# New Fuzzy Numerical Methods for Solving Cauchy Problems
## 2 Basic Concepts  
In this section, we give some definitions and introduce the necessary notation in [10], 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 \(x_{i}\in\left[a,b\right],\ i=1,\ldots,n,\) be fixed nodes such that \(a=x_{1}<\ldots<x_{n}=b,\ n\geq 2\). 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 for every \(i=1,\ldots,n\) there exists \(h>0\) such that \(x_{0}=x_{1},\ x_{n}=x_{n+1},\)\(\left[x_{i}-h,\ x_{i}+h\right]\subseteq\left[a,b\right]\) and the following conditions are fulfilled:_  
1. _(positivity and locality) -_ \(A_{i}\left(x\right)>0\) _if_ \(x\in\left(x_{i-1},x_{i+1}\right)\) _and_ \(A_{i}\left(x\right)=0\) _if_ \(x\in\left[a,b\right]\setminus\left(x_{i-1},x_{i+1}\right)\)_;_
2. _(continuity) -_ \(A_{i}\) _is continuous on_ \(\left[x_{i-1},x_{i+1}\right]\)_;_
3. _(covering) - for_ \(x\in\left[a,b\right],\)__\(\sum_{i=1}^{n}A_{i}\left(x\right)\ >0\)_._  
_Fuzzy sets \(A_{1},\ldots,\)\(A_{n}\) are called basic functions. It is important to remark that by conditions of locality and continuity, \(\int_{a}^{b}A_{i}(x)dx>0\). A generalized of uniform fuzzy partition of \(\left[a,b\right]\) is defined for equidistant nodes, i.e., for all \(i=1,\ldots,n-1,\ x_{i}\ =\ x_{i+1}+h\), where \(h\ =\ \left(b-a\right)/\left(n-1\right)\) and two additional properties are satisfied,_  
1. \(A_{i}\left(x_{i}-x\right)=A_{i}\left(x_{i}+x\right)\) _for all_ \(x\in\left[0,h\right],\ i=2,\ldots,n-1\)_;_
2. \(A_{i}\left(x\right)=A_{i-1}\left(x-h\right)\) _and_ \(A_{i+1}\left(x\right)=A_{i}\left(x-h\right)\) _for all_ \(x\in\left[x_{i},\ x_{i+1}\right],\ i=2,\ldots,n-1\)_;_  
_then the fuzzy partition is called h-uniform generalized fuzzy partition. Throughout this paper, we will write generalized uniform fuzzy partition instead of 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 \(K\left(x\right)>0\) if \(x\in\left(-1,1\right)\). The function \(K:\left[-1,1\right]\rightarrow\mathbb{R}\) is even if for all \(x\in\left[0,1\right],\)\(K\left(-x\right)=K\left(x\right)\)._  
The following definition recall the concept of 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 generalized uniform fuzzy partition of interval \([a,b]\), determined by the triplet \((K,\,h,\,a)\), can be defined using generating function \(K\) (Definition 2). Then, basic functions of a generalized uniform fuzzy partition are shifted copies of \(K\) defined by_  
\[A_{i}\left(x\right)=K\left(\frac{x-x_{i}}{h}\right)\,,\,\,\,x\in\left[x_{i}-h, \,x_{i}+h\right],\]  
_for all \(i=1,\ldots,n\). The parameter \(h\) is called the bandwidth or the shift of the fuzzy partition and the nodes \(x_{i}=a+ih\) are called the central point of the fuzzy sets \(A_{1},\ldots,\,A_{n}\)._  
**Remark 1**.: _A fuzzy partition is called Ruspini if the following condition_  
\[A_{i}\left(x\right)+A_{i+1}\left(x\right)=1,\,i=1,\ldots,n-1, \tag{1}\]  
_holds for any \(x\in\left[x_{i},x_{i+1}\right]\). This condition is often called Ruspini condition._  
**3. New Representations of Basic Functions for Particular Cases**  
In this section, we propose two subsection, new representations of basic functions constitute a generalized uniform fuzzy partition of interval \([a,b]\) and then FT technique based on new representations of basic functions.