# A Meshless Radial Point Interpolation Method for Solving Fractional Navier-Stokes Equations
## 3 RPI Methodology for Fractional-Order Navier-Stokes Equations  
Consider the time interval \([0,T]\) uniformly partitioned into \(M\) subintervals, where \(\Delta t=\frac{T}{M}\) and \(t_{k}=k\Delta t\). Using the method in [41; 42; 43], Caputo fractional derivative operator (2) can be approximated by the quadrature formula  
\[\mathcal{D}_{t}^{\alpha}u(x,t_{k}) \approx D_{\Delta t}^{\alpha}u(x,t_{k})\] \[=b_{\alpha,\Delta t}\Bigg{(}u(x,t_{k})-\sum_{j=1}^{k-1}(\omega_{ \alpha,k-j-1}-\omega_{\alpha,k-j})u(x,t_{j})-\omega_{\alpha,k-1}u(x,t_{0}) \Bigg{)}, \tag{4}\]  
where  
\[\omega_{\alpha,j}:=(j+1)^{1-\alpha}-j^{1-\alpha},\quad j=1,2,\ldots,k,\]  
and  
\[b_{\alpha,\Delta t}:=\frac{\Delta t^{-\alpha}}{\Gamma(2-\alpha)}.\]  
Moreover, the truncation error is also determined by (see [41])  
\[|\mathcal{D}_{t}^{\alpha}u(x,t_{k})-D_{\Delta t}^{\alpha}u(x,t_{k})|\leq c_{ \alpha}\Delta t^{2-\alpha}\max_{0\leq t\leq t_{k}}\bigg{|}\frac{\partial^{2}u (x,t_{k})}{\partial t^{2}}\bigg{|},\]  
where \(c_{\alpha}\) is a constant that depends only on \(\alpha\). This indicates that the scheme has an order of \(\Delta t^{2-\alpha}\), provided the function \(u\) is twice continuously differentiable in \(t\). Using Equation (4), the discretized system of nonlinear algebraic equations corresponding to Equation (1) is  
\[b_{\alpha,\Delta t}A\sum_{k=1}^{n}\omega_{\alpha,k}(U^{n-k+1}-U^{n-k})+B^{n} U^{n}=F^{n}, \tag{5}\]  
where  
\[A= \begin{pmatrix}I&0&0\\ 0&I&0\\ 0&0&I\end{pmatrix} ,\quad B= \begin{pmatrix}B_{11}&0&B_{13}\\ 0&B_{22}&B_{23}\\ B_{31}&B_{32}&0\end{pmatrix} ,\quad F= \begin{pmatrix}f_{x}\\ f_{y}\\ 0\end{pmatrix} ,\quad U= \begin{pmatrix}\widehat{u}\\ \widehat{v}\\ \widehat{p}\end{pmatrix} ,\]  
Figure 1: A basic illustration of an interpolated scalar function via RPI using the combination of Radial Basis Functions (RBFs). Black dots represent interpolation points \(x_{1},\ldots,x_{6}\), while the green, blue, and red curves show RBFs centered at \(x_{2}\), \(x_{4}\), and \(x_{6}\), respectively. The thick black curve represents the final interpolated function. \[\varphi_{ij}(u,v) =u(x_{i})\phi_{j,x}(x_{i})+v(x_{i})\phi_{j,y}(x_{i})-\frac{1}{Re}( \phi_{jxx}(x_{i})+\phi_{jyy}(x_{i})), \tag{6a}\] \[B_{11} =[\varphi_{ij}(u,v)],\quad B_{13}=[\phi_{j,x}(x_{i})],\quad B_{22} =B_{11},\] (6b) \[B_{23} =[\phi_{j,y}(x_{i})],\quad B_{31}=B_{13},\quad B_{32}=B_{23},\] (6c) \[F_{x} =[f_{x1},f_{x2},\ldots,f_{xN}]^{\top},\quad F_{y}=[f_{y1},f_{y2}, \ldots,f_{yN}]^{\top},\] (6d) \[\widehat{U} =[\widehat{u_{1}},\widehat{u_{2}},\ldots,\widehat{u_{N}}],\] (6e) \[\widehat{V} =[\widehat{v_{1}},\widehat{v_{2}},\ldots,\widehat{v_{N}}],\] (6f) \[\widehat{P} =[\widehat{p_{1}},\widehat{p_{2}},\ldots,\widehat{p_{N}}], \tag{6g}\]  
in which \(I\) is the \(N\times N\) identity matrix and \(0\) is the \(N\times N\) zero matrix. Thus, the system can be rewritten as  
\[b_{\alpha,\Delta t}A(U^{n}-U^{n-1})+B^{n}U^{n}=-b_{\alpha,\Delta t}\sum_{k=2}^ {n}\omega_{\alpha,k}(U^{n-k+1}-U^{n-k})+F^{n}, \tag{7}\]  
such that at the first time level (\(n=1\)), it can be determined by  
\[(b_{\alpha,\Delta t}A+B^{1})U^{1}=b_{\alpha,\Delta t}A\Delta U_{0}+F^{1}, \tag{8}\]  
and for \(n\geq 2\), it yields  
\[\varphi_{ij}(u,v) =u(x_{i})\phi_{j,x}(x_{i})+v(x_{i})\phi_{j,y}(x_{i})-\frac{1}{Re}( \phi_{jxx}(x_{i})+\phi_{jyy}(x_{i})), \tag{6a}\]0  
which can be expressed more conveniently as  
\[\varphi_{ij}(u,v) =u(x_{i})\phi_{j,x}(x_{i})+v(x_{i})\phi_{j,y}(x_{i})-\frac{1}{Re}( \phi_{jxx}(x_{i})+\phi_{jyy}(x_{i})), \tag{6a}\]1  
with  
\[\varphi_{ij}(u,v) =u(x_{i})\phi_{j,x}(x_{i})+v(x_{i})\phi_{j,y}(x_{i})-\frac{1}{Re}( \phi_{jxx}(x_{i})+\phi_{jyy}(x_{i})), \tag{6a}\]2  
Algorithm 1 presents the proposed RPIM for solving (1). The algorithm starts by initializing parameters and setting up data points and centers. It then discretizes the time domain, establishes initial conditions, and computes the approximate solution. In each iteration, shape functions are updated, and the solution is refined for each time step.