# A Meshless Radial Point Interpolation Method for Solving Fractional Navier-Stokes Equations
## 2 Basic Definitions for Radial Point Interpolation  
This section introduces the fundamental definitions and concepts that form the basis for the subsequent development and analysis. These preliminaries include key math ematical formulations and theoretical underpinnings necessary for understanding the proposed method. Radial basis functions are a key component of the RPI method, providing a flexible and powerful means for interpolating scalar fields across a domain [39; 40]. The RPI of a scalar function \(u(\mathbf{x}):\mathbb{R}^{d}\mapsto\mathbb{R}\) in the domain \(\Omega\subset\mathbb{R}^{d}\) at \(n\) points is given by  
\[u(\mathbf{x})=\sum_{i=1}^{N}R_{i}(\mathbf{x})a_{i}+\sum_{j=1}^{m}p_{j}( \mathbf{x})b_{j}=\{R(\mathbf{x})^{\top},p(\mathbf{x})^{\top}\}\{a,b\}^{\top}, \tag{3}\]  
where \(R_{i}(\mathbf{x})\) represents the RBFs associated with interpolation, and \(p_{j}(\mathbf{x})\) represents the polynomial terms used in conjunction with the RBFs to enhance stability and accuracy. In this formulation, \(N\) denotes the number of RBFs used, and \(m\) represents the number of polynomial terms in the spatial coordinates \(\mathbf{x}^{\top}=[x,y]\). The RBFs, \(R_{i}(\mathbf{x})\), are defined based on the Euclidean distance between the interpolation points, allowing the method to handle scattered data and complex geometries without requiring a structured grid. Some common choices for RBFs include Gaussian, Multi-quadrics, Inverse Multiquadrics, and Thin Plate Splines, and the detailed descriptions of these RBFs can be found in [39; 40]. The number of points \(n\) in the support domain of \(\mathbf{x}\) directly influences the interpolation accuracy and determines the number of RBFs required for a specific problem. The polynomial term, \(p_{j}(\mathbf{x})\), is often added to ensure the invertibility of the interpolation matrix, which is crucial for solving the resulting system of equations. Typically, \(m\ll n\), the number of polynomial terms, is much smaller than the number of RBFs. However, including these monomial terms can enhance the accuracy of interpolation, particularly near the domain's boundaries. When the polynomial terms are omitted (i.e., \(m=0\)), the method uses pure RBF interpolation, which can still provide accurate results for many applications, albeit with a higher sensitivity to the placement of nodes and the selection of the RBF's shape parameter. The coefficients \(a_{i}\) and \(b_{j}\) are unknowns that need to be determined by enforcing the interpolation condition at \(n\) distinct nodes within the support domain of the point of interest \(\mathbf{x}\). These coefficients are typically computed by solving a system of linear equations formed by applying Equation (3) at the \(n\) nodes  
\[u(\mathbf{x}_{k})=\sum_{i=1}^{N}R_{i}(\mathbf{x}_{k})a_{i}+\sum_{j=1}^{m}p_{j} (\mathbf{x}_{k})b_{j},\quad k=1,2,\ldots,n.\]  
This results in a system of linear equations that can be expressed in matrix form  
\[\mathbf{Ac}=\mathbf{f},\]  
where \(\mathbf{A}\) is the interpolation matrix, \(\mathbf{c}=[a_{1},a_{2},\ldots,a_{N},b_{1},b_{2},\ldots,b_{m}]^{\top}\) is the vector of unknown coefficients, and \(\mathbf{f}\) is the vector of known function values at the interpolation points. This matrix system can then be solved to determine the coefficients \(a_{i}\) and \(b_{j}\), providing the desired interpolated solution. Figure 1 demonstrates the basic concept of RBFs in a scalar problem and how they are combined to form an interpolated function through RPI. The black dots on the \(x\)-axis represent the interpolation points \(x_{1},\ldots,x_{6}\), where the function \(u(x)\) is evaluated. The combination of RBFs allows the method to flexibly adapt to scattered data points, making RPI an effective meshless technique for solving partial differential equations in domains with complex geometries. The green, blue, and red curves represent three RBFs centered at the interpolation points \(x_{2}\), \(x_{4}\), and \(x_{6}\), respectively.