# A Meshless Radial Point Interpolation Method for Solving Fractional Navier-Stokes Equations
## 4 Numerical Results
As \(m\) increases from 1 to 3, a general trend of enhanced accuracy is observed, particularly for coarser meshes. The improvement is most notable when transitioning from \(m=1\) to \(m=2\), but the gains from \(m=2\) to \(m=3\) are less significant, especially for finer meshes. This suggests that higher-order polynomial approximations can significantly enhance solution quality, though there may be diminishing returns beyond a certain point. Figure 3: Visualization of the test problem solution for FNSE. **Left**: three-dimensional surface plots of the velocity components \((u,v)\) and pressure \((p)\). **Right**: Absolute Error distribution for each variable, showing the accuracy of the numerical solution compared to the exact solution. Table 3 shows that while \(\alpha\) influences error magnitude, it does not alter the error order regardless of the node density. The relationship between \(\alpha\) and discretization parameters is complex, with certain \(\alpha\) values improving accuracy for specific mesh resolutions and time steps. In all scenarios, the error magnitudes for the pressure field (\(p\)) do not consistently exceed those for the velocity components (\(u\) and \(v\)). Mesh refinement improves the \(u\) and \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Node Density** & **Time Step** & \(\|\mathbf{u}-\mathbf{u}_{app}\|_{\infty}\) & \(\|\mathbf{v}-\mathbf{v}_{app}\|_{\infty}\) & \(\|\mathbf{p}-\mathbf{p}_{app}\|_{\infty}\) \\ \hline \(4\times 4\) & \(\Delta t\) & \(3.4938\times 10^{-4}\) & \(4.8785\times 10^{-4}\) & \(2.2210\times 10^{-4}\) \\  & \(\frac{\Delta t}{4}\) & \(3.1950\times 10^{-4}\) & \(4.0935\times 10^{-4}\) & \(2.1570\times 10^{-4}\) \\  & \(\frac{\Delta t}{4}\) & \(3.0040\times 10^{-4}\) & \(3.5558\times 10^{-4}\) & \(2.0275\times 10^{-4}\) \\ \hline \(8\times 8\) & \(\Delta t\) & \(3.3457\times 10^{-4}\) & \(4.1190\times 10^{-4}\) & \(2.2721\times 10^{-4}\) \\  & \(\frac{\Delta t}{4}\) & \(3.2457\times 10^{-4}\) & \(4.0380\times 10^{-4}\) & \(2.0791\times 10^{-4}\) \\  & \(\frac{\Delta t}{4}\) & \(2.9563\times 10^{-4}\) & \(9.4547\times 10^{-5}\) & \(2.0287\times 10^{-4}\) \\ \hline \(16\times 16\) & \(\Delta t\) & \(2.2121\times 10^{-4}\) & \(4.6207\times 10^{-5}\) & \(2.8227\times 10^{-4}\) \\  & \(\frac{\Delta t}{4}\) & \(2.1790\times 10^{-4}\) & \(4.4070\times 10^{-5}\) & \(2.7673\times 10^{-4}\) \\  & \(\frac{\Delta t}{4}\) & \(2.0264\times 10^{-4}\) & \(4.2373\times 10^{-5}\) & \(2.6129\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The numerical results for the \(u\), \(v\), and \(p\) fields in Example 1 with \(m=1\), \(\alpha=0.75\), and \(\Delta t=0.01\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of \(m\)** & **Node Density** & \(\|\mathbf{u}-\mathbf{u}_{app}\|_{\infty}\) & \(\|\mathbf{v}-\mathbf{v}_{app}\|_{\infty}\) & \(\|\mathbf{p}-\mathbf{p}_{app}\|_{\infty}\) \\ \hline \multirow{4}{*}{\(m=1\)} & \(4\times 4\) & \(6.0938\times 10^{-4}\) & \(6.8785\times 10^{-4}\) & \(2.2210\times 10^{-4}\) \\  & \(8\times 8\) & \(5.1950\times 10^{-4}\) & \(5.0935\times 10^{-4}\) & \(2.0570\times 10^{-4}\) \\  & \(16\times 16\) & \(3.3040\times 10^{-4}\) & \(3.5558\times 10^{-4}\) & \(1.1275\times 10^{-4}\) \\ \hline \multirow{4}{*}{\(m=2\)} & \(4\times 4\) & \(4.8792\times 10^{-4}\) & \(5.4894\times 10^{-4}\) & \(1.9232\times 10^{-4}\) \\  & \(8\times 8\) & \(2.7651\times 10^{-4}\) & \(2.9812\times 10^{-4}\) & \(1.5720\times 10^{-4}\) \\  & \(16\times 16\) & \(1.4652\times 10^{-4}\) & \(1.5618\times 10^{-4}\) & \(1.0450\times 10^{-4}\) \\ \hline \multirow{4}{*}{\(m=3\)} & \(4\times 4\) & \(3.7825\times 10^{-4}\) & \(4.2658\times 10^{-4}\) & \(1.5723\times 10^{-4}\) \\  & \(8\times 8\) & \(1.6352\times 10^{-4}\) & \(1.7869\times 10^{-4}\) & \(1.1124\times 10^{-4}\) \\ \cline{1-1}  & \(16\times 16\) & \(8.6543\times 10^{-5}\) & \(9.1982\times 10^{-5}\) & \(8.2310\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table}