## 4 Partition of Unity for Taylor Series Radial Basis Function (RBF-TPU)
The \(m\) parameter is introduced to control the width of the function. As \(m\) approaches zero, the weight becomes flat and approaches one (the simplest weight). The compactly supported weight function can be emulated by changing the value \(m\) in the weights. The partition of unity approach can be applied to Equations (8) and (9) to find approximations of derivatives of \(\phi\). \[\frac{\partial\phi(r,\varepsilon)}{\partial x_{a}}=\ \sum_{k=1}^{N_{e}}W( \varepsilon,\varepsilon_{k})\,\sum_{n=0}^{O}\frac{\partial^{n+1}\phi(r, \varepsilon)}{\partial\varepsilon^{n}\partial x_{a}}\bigg{|}_{\varepsilon= \varepsilon_{k}}\frac{(\varepsilon-\varepsilon_{k})^{n}}{n!} \tag{14}\]  
\[\frac{\partial^{2}\phi(r,\varepsilon)}{\partial x_{a}\partial x_{g}}=\ \sum_{k=1}^{N_{e}}W( \varepsilon,\varepsilon_{k})\,\sum_{n=0}^{O}\frac{\partial^{n+2}\phi(r, \varepsilon)}{\partial\varepsilon^{n}\partial x_{a}\partial x_{g}}\bigg{|}_{ \varepsilon=\varepsilon_{k}}\frac{(\varepsilon-\varepsilon_{k})^{n}}{n!} \tag{15}\]  
Figure 3 illustrates the effect of the weight function; more specifically, Figure 3a shows the effect of the first weight function (\(w=1\)), while the second weight function is presented in the rest of the sub-figures for dif-ferent value of \(m\) (i.e., Figure 3b for \(m=0.01\), Figure 3c for \(m=4\), and Figure 3d for \(m=8\)). In Figure 3, the arrangement of rows and columns of sub-figures is similar to that of Figure 2; however, there are two main differences. Firstly, the results are obtained from the partition of unity approximations (Equations (12), (14) and (15)) rather than the Tay-lor series. Secondly, the vertical axes in Figure 3 are the number  
of points to equally divide the \(\varepsilon_{k}\) range (\(N_{e}\)), as the partition of unity approximation is contingent on a range of \(\varepsilon_{k}\) values (taken as \([0,1]\)) instead of a single value (\(\varepsilon_{0}\)). As shown in Figure 3(a) (\(w=1\)) and Figure 3(b) (\(m=0.01\)), the results are almost identical because \(m\) is a very small value. The results can be confirmed analytically by taking the limit of \(exp(-m(\varepsilon-\varepsilon)^{2})\) as \(m\) approaches zero, which results in one (the same weight as the one used for Figure 3(a)). The minimum error occurs at approximately \(\varepsilon=0.5\), which denotes the mean value of the considered range. This is expected as the weight is flat when \(m\) is low, which causes the average value to show up in the middle of the range. The area of the middle region with the minimum error increases with an increase in the truncation order. This indicates that, as \(O\) becomes higher, the present new basis and Gaussian RBF become identical. Figure 3: The log10 of the difference between Gaussian RBF and its partition of unity for Taylor series RBF (RBF-TPU) approximation as a function of \(\varepsilon\) and the number of points of \(\varepsilon_{k}\) (\(N_{e}\)) for Gaussian RBF and different truncation orders. The rows are for function, first derivative, and second derivative approximations of RBF, while the columns are for the zeroth (\(O=0\)), first (\(O=1\)), and second (\(O=2\)) orders of truncation. Figure 3c (\(m=4\)) illustrates that the region with the lowest error is deviated from \(\varepsilon=0.5\), unlike the results of Figure 3b. The region is wider when compared with the results of the lower \(m\) values. That is attributed to the weight, which becomes a narrow bell shape as \(m\) increases. With a view to ascertaining this observation, the results for an even higher \(m\) value (\(m=8\)) are plotted in Figure 3d. The figure shows an even larger accurate region and lower error than Figure 3c. These results indicate that the variable \(m\) controls how the distances (\(r\)) and shape parameter (\(\varepsilon\)) are related. Furthermore, these results do not reveal any significant change when \(N_{e}\) is greater than 5, which is why \(N_{e}=5\) is used in this study unless stated otherwise. The present new bases are validated in the next section.