## 6 Conclusions  
In this work, a proposal is made to apply the partition of unity approach to the Taylor series of RBF method in order to improve the basis of the method when using low shape parameter values (\(\varepsilon\)). The Gaussian function is used as a weighting function for the partition of unity approximation and a constant referred to as \(m\) is used to control the width of the Gaussian function. Subsequently, the effect of changing \(m\) is presented. The obtained results from comparing the present basis and the Gaussian RBF reveal that, as \(m\) approaches zero, the present basis approaches the mean value of Gaussian RBF over the reference shape parameter interval (\(\varepsilon_{0}\)). The proposed model is benchmarked and compared to a simple function. According to the results of this study, the proposed model has a good accuracy that is independent of \(\varepsilon\) and \(N_{\varepsilon}\) (the number of points to divide \(\varepsilon_{0}\) interval equally) when \(m\) is constant. Meanwhile, the effect of varying \(m\) instead of \(N_{\varepsilon}\) is also studied. The results indicate that \(m\) and \(\varepsilon\) are inversely proportional to each other at a constant accuracy. The accuracy can be made independent of \(m\) and \(\varepsilon\) by altering the interval of \(\varepsilon_{0}\). The results also reveal that an interval of \([-1,1]\) is an excellent choice because it produces accuracy that is comparable to the maximum accuracy of the Gaussian RBF and is also independent of \(\varepsilon\), \(m\), and \(N_{\varepsilon}\). The main novelty of the current work is in introducing a way to get a reasonable accuracy using RBF interpolation without the need to pinpoint a specific value for the shape parameter.  
Figure 10: The log10 of the norm of the error of approximating the given function \(f(x,y)\) and its first derivatives using Gaussian RBF-TPU with zeroth truncation order for a range of \(\varepsilon\) and \(m\) values. The rows are for function, first x-derivative, and first y-derivative approximations. The \(\varepsilon_{k}\) range is \([-1,1]\) and \(N_{\varepsilon}=5\).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Mesh Distribution**} & \multirow{2}{*}{**Model**} & \multicolumn{3}{c}{**Min log10[Norm(Error)]**} \\ \cline{3-5}  & & \(f\) & \(\frac{\partial f}{\partial\varepsilon}\) & \(\frac{\partial f}{\partial y}\) \\ \hline \multirow{2}{*}{Regular} & RBF & \(-0.707\) & \(0.66\) & \(0.66\) \\  & RBF-TPU & \(-0.713\) & \(0.65\) & \(0.65\) \\ \hline \multirow{2}{*}{Stretched (i.e., Equation (20))} & RBF & \(-1.840\) & \(-0.260\) & \(-0.260\) \\  & RBF-TPU & \(-1.117\) & \(0.132\) & \(0.145\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of the minimum error for approximating derivatives between Gaussian RBF and RBF-TPU for the 2D case (Equations (24)–(26)).  
**Author Contributions:** Conceptualization, S.A.B.; Formal analysis, S.A.B.; Investigation, S.S.B. and S.A.B.; Methodology, S.A.B.; Supervision, A.A.M.; Validation, S.S.B. and S.A.B.; Visualization, S.S.B. and S.A.B.; Writing--original draft, S.A.B.; Writing--review & editing, S.S.B. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not available.  
**Conflicts of Interest:** The authors declare no conflict of interest.