# A Meshless Radial Point Interpolation Method for Solving Fractional Navier-Stokes Equations
Meshless methods solve PDEs in strong and weak forms by applying arbitrarily distributed collocations in the solution domain [35]. These points are approximated by assumed global or local basis functions. Weak form methods create weak governing equations on local subdomains, allowing task decoupling and trial and test space selection. Most of the weak-based mesh-free methods suffer from the cost of numerical integrations. For instance, the MLPG method, as a weak form method, utilizes a local symmetric weak form and shape functions from the MLS approximation to solve boundary value problems and partial differential equations effectively [30]. The MLPG method does not require any background integration cells, and all the integrations are carried out locally over the small quadrature domains of regular shapes. Hence, many numerical integrations need to be computed over the complicated MLS shape functions, resulting in an expensive numerical scheme [30]. In contrast, strong form methods are meshless, straightforward, and have good convergence when using simple algorithms and coding. For instance, the RPI method avoids expensive integration by using a smart modification for which more accurate ones have replaced shape functions. The RPI method has successfully solved heat conduction [36; 37], two and three-dimensional potential, and anisotropic diffusion problems [38]. This paper focuses primarily on developing a cost-effective and efficient mesh-free numerical methodology for two-dimensional FNSE problems in a conservation form, incorporating the stream function, vorticity, and advective terms used to model flow in a cavity [5; 11]. That is,  
\[D^{\alpha}\mathbf{u}(t;\mathbf{x}) = -\
abla p(t;\mathbf{x})+\frac{1}{\text{Re}}\
abla^{2}\mathbf{u}(t ;\mathbf{x})+\mathbf{f}(t), \tag{1}\] \[\
abla\mathbf{u}(t;\mathbf{x}) = 0,\]  
where \(\mathbf{u}(t;\mathbf{x})\) is the flow velocity in a two-dimensional space \(\mathbf{x}^{\top}=[x,y]\), \(p(t;\mathbf{x})\) is the pressure, the Reynolds number is denoted by \"Re\", the function \(\mathbf{f}\) is a source vector, divergence is denoted by \(\
abla\), and \(D^{\alpha}\) is called fractional material derivative, which is defined as  
\[\mathcal{D}_{t}^{\alpha}(\cdot)+\mathbf{u}(t;\mathbf{x})\cdot\
abla(\cdot)\]  
in which \(\mathcal{D}_{t}^{\alpha}u\) is the Caputo fractional derivative of order \(\alpha\), \(0<\alpha<1\), defined by  
\[\mathcal{D}_{t}^{\alpha}u=\frac{1}{\Gamma(1-\alpha)}\int\limits_{0}^{T}\frac{u ^{\prime}(\tau)}{(t-\tau)^{\alpha}}d\tau, \tag{2}\]  
where \(\Gamma(\cdot)\) is the gamma function. The results obtained in this investigation discuss the challenges of the accurate solution of the FNSE through a direct comparison with the analytical solution of classical Navier-Stokes equations, particularly for a fractional order close to 1. Numerical tests are performed for the benchmark problem of a Taylor-Green vortex, able to underline the performance and accuracy of such a method with both strengths and weaknesses. These parameters have been found to strongly interact with each other in a complicated way to provide the accuracy of the solution. The computations of the pressure field are of particular concern. In such a case, the present work emphasizes the importance of selecting the parameters for better accuracy and efficiency of simulations within computational fluid dynamics. The scheme of this paper is as follows: Section 2 describes all accompanying definitions and theories that are used later on; the RPI shape functions are also introduced in this section, where the approximation of the time-fractional derivative is approached by the usual finite difference method. The proposed RPI method is developed in Section 3. In contrast, the efficiency of the proposed method is demonstrated for two illustrative problems in Section 4. Finally, in Section 5, conclusions of this present study are drawn.