# A Meshless Radial Point Interpolation Method for Solving Fractional Navier-Stokes Equations
## 4 Numerical Results
Higher-order approximations yield slight improvements, regardless of the choice of node densities. As the order of the polynomial basis functions increases, the accuracy of the solution improves slightly. This effect is especially pronounced in coarser mesh settings, where higher-order polynomials capture finer details of the solution than lower-order approximations miss. Table 6 highlights the impact of the fractional-order \(\alpha\) on solution accuracy for time-fractional Navier-Stokes equations. Similar to the previous example, it shows that \(\alpha\) affects error magnitude but does not change the error order regardless of node density. The interplay between \(\alpha\) and discretization parameters is crucial, with certain values of \(\alpha\) enhancing accuracy for specific mesh resolutions and time steps. This suggests the need for an adaptive approach to parameter selection to balance accuracy and computational efficiency. Figure 4: 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. Although the results from the RPI method were promising, some theoretical challenges emerged, particularly regarding the optimal choice of spatial form parameters, especially with a higher mesh resolution. Moreover, it is known that PRI methods perform poorly for highly irregular geometries, steep gradients, and discontinuities. They suffer from the curse of dimensionality in high-dimensional problems. The choice of appropriate support radius is a nontrivial task in compactly supported RBFs, where locality and overlap balance each other. The technique also requires heavy implementation in numerical integration and parameter tuning. Moreover, there is no theoretical guarantee of convergence. They are sensitive to multiple parameters requiring laborious tuning, which could be the major \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**\(\mathbf{\alpha}\)** & **Node Density** & \(\|\mathbf{u}-\mathbf{u}_{app}\|_{\infty}\) & \(\|\mathbf{v}-\mathbf{v}_{app}\|_{\infty}\) & \(\|\mathbf{p}-\mathbf{p}_{app}\|_{\infty}\) \\ \hline \multirow{4}{*}{\(\alpha=\frac{1}{4}\)} & \(4\times 4\) & \(8.9736\times 10^{-4}\) & \(9.7654\times 10^{-4}\) & \(5.3210\times 10^{-4}\) \\  & \(8\times 8\) & \(5.2415\times 10^{-4}\) & \(5.1144\times 10^{-4}\) & \(4.1570\times 10^{-4}\) \\  & \(16\times 16\) & \(2.4868\times 10^{-4}\) & \(2.6389\times 10^{-4}\) & \(3.0275\times 10^{-4}\) \\ \hline \multirow{4}{*}{\(\alpha=\frac{1}{2}\)} & \(4\times 4\) & \(7.1304\times 10^{-4}\) & \(7.5735\times 10^{-4}\) & \(4.2145\times 10^{-4}\) \\  & \(8\times 8\) & \(3.3406\times 10^{-4}\) & \(3.2145\times 10^{-4}\) & \(3.5781\times 10^{-4}\) \\  & \(16\times 16\) & \(1.3426\times 10^{-4}\) & \(1.2351\times 10^{-4}\) & \(3.1791\times 10^{-4}\) \\ \hline \multirow{4}{*}{\(\alpha=\frac{2}{3}\)} & \(4\times 4\) & \(5.4527\times 10^{-4}\) & \(5.6652\times 10^{-4}\) & \(3.8275\times 10^{-4}\) \\  & \(8\times 8\) & \(2.2104\times 10^{-4}\) & \(2.5668\times 10^{-4}\) & \(3.4145\times 10^{-4}\) \\  & \(16\times 16\) & \(9.3356\times 10^{-5}\) & \(9.2415\times 10^{-5}\) & \(3.1791\times 10^{-4}\) \\ \hline \multirow{4}{*}{\(\alpha=\frac{3}{4}\)} & \(4\times 4\) & \(4.4567\times 10^{-4}\) & \(4.6789\times 10^{-4}\) & \(3.6275\times 10^{-4}\) \\  & \(8\times 8\) & \(3.1234\times 10^{-4}\) & \(3.2901\times 10^{-4}\) & \(3.3145\times 10^{-4}\) \\ \cline{1-1}  & \(16\times 16\) & \(2.8870\times 10^{-4}\) & \(2.8740\times 10^{-4}\) & \(3.1910\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The numerical results for \(u\), \(v\), and \(p\) fields in Example 2 using different fractional orders (a) with \(\Delta t=0.01\).