## 4 Partition of Unity for Taylor Series Radial Basis Function (RBF-TPU)  
As evidenced in the previous section, the coupling between \(r\) and \(\varepsilon\) becomes weak as \(\varepsilon\) moves away from \(\varepsilon_{0}\). In order to resolve this issue, \(N_{\varepsilon}\) different reference values of shape parameter (\(\varepsilon_{k}\)) are used, which help strengthen the coupling at \(\varepsilon_{k}\) values of the shape parameter. \[\phi(r,\varepsilon)=\sum_{n=0}^{O}\frac{\partial^{n}\phi(r,\varepsilon)}{ \partial\varepsilon^{n}}\bigg{|}_{\varepsilon=\varepsilon_{0}}\frac{( \varepsilon-\varepsilon_{0})^{n}}{n!} \tag{10}\]  
This equation (i.e., Equation (10)) is identical to \(\phi\) at \(\varepsilon=\varepsilon_{k}\) values. In order to find a proper approximation of \(\phi\) in between \(\varepsilon_{k}\) values, the set of equations (i.e., Equation (10)  
Figure 2: The log10 of the difference between Gaussian radial basis function (RBF) and its Taylor approximation (RBF-TS) as a function of \(\varepsilon\) and \(\varepsilon_{0}\) for different truncation orders. The rows are for function, first derivative, and second derivative approximations of RBF, while the columns are for zeroth (\(O=0\)), first (\(O=1\)), and second (\(O=2\)) orders of truncation. with different values of \(\varepsilon_{k}\)) need to be somehow linked. This can be achieved by using the partition of unity approach with respect to the shape parameter, where each equation is weighted with an appropriate weight function (\(w(\varepsilon,\varepsilon_{k})\)). The results are summed over \(k\). \[\phi(r,\varepsilon)\sum_{k=1}^{N_{e}}w(\varepsilon,\varepsilon_{k})=\sum_{k=1} ^{N_{e}}w(\varepsilon,\varepsilon_{k})\,\sum_{n=0}^{O}\frac{\partial^{n}\phi(r,\varepsilon)}{\partial e^{n}}\bigg{|}_{\varepsilon=\varepsilon_{0}}\frac{( \varepsilon-\varepsilon_{k})^{n}}{n!} \tag{11}\]  
Now, \(\phi\) could be isolated on the left-hand side by dividing both sides of Equation (11) by \(\sum_{k=1}^{N_{e}}w(\varepsilon,\varepsilon_{k})\), which leads to a better approximation for \(\phi\) over a wider range of the shape parameter. \[\phi(r,\varepsilon)=\sum_{k=1}^{N_{e}}W(\varepsilon,\varepsilon_{k})\,\sum_{ n=0}^{O}\frac{\partial^{n}\phi(r,\varepsilon)}{\partial\varepsilon^{n}} \bigg{|}_{\varepsilon=\varepsilon_{0}}\frac{(\varepsilon-\varepsilon_{k})^{ n}}{n!}, \tag{12}\]  
where the partition weight (\(W(\varepsilon,\varepsilon_{k})\)) is expressed as follows:  
\[W(\varepsilon,\varepsilon_{k})=\frac{w(\varepsilon,\varepsilon_{k})}{\sum_{ j=1}^{N_{e}}w(\varepsilon,\varepsilon_{j})}. \tag{13}\]  
It should be emphasized that the primary purpose of Equation (12) is to find a new basis that is not necessarily identical to the Gaussian basis. This new basis should have a weak dependency on the shape parameter and accuracy that is comparable to the Gaussian basis for the interpolation as well as the derivative approximation. To use the new basis (Equation (12)), the considered range of shape parameter (\(\varepsilon\)) should be divided into \(N_{e}\) slightly overlapped sub-ranges, where non-negative weights for the partition of unity approximation (\(w\)) should be selected [55]. Regardless of the selected weight function, the sum of partition weights is one (\(\sum_{k=1}^{N_{e}}W(\varepsilon,\varepsilon_{k})=1\)), which is where the name of the partition of unity comes from. In this work, two simple weight functions will be considered. The first and simplest weight function is \(w=1\), which can be interpreted as averaging \(\phi\) over a given shape parameter range. Bell shape functions are generally chosen for a partition of unity approximation. They approximate cardinal condition (unity at the reference point and monotonically decreasing till approaches zero at neighbors), which is a desired property. This property makes the Gaussian function (\(w(\varepsilon,\varepsilon_{0})=exp(-m(\varepsilon-\varepsilon_{0})^{2})\)) a good candidate for the weight function. Hence, the Gaussian function is considered as a second weight function in this study.