# 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  
Let us recall the \(h\)-uniform generalized fuzzy partition of real line can be defined using generating function \(K\). Then, basic functions of the \(h\)-uniform generalized fuzzy partition are shifted copies of \(K\). On the basis of Definition 1 can be also defined using a generating function \(\lambda\beta K(t)\) where \(\beta=1/K(0)\), \(K(0)\
eq 0\), \(\beta>0\) and \(\lambda>0\) (in general, not necessarily satisfying normal and Ruspini condition) which is that \(K(t)\) assumed to be even, continuous and \(K\left(t\right)>0\) if \(t\in(-1,1)\). Therefore, we will modify the basic functions of the \(h\)-uniform generalized fuzzy partition so that they are shifted copies of \(\lambda\beta K\) defined by \[A_{k}\left(t,t_{0}\right)=\lambda\beta K\left(\frac{t-t_{0}}{h}-k\right),\ t\in \left[t_{k}-h,t_{k}+h\right],\ k\in\mathbb{Z}. \tag{4}\]  
The parameter \(h\) is bandwidth of the fuzzy partition and \(t_{0}+kh=t_{k}\). The concept of the \(h\)-uniform generalized fuzzy partition can be easily extended to the interval \(\left[a,b\right]\) as follows. **Definition 4**. : _Let \(t_{1}<\ldots<t_{n}\) be fixed nodes within \(\left[a,b\right]\subset\mathbb{R}\), such that \(t_{1}=a\), \(t_{n}=b\) and \(n\geq 2\). We consider nodes \(t_{1},\ldots,\ t_{n}\) are equidistant, with distance (shift) \(h=\left(b-a\right)/\left(n-1\right)\). A system of fuzzy sets \(B_{1},\ldots,\ B_{n}\ \left[a,b\right]\rightarrow\left[0,1\right]\) be a generalized uniform fuzzy partitions of \(\left[a,b\right]\) if it is defined by_  
\[B_{k}\left(t\right) = \begin{cases}A_{k}(t,a),&t\in\left[a,b\right],\\ 0,&\text{otherwise}.\end{cases}  = \begin{cases}\lambda\beta K\left(\frac{t-t_{k}}{h}\right),&t \in\left[a,b\right],\\ 0,&\text{otherwise}.\end{cases}  \tag{5}\]  
_where \(t_{k}=a+(k-1)h\), \(\beta=1/K(0)\), \(K(0)\
eq 0\), \(\beta>0\) and \(\lambda>0\). In the sequel, a generating function denote by \(K\) and basic functions of \(FzT\) denote by \(B_{k},\ k=1,\ldots,n\)._  
**Lemma 1**. : _If basic functions \(B_{k},\ k=1,\ldots,n,\) of a \(h\)-uniform generalized fuzzy partition are shifted copies of \(\lambda\beta K\) defined by (5). Then, for each \(k=1,\ldots,\ n,\ B_{k}(t_{k})=\lambda,\ t_{k}\in\left[t_{k}-h,t_{k}+h\right]\)._  
Proof. : By (5), we get \(B_{k}\left(t_{k}\right)=\lambda\beta K\left(\frac{t_{k}-t_{k}}{h}\right)=\lambda\). Simpler Form of F-Transform Components Based on Generalized Uniform Fuzzy Partitions with the Generalized Normal Case  
In this subsection, we present the main principles of FzT with respect to new representations of \(h\)-uniform generalized fuzzy partition. Further, we will show that FzT components with respect to new representations of \(h\)-uniform generalized fuzzy partition can be simplified and approximated of an original function, say \(f\). **Definition 5**. : _Let \(f\) be a continuous function on \(\left[a,b\right]\) and \(B_{k}(t)\), \(k=1,\ldots,n\), be \(h\)-uniform generalized fuzzy partition of \(\left[a,b\right]\), \(n\geq 2\). A vector of real numbers \(F[f]=\left(F_{1},\ F_{2},\ldots,\ F_{n}\right)\) given by_  
\[F_{k}=\frac{\int_{a}^{b}f\left(t\right)B_{k}(t)\,dt}{\int_{a}^{b}B_{k}(t)\,dt}, \tag{6}\]  
_for \(k=1,\ldots,n\) is called the direct FzT of \(f\) with respect to \(B_{k}\)._  
In the following, we will simplify the representation (6). **Lemma 2**. : _Let \(f\in C\left(\left[a,b\right]\right)\) and according to Definition 4, fuzzy sets \(B_{k},\ k=1,\ldots,n,\ n\geq 2\), be a \(h\)-uniform generalized fuzzy partition of \(\left[a,b\right]\) with a generating function \(K\), then representation (6) of direct FzT can be simplified for \(k=1,\ldots,n\) as follows_  
\[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}. \tag{7}\]  
Proof.