# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II
## 2 Basic Concepts  
In this section, we give some definitions and introduce the necessary notation following [38], which will be used throughout the paper. Throughout this section, we deal with an interval \(\left[a,b\right]\subset\mathbb{R}\) of real numbers. **Definition 1**. : (_generalized uniform fuzzy partition) Let \(t_{i}\in\left[a,b\right],\ i=1,\ldots,n\), be fixed nodes such that \(a=t_{1}<\ldots<t_{n}=b\), \(t_{0}=t_{1}\), \(t_{n}=t_{n+1}\), \(n\geq 2\) and \(\left[t_{i}-h,\,t_{i}+h\right]\subseteq\left[a,b\right]\). We say that the fuzzy sets \(A_{i}:\left[a,b\right]\rightarrow\left[0,1\right]\) constitute a generalized fuzzy partition of \(\left[a,b\right]\) if the following conditions are fulfilled:_  
1. _(positivity and locality)--_\(A_{i}\left(t\right)>0\) _if_ \(t\in\left(t_{i-1},t_{i+1}\right)\) _and_ \(A_{i}\left(t\right)=0\) _if_ \(t\in\left[a,b\right]\setminus\left(t_{i-1},t_{i+1}\right)\)_;_
2. _(continuity)--_\(A_{i}\) _is continuous on_ \(\left[t_{i-1},t_{i+1}\right]\)_;_
3. _(covering)--_\(t\in\left[a,b\right],\)__\(\sum_{i=1}^{n}A_{i}\left(t\right)\ >0\)_._  
_Fuzzy sets \(A_{1},\ldots,\)\(A_{n}\) are called basic functions. It is important to remark that by conditions of locality and continuity, \(\int_{a}^{b}A_{i}(t)dt>0\). A generalized uniform fuzzy partition of \(\left[a,b\right]\) is defined for equidistant nodes, i.e., for all \(i=1,\ldots,n-1,\ t_{i}=t_{i+1}+h\), where \(h=\left(b-a\right)/\left(n-1\right)\) and two additional properties are satisfied,_  
1. \(A_{i}\left(t_{i}-t\right)=A_{i}\left(t_{i}+t\right)\) _for all_ \(t\in\left[0,h\right],\)__\(i=2,\ldots,n-1\)_;_
2. \(A_{i}\left(t\right)=A_{i-1}\left(t-h\right)\) _and_ \(A_{i+1}\left(t\right)=A_{i}\left(t-h\right)\) _for all_ \(t\in\left[t_{i},\,t_{i+1}\right],\)__\(i=2,\ldots,n-1\)_;_  
_then the fuzzy partition is called h-uniform generalized fuzzy partition._  
**Definition 2**. : (_generating function_) _A function \(K:\left[-1,1\right]\rightarrow\left[0,1\right]\) is called a generating function if it is assumed to be even, continuous and \(K\left(t\right)>0\) if \(t\in\left(-1,1\right)\). The function \(K:\left[-1,1\right]\rightarrow\mathbb{R}\) is even if for all \(t\in\left[0,1\right],\)\(K\left(-t\right)=K\left(t\right)\)._  
The following definition recalls the concept of the generalized fuzzy partition which can be easily extended to the interval \(\left[a,b\right]\). We assume that \(\left[a,b\right]\) is partitioned by \(A_{1},\ldots,\)\(A_{n}\), according to Definition 1. **Definition 3**. : _A h-uniform generalized fuzzy partition of interval \(\left[a,b\right]\), determined by the triplet \(\left(K,\,h,\,a\right)\), can be defined using generating function \(K\) (Definition 2). Then, basic functions of a h-uniform generalized fuzzy partition are shifted copies of \(K\) defined by_  
\[A_{i}\left(t\right)=K\left(\frac{t-t_{i}}{h}\right),\ t\in\left[t_{i}-h,\,t_{ i}+h\right],\]_for all \(i=1,\ldots,n\). The parameter \(h\) is called the bandwidth or the shift of the fuzzy partition and the nodes \(t_{i}=a+ih\) are called the central point of the fuzzy sets \(A_{1},\ldots,\,A_{n}\)._  
**Remark 1**. : _A \(h\)-uniform fuzzy partition is called Ruspini if the following condition_  
\[A_{i}\left(t\right)+A_{i+1}\left(t\right)=1,\;i=1,\ldots,n-1, \tag{1}\]  
_holds for any \(t\in[t_{i},t_{i+1}]\). This condition is often called Ruspini condition._  
_New Iterative Method_  
NIM have proposed by [2] for solving linear and nonlinear functional equations of the form  
\[u=f_{1}+N\left(u\right), \tag{2}\]  
where \(f_{1}\) is a known function and \(N\) a non linear operator. Solutions obtained by this method are in the form of rapidly converging infinite series which can be effectively approximated by calculating only the first few terms. In this method non linear operator \(N\) is decomposed as \(N(u)=N(u_{0})+\sum_{i=1}^{\infty}\left\{N\left(\sum_{n=0}^{i}u_{n}\right)-N \left(\sum_{n=0}^{i-1}u_{n}\right)\right\}.\) In [2], the authors were defined the recurrence relation:  
\[