## 5 Numerical Results
### General Comparison  
In the previous sections, it was claimed that the proposed model improves accuracy and stability over a wider range of shape parameter. In order to verify this claim, the model function and its derivatives (Equations (16)-(18)) are approximated with 41 mesh points. The error of the approximations is calculated. To compare for a range of \(\varepsilon\) values, the norm of the error is considered to identify a single value for each case. Additionally, the log10 is applied to the norm to show the number of significant digits of the error. Accordingly, the final formula to calculate the error can be expressed as follows:  
\[\log 10\Big{[}norm\Big{(}F_{approx}-F_{analytical}\Big{)}\Big{]}, \tag{19}\]  
where \(F\) could be \(f(x)\), \(\partial f/\partial x\), or \(\partial^{2}f/\partial x^{2}\). The results of the conventional approach are plotted in Figure 4. It is noteworthy that the error increases as \(\varepsilon\) decreases. This is expected because, at low \(\varepsilon\), the generated matrix of the RBF is ill-conditioned, which has a negative effect on the accuracy. It is shown that the minimum interpolation error decreases for \(\varepsilon\geq 1.43\) when \(\varepsilon\) increases, while it is the opposite for the derivatives. This is a clear indication of the undesired Runge's phenomenon, which affects the accuracy of the interpolation for any other locations set than the given one. In Figure 4, the symbols denote the results evaluated at different locations (the mid-points between nodes) than the one used for evaluating the interpolation coefficients. It is shown that the results of the derivatives are not affected by location. Meanwhile, the accuracy of the interpolation decreases when \(\varepsilon\geq 1.43\), which is consistent with the derivatives. The minimum errors of the derivatives are \(-5.37\) and \(-3.58\) for \(\frac{\partial f}{\partial x}\) and \(\frac{\partial^{2}f}{\partial x^{2}}\), respectively, which are located at \(\varepsilon\approx 1.43\). These results will act as reference results with which to compare For the sake of comparison, RBF-TS is used to approximate the same function used in the conventional approach. The results are shown in Figure 5. To determine the effect of the truncation order on the accuracy, the sub-figures are compared column-wise. It can be deduced that the order of truncation does not have a significant effect on the accuracy as the results do not vary greatly when they are compared column-wise. In each sub-figure in Figure 5, the accuracy does not change significantly with \(\varepsilon\); however, it changes with \(\varepsilon_{0}\). This indicates that the chosen approach is unfeasible because it switches the dependency from \(\varepsilon\) to \(\varepsilon_{0}\). Nevertheless, it will give a chance to introduce another novel approach (i.e., RBF-TPU). Figure 4: The error of approximating the given function \(f(x)\) and its first two derivatives using the conventional RBF approach as it varies with \(\varepsilon\). The Gaussian radial basis function is used and the number of mesh points is fixed as 41 regularly distributed points. Figure 5: The log10 of the norm of the error of approximating the given function \(f(x)\) and its first two derivatives using Gaussian RBF-TS with different truncation orders for a range of \(\varepsilon\) and \(\varepsilon_{0}\) values. The rows are for function, first derivative, and second derivative approximation, while the columns are for the zeroth, first, and second orders of truncation. In RBF-TPU, the range of \(\varepsilon_{k}\) is fixed between [0, 1], because we are interested in the region of small \(\varepsilon\). However, the number of regular points to divide that range (\(N_{e}\)) is varied. The same model function is used as in RBF-TS and regular RBF. The results are plotted in Figure 6, which show that the accuracy does not change greatly with the truncation order or with \(\varepsilon\).