Kolmogorov equation [35], stream function formulation [36], and elastic wave [37]. In this research, the space-time GFDM (ST-GFDM) was applied as the foundation of the proposed meshless numerical scheme, and it is an extended meshless method from the GFDM. The space-time (ST) coupled approach is a numerical technique that treats the temporal derivative as one of the spatial derivatives. Prior to the use of the ST-coupled approach, time-dependent problems were typically solved by combining spatial discretization methods with temporal discretization methods. However, this approach often led to challengesin defining the accuracy and stability of the numerical scheme. The ST-coupled approach transforms a time-dependent problem of dimension (n) into a steady-state problem of dimension (n + 1). As a result, all the partial derivatives in the governing equation can be discretized using numerical methods designed for spatial discretization. This simplification enables a clearer determination of the properties of the numerical scheme, such as accuracy and stability. Recently, the ST coupled approach has been widely combined with meshless methods such as the ST localized RBFCM [38], ST kernel-based method [39], ST localized method of fundamental solutions [40; 41], ST Trefftz Method [42; 43], ST backward substitution method [44], and ST-GFDM [45; 46; 47; 48; 49; 50; 51; 52; 53; 54]. Based on the flexibility of the ST-GFDM, researchers have applied this meshless method for engineering problems in the past few years. Examples of these applications include heat conduction problems [45; 54], Burgers' equations [46], parabolic PDEs [47], unsteady double-diffusive natural convection [48], thin elastic plate bending problems [49], thermoelasticity problems [50], equal-width equation [51], Zakharov-Kuznetsov-Modified equal-width equation [52], and hyperbolic PDEs [53], etc. The aforementioned advancements demonstrate that ST-GFDM is a promising meshless method with potential for engineering applications. In this study, the ST-GFDM is utilized in combination with the two-step Newton's method and the time-marching approach to tackle the problem of solving dispersive shallow water waves. After providing the motivation for the research. The governing equations of the dispersive shallow water wave are represented in Section 2.