# A Meshless Radial Point Interpolation Method for Solving Fractional Navier-Stokes Equations
They have been applied quite effectively to describe anomalous diffusion processes in which the mean square displacement departs from the linear growth characteristics of classical diffusion [5; 11]. By systematically studying the properties and behavior of fractional Navier-Stokes equations (FNSE), much work has been carried out: the study of the well-posedness of mild solutions in Besov spaces [12], the decay properties analysis of weak solutions by using the Fourier splitting method [13], and the extension of [PERSON]'s classical results on energy equality for distributional solutions [14]. Others have investigated the approximate controllability of mild solutions with delay [15], introduced an approximate analytical solution via a generalized Laplace residual power series method [16], and proposed a reliable algorithm using the homotopy perturbation transform method [17]. The controlled Picard approach combined with the Laplace transform will be applied to the multi-dimensional time-FNSE driven from incompressible fluid flow, and we will propose an efficient method that can solve nonlinear fractional differential types without using Lagrange multipliers or Adomian polynomials [18]. Other results include the determination of critical values in fractional three-dimensional Navier-Stokes equations [19], an investigation on averaging between two-dimensional FNSE with singularly oscillating external forces [20], and research into the existence of uniform analytic solutions of the Cauchy problem in the critical Fourier-Herz spaces [21]. The numerical solution of FDEs is different from classical integer-order differential equations due to the type of problem. Unlike the solutions of classical differential equations, which are local to function shapes and information, solving FDEs requires a buffer that stores the solution history to obtain a horizon solution. This is because the fractional derivatives are non-local and dependent on memory; historical information must be considered to model the system's behavior accurately. Several works have been proposed for efficiently solving FDEs and coping with their intensive computational needs, which usually involves a trade-off between accuracy and required memory [22; 23]. The most popular numerical approaches, so far, to solve the system of FNSE are finite difference method [24; 25], homotopy-based approaches [26; 27], Laplace transform [28], Adomian decomposition method [29], and fractional-step methods [30; 31]. Each has clear advantages in tackling fractional derivatives, nonlinearities, and multidimensional problems. The choice of method largely depends on problem formulation, the desired accuracy, and computational resources. Despite the diversity within this family of methods, they all share one basic principle: the solution representation at discrete grid points and approximations of derivatives are shown as the differences between these values. Thus, the solutions obtained using such methods result in expensive computational costs, besides difficulties in maintaining accuracy near boundaries. Moreover, their inability to efficiently handle complex, spatially intricate boundary geometries--where grid points must conform to intricate boundaries--presents a significant drawback. In the recent couple of years, there has been an increasing interest in seeking alternative methods, for example, mesh-free approaches to solve spatial fractional partial differential equations (PDEs), since they do not require any predefined mesh structure; hence, these types of methods are suitable for problems involving complicated geometries or time-evolving domains [24]. The so-called mesh-free methods, representing radial basis function (RBF) methods [32], radial point interpolation (RPI) [25], moving least squares (MLS) [33], and the meshless local Petrov-Galerkin (MLPG) method [34], are themselves promising alternatives. The advantages of such methods include the flexibility in handling complicated geometries and avoiding the limitations arising in the case of mesh-based methods.