# 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
: By Definition 4, we get  
\[B_{k}\left(t\right)=\lambda\beta K\left(\frac{t-t_{k}}{h}\right),\ \ t\in\left[t_{k}-h,t_{k}+h\right],\]  
for \(k=1,\ldots,n\), \(t_{0}=t_{1},\ t_{n+1}=t_{n}\), and substituting \(u=\frac{t-t_{k}}{h}\) and then substituting \(t=s/h\). Thus, we get \[\int_{t_{k-1}}^{t_{k+1}}\,f\left(t\right)B_{k}(t)\,dt=\lambda\beta h\int_{-1}^{1}f \left(th+t_{k}\right)K(t)\,dt=\lambda\beta\int_{-h}^{h}\,f\left(t+t_{k}\right)K( \frac{t}{h})\,dt,\]  
\[\int_{t_{k-1}}^{t_{k+1}}\,B_{k}(t)\,dt=\lambda\beta h\int_{-1}^{1}K(t)\,dt= \lambda\beta\int_{-h}^{h}\,K(\frac{t}{h})\,dt,\]  
and its corresponding results with representation (6). \(\qed\)  
If \(\lambda>0\), the Lemma 1 still hold by choosing suitable constant \(\lambda\), satisfying \(\lambda=1/\left(\int_{-1}^{1}\beta K(t)dt\right)\), where \(\int_{-1}^{1}\beta K(t)dt>0\). So, we will restrict ourselves to \(h\)-uniform generalized fuzzy partition with \(0<\lambda=1/\left(\int_{-1}^{1}\beta K(t)dt\right)\), where \(\int_{-1}^{1}\beta K(t)dt\
eq 0\). In the following, we will simplify the above given expressions for the coefficients \(F[f]=\left(F_{1},\ F_{2},\ldots,\ F_{n}\right)\) in the representation (6). This fact is very important for applications which are more flexible and consequently easier to use. **Corollary 1**. : _Let the assumptions of Lemma 2 be fulfilled and \(0<\lambda=1/\left(\int_{-1}^{1}\beta K(t)dt\right)\), where \(\int_{-1}^{1}\beta K(t)dt\
eq 0\). Then, the coefficients \(F[f]=\left(F_{1},\ F_{2},\ldots,\ F_{n}\right)\) in the expression (6) of the FzT component \(F_{k}\) of \(f\) as follows:_  
\[F_{k}=\frac{1}{h}\int_{a}^{b}f\left(t\right)B_{k}(t)\,dt=\frac{\lambda\beta}{h }\int_{a}^{b}f\left(t\right)K\left(\frac{t-t_{k}}{h}\right)\ dt, \tag{8}\]  
_for \(k=1,\ldots,\,n\), where interval \([a,b]\) is partitioned by the \(h\)-uniform generalized fuzzy partition \(B_{1},\ldots,\ B_{n}\)._  
**Proof. ** Let \(k\in\{1,\ldots,\,n\}\) and consider set of fuzzy sets \(B_{k}(t)\) be the \(h\)-uniform generalized fuzzy partition of \([a,b]\) defined by (5). Using the proof of Lemma 2, we get  
\[\int_{t_{k-1}}^{t_{k+1}}B_{k}(t)\,dt=\int_{t_{k-1}}^{t_{k+1}}A_{k}(t,a),dt= \int_{t_{k}-h}^{t_{k}+h}\lambda\beta K\left(\frac{t-t_{k}}{h}\right)\ dt=h \lambda\int_{-1}^{1}\beta K\left(t\right)\,dt=h, \tag{9}\]  
where \(0<\lambda=1/\left(\int_{-1}^{1}\beta K(t)dt\right)\), \(\int_{-1}^{1}\beta K(t)dt\
eq 0\), \(h\) is bandwidth of the fuzzy partition and \(t_{k}=a+(k-1)h\) and then its corresponding in the expression (6). \(\qed\)  
**Lemma 3**. : _Let \(f\in C\left[a,b\right]\). Then for any \(\varepsilon>0\) there exist \(n_{\varepsilon}\in\mathbb{N}\) and \(B_{1},\ldots,\ B_{n_{\varepsilon}}\) be basic functions form the \(h\)-uniform generalized fuzzy partition of \([a,b]\). Let \(F_{k},\ k=1\ldots,\ n\), be the integral FzT components of \(f\) with respect to \(B_{1},\ldots,\ B_{n_{\varepsilon}}\). Then for each \(k=1\ldots,\ n_{\varepsilon}-1\) the following estimations hold: \(\left|f(t)-F_{i}\right|\leq\varepsilon\) for each \(t\in[a,b]\cap[t_{k},t_{k+1}]\) and \(i=k\), \(k+1\)._  
**Proof. ** see [5]. \(\qed\)  
**Corollary 2**. : _Let the conditions of Lemma 3 be fulfilled. Then for each \(k=1\ldots,\ n_{\varepsilon}-1\) the following estimations hold: \(\left|F_{k}-F_{k+1}\right|<\varepsilon\)._  
**Proof. ** According to [5,39], let \(t\in[a,b]\cap[t_{k},t_{k+1}]\). Then by Lemma 3, for any \(k=1,\ldots,n-1\) we obtain that \(\left|f\left(t\right)-F_{k}\right|<\varepsilon/2\) and \(\ \left|f\left(t\right)-F_{k+1}\right|<\varepsilon/2\). Thus,  
\[\left|F_{k}-F_{k+1}\right|\leq\left|f\left(t\right)-F_{k}\right|+\left|f\left(t \right)-F_{k+1}\right|<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon.\qed\]  
The following theorem estimates the difference between the original function and its direct FzT with respect to the \(h\)-uniform generalized fuzzy partition. **Theorem 1**. : _Let \(f\left(t\right)\in C^{2}\left[a,b\right]\) and the conditions of Lemma 2 be fulfilled.