# 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 for \(k=1,\ldots,n\)_  
\[F_{k}=\lambda f\left(t_{k}\right)+\mathcal{O}\left(h^{2}\right), \tag{10}\]_where \(0<\lambda=1/\left(\int_{-1}^{1}\beta K(t)dt\right)\) and \(\int_{-1}^{1}\beta K(t)dt\
eq 0\)._  
Proof. : By locality condition of definition of \(h\)-uniform generalized fuzzy partition, Corollary 1, Lemma 1, and according to [17], using the trapezoid formula with nodes \(t_{k-1}\), \(t_{k}\), \(t_{k+1}\) to the numerical computation of the integral, we get for \(k=1,\ldots,n\) and \(0<\lambda=1/\left(\int_{-1}^{1}\beta K(t)dt\right)\)  
\[F_{k} =\frac{1}{h}\int_{t_{k-1}}^{t_{k+1}}f\left(t\right)B_{k}(t)\,dt,\] \[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}. \tag{7}\]0 \[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}. \tag{7}\]1  
**Corollary 3**. : _Let \(f\left(t\right)\in C^{2}\left[a,b\right]\) and the conditions of Lemma 2 be fulfilled. Let moreover, \(f\) be Lipschitz continuous with respect to \(t\), i.e., there exists a constant \(L\in\mathbb{R}\), such that for all \(t\in\left[a,\,b\right]\) and \(t,t^{\prime}\in\mathbb{R}\),_  
\[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}. \tag{7}\]2  
_Then for \(k=1,\ldots,n\)_  
\[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}. \tag{7}\]3  
_where \(0<\lambda=1/\left(\int_{-1}^{1}\beta K(t)dt\right)\), \(\int_{-1}^{1}\beta K(t)dt\
eq 0\), \(M=\max\limits_{t\in\left[t_{k-1},\,t_{k+1}\right]}\left|f^{\prime\prime}\left( t\right)\right|\) and \(\left|t-t_{k}\right|<h\) whenever \(t\in\left[t_{k-1},\,t_{k+1}\right]\)._  
Proof. : By the assumption \(f\) has continuous second order derivatives on \(\left[a,b\right]\) and is Lipschitz continuous with respect to \(t\). Therefore, using the trapezoid rule and let us choose a value of \(k\) in the range \(1\leq k\leq n\) and \(t\in\left[t_{k-1},\,t_{k+1}\right]\), we get for \(0<\lambda=1/\left(\int_{-1}^{1}\beta K(t)dt\right)\)  
\[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}. \tag{7}\]4 \[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}. \tag{7}\]5 \[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}. \tag{7}\]6 \[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}. \tag{7}\]7  
where \(\xi_{k-1}\in\left(t_{k-1},t_{k}\right)\), \(\xi_{k+1}\in\left(t_{k},t_{k+1}\right)\) and \(M=\max\limits_{t\in\left[t_{k-1},\,t_{k+1}\right]}\left|f^{\prime\prime}\left( t\right)\right|\). **Remark 2**. : _In view of (13), if \(0<\lambda\leq 1\). Then, \(\left|f\left(t\right)-\frac{1}{\lambda}F_{k}\right|\leq Lh+\frac{h^{2}}{6}M\)._  
**Definition 6**. : _Let \(F[f]=\left(F_{1},\,F_{2},\,\ldots,\,F_{n}\right)\) be direct FzT of a function \(f\in C\left[a,b\right]\) with respect to the fuzzy partition \(B_{k}(t)\), \(k=1,\ldots,n\) of \(\left[a,b\right]\).