# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II
## 3 New Representations for Basic Functions of FzT
### Generalized Uniform Fuzzy Partitions with the Generalized Normal Case
Then, the function \(\hat{f}\) defined on \(\left[a,b\right]\)_  
\[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}. \tag{7}\]8  
_is called the inverse FzT of \(f\)._  
The following lemma estimates the difference between the original function and its inverse FzT. **Lemma 4**: _Let the assumptions of Theorem 1 and let \(\hat{f}\left(t\right)\) be the inverse FzT of \(f\) with respect to the fuzzy partition of \(\left[a,b\right]\) is given by Definition 4. Then, the following estimation holds for \(t\in\left[a,b\right]\) and \(k=1,\ldots,n\)_  
\[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}. \tag{7}\]9  
_where \(0<\lambda=1/\left(\int_{-1}^{1}\beta K(t)dt\right)\) and \(\int_{-1}^{1}\beta K(t)dt\
eq 0\)._  
Let \(t\in\left[a,b\right]\) so that \(t\in\left[t_{k},t_{k+1}\right]\) for some \(k=1,\ldots,n\). By Theorem 1,  
\[B_{k}\left(t\right)=\lambda\beta K\left(\frac{t-t_{k}}{h}\right),\ \ t\in\left[t_{k}-h,t_{k}+h\right],\]0 \[B_{k}\left(t\right)=\lambda\beta K\left(\frac{t-t_{k}}{h}\right),\ \ t\in\left[t_{k}-h,t_{k}+h\right],\]1  
Let \(f\in C\left[a,b\right]\). Thus, for any \(\varepsilon>0\) there exist \(n_{\varepsilon}\in\mathbb{N}\) and \(B_{1},\ldots,B_{n_{\varepsilon}}\) be the \(h\)-uniform generalized fuzzy partition of \(\left[a,b\right]\) defined by (5). Then, the following estimations hold \(\left|\hat{f}\left(t\right)-f\left(t\right)\right|<\varepsilon\) for each \(t\in\left[a,b\right]\cap\left[t_{k},t_{k+1}\right]\). From the proof of Lemma 4 and then using Lemma (3) in the sense that for all \(k=1,\ldots,n\),  
\[B_{k}\left(t\right)=\lambda\beta K\left(\frac{t-t_{k}}{h}\right),\ \ t\in\left[t_{k}-h,t_{k}+h\right],\]2  
According to Definition (4), it is easy to see that the inverse \(FzT\hat{f}\left(t_{k}\right)=F_{k}\) for all \(k=1,\ldots,n\). On the basis of Definition 4, necessary steps of a new method to construct generalized uniform fuzzy partitions of \(\left[-1,1\right]\) for solve case \(K\) is not normal in the following. 1. Select the generating function \(K\) which is assumed to be even, continuous and \(K\left(t\right)>0\) if \(t\in\left(-1,1\right)\). 2. Specify the value \(\beta=1/K(0)\), where \(K(0)\
eq 0\) to get the normal generating function \(K\) and then compute the value \(\lambda=1/\left(\int_{-1}^{1}\beta K(t)dt\right)\), where \(\int_{-1}^{1}\beta K(t)dt\
eq 0\). 3. If conditions \(\beta>0\) and \(\lambda>0\) holds, then construct generalized uniform fuzzy partitions of \(\left[-1,1\right]\) by \(\lambda\beta K\left(t\right)\). Let \(K:\mathbb{R}\rightarrow\left[0,1\right]\) be defined by  
\[B_{k}\left(t\right)=\lambda\beta K\left(\frac{t-t_{k}}{h}\right),\ \ t\in\left[t_{k}-h,t_{k}+h\right],\]3  
One can see in Table 1 the \(h\)-uniform generalized fuzzy partition of \(\left[a,b\right]\) determined by Definition 4. The following remark is for modified Trapezoidal rule based on FzT and NIM to solve SODEs. In view of Equation (9), \(\int_{t_{k-1}}^{t_{k+1}}B_{k}(t)\,dt=h\). This means that \(\int_{t_{k}}^{t_{k+1}}B_{k}(t)\,dt=\frac{h}{2}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{K}_{\mathbf{z}}^{\mathbf{m}}\left(\mathbf{t}\right)\) & \(\beta\) & \(\lambda\) & \(B_{k}=\lambda\beta K\left(\frac{t-t_{k}}{h}\right)\) \\ \hline \(\left(1+\cos\left(\pi t\right)\right)^{m}\) & \(\frac{1}{2^{m}}\) & \(\frac{\sqrt{\pi}\Gamma\left(m+1\right)}{2^{\Gamma}\left(m+\frac{1}{2}\right)}\) & \(\left(\frac{\sqrt{\pi}\Gamma\left(m+1\right)}{2^{\Gamma}\left(m+\frac{1}{2} \right)}\right)\frac{1}{2^{m}}\left(1+\cos\left(\pi\frac{t-t_{k}}{h}\right) \right)^{m}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example 1.