# A Meshless Radial Point Interpolation Method for Solving Fractional Navier-Stokes Equations
## 4 Numerical Results  
A series of numerical experiments assessed the proposed method's accuracy and empirical convergence in solving two-dimensional Navier-Stokes equations. The computational implementation was performed using MATLAB R2019a on a system equipped with an Intel(R) Core(TM) i5-2410M 2.30 GHz processor and 4 GB memory. For this study, two test problems from the existing literature were selected to rigorously evaluate the approach's performance. A thorough analysis of these examples demonstrates the effectiveness of the proposed method, with results presented in tabular and graphical formats. These numerical experiments showcase the method's ability to handle various scenarios encountered in two-dimensional Navier-Stokes equations, providing insights into its practical applicability and computational efficiency. **Example 1**. : _Consider the following test problem for the FNSE (1):_  
\[u(x,y) = e^{-t}(2x^{2}y(1-x^{2})(1-y)(1-2y)), \tag{11a}\] \[v(x,y) = e^{-t}(-2 xy^{2}(1-x)(1-2x)(1-y)^{2}),\] (11b) \[p(x,y) = e^{-t}(x^{2}-y^{2}). \tag{11c}\]  
Since this equation is challenging to solve analytically, a reverse engineering approach is proposed, where \(u(x,y)\), \(v(x,y)\), and \(p(x,y)\) in Equation (11) are substituted directly into FNSE (1), and the source vector analytical is obtained as a function of \(\alpha\) and time, i.e., \(f(\alpha;t)\), to evaluate the obtained numerical solutions. To conserve space, the obtained term, being highly nonlinear and lengthy, was omitted from this article. In addition, multi-quadric functions were used as shape functions in the RPI functions. Figure 2 shows the Hammersley point set for interpolating scattered data. This quasi-random sequence of points achieves a uniform distribution across the domain while balancing computational efficiency and solution accuracy. The clear distinction between interior (blue circles) and boundary points (red crosses) demonstrates the method's ability to deal with complex geometries and boundary conditions. Figure 3 comprehensively visualizes the test problem solution. The top row contains 3D surface plots of the velocity components \((u,v)\) and pressure (\(p\)), which offer intuitive insights into the solution's spatial distribution and overall behavior. The Absolute Error (AE) distribution for each variable is shown in the bottom row, allowing for an evaluation of the numerical solution's accuracy compared to the exact solution. These error distributions highlight areas of high and low accuracy within the computational domain, guiding future numerical method refinements. Table 1 presents the effect of mesh refinement and time step reduction on solution accuracy. For \(m=1\) and \(\alpha=0.75\), mesh refinement improves the accuracy in the \(u\) and \(v\) fields while slightly increasing the error in the \(p\) field. Reducing the time step from \(\Delta t\) to \(\Delta t/4\) generally enhances precision, though the improvement is not uniformly monotonic. Node density refers to the distribution and concentration of nodes within the computational domain, directly impacting the accuracy and convergence of the solution. It is used as a quantity aligned with the fixed-step size in finite difference methods, serving a similar role in controlling the discretization level of the problem. Mesh refinement consistently enhances accuracy in the \(u\) field and shows significant improvement in the \(v\) field. Conversely, the \(p\) field exhibits a slight increase in error as the node density decreases. Time step reduction generally improves accuracy, but the effect varies with field and node density. In finer meshes, decreasing the time step reduces the error across all variables, whereas in coarser meshes, some variables show substantial improvement with a smaller time step. Figure 2: A 2D Hammersley set with boundary values, consisting of 272 points. Blue circles represent the Hammersley points, and red crosses denote the boundary points. Table 2 illustrates the influence of the number of polynomial basis functions on the results.