## 5 Numerical Results
### Effect of Changing the Model Function, Mesh Size, and Mesh Distribution  
The findings of the previous sections are summarized in Table 1. It is shown that RBF-TPU yields results that are comparable to those of Gaussian RBF, which are also independent of \(\varepsilon\), \(N_{e}\), and \(m\). However, the previous results are obtained for a single model function (i.e., Equation (21)), a single mesh size (21), and a single mesh distribution (Uniform). Importantly, these parameters should be varied to investigate the accuracy over a wider range of parameters so as to validate the optimality of the chosen range of \(\varepsilon_{0}\) ([\(-1\), \(1\)]). The mesh sizes of 21, 41, and 61 are considered, and uniform and stretched mesh distributions are also studied. The following formula is used to map a uniform mesh to a stretched one:  
\[x=x_{min}+\left[\frac{\mathrm{erf}\Big{(}2|\pi\frac{x-x_{min}}{x_{max}-x_{min}} -\pi\Big{)}}{2|\mathrm{erf}(\pi)}+\frac{1}{2}\right](x_{max}-x_{min}) \tag{20}\]  
where erf denotes the error function, \(x_{max}=1\), and \(x_{min}=-1\). \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**Min log10[Norm(Error)]**} & \multirow{2}{*}{**Notes**} \\ \cline{2-3} \cline{5-4}  & \(\frac{\partial f}{\partial x}\) & \(\frac{\partial^{2}f}{\partial x^{2}}\) & \\ \hline RBF & \(-5.37\) & \(-3.58\) & \multirow{2}{*}{Depends on \(\varepsilon\) and has [PERSON]’s phenomenon.} \\ RBF-TPU & \(-4.70\) & \(-3.00\) & & Does not depend on \(\varepsilon\), \(m\), or \(N_{e}\); once the range of \(\varepsilon_{k}\) is optimized, which should be done once. \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison of the minimum error for approximating derivatives (Equations (17) and (18)) between Gaussian radial basis function (RBF) and partition of unity for Taylor series RBF (RBF-TPU). The results of RBF-TPU are obtained from Figure 8.  
Figure 8: The log10 of the norm of the error of approximating the given function \(f(x)\) and its first two derivatives using Gaussian RBF-TPU with zeroth truncation order for a range of \(\varepsilon\) and m values. The rows are for function, first derivative, and second derivative approximations. The \(\varepsilon_{k}\) range is [\(-1\), \(1\)] and \(N_{e}=5\). A second model function is considered in addition to the first model function and its derivatives (Equations (21)-(23)). \[f(x)=\frac{\left(x^{2}-1\right)^{2}}{\sqrt{x^{2}+1}} \tag{21}\]  
\[\frac{\partial f}{\partial x}=\frac{3x^{5}+2x^{3}-5x}{\left(x^{2}+1\right)^{3/2}} \tag{22}\]  
\[\frac{\partial^{2}f}{\partial x^{2}}=\frac{6x^{6}+15x^{4}+16x^{2}-5}{\left(x^{ 2}+1\right)^{5/2}} \tag{23}\]  
RBF-TPU and Gaussian RBF are compared by varying the mesh size and mesh distribution for the two model functions. The results are depicted in Table 2. For all cases reported in Table 2, the accuracy is independent of \(m\) and \(N_{e}\), which helps verify the results of the previous section. Table 2 illustrates that the accuracy of Gaussian RBF is maximum at a certain value of \(\varepsilon\); however, the accuracy of RBF-TPU is consistent over the studied range of \(\varepsilon\) ([0, 2]). Therefore, it can be concluded that the accuracy of RBF-TPU is always comparable to the maximum accuracy that is achieved with Gaussian RBF, although Gaussian RBF has a marginal better maximum accuracy at a certain \(\varepsilon\) value. This marginal difference is the price paid to achieve a consistent accuracy that is independent of the shape parameter.