# New Fuzzy Numerical Methods for Solving Cauchy Problems
## 2 Basic Concepts
### Power of the Triangular and Raised Cosine Generalized Uniform Fuzzy Partition
On the basis of Definitions 4 and 5, Lemma 1, and according to Definition 3, we can also be defined using generating function \(\alpha K\) for \(\alpha>0\) (in general, not necessarily satisfy Ruspini condition). Thus, basic functions of a generalized uniform fuzzy partition are shifted copies of \(\alpha K\) defined by  
\[A_{k}\left(x,x_{0}\right)=\alpha K\left(\frac{x-x_{0}}{h}-k\right),\ x\in\left[x _{i-1},x_{i+1}\right]. \tag{5}\]  
In particular, let \(K_{T_{1}^{m}},\ K_{T_{2}^{m}},\) (and \(K_{C^{m}}\)) be power of the triangular (and raised cosine) generating function defined above. We will say that a generalized uniform fuzzy partition is power of a triangular (or of raised cosine) generalized uniform fuzzy partition if its generating function \(K\) belongs to \(\alpha K_{T_{1}^{m}},\)\(\alpha K_{T_{2}^{m}},\) (or \(\alpha K_{C^{m}}\)) whenever \(\alpha=1/\left(\int_{-1}^{1}K(t)dt\right)\). Indeed, the equality \(\alpha\) immediately follows from \(\int_{-1}^{1}\alpha K_{T_{1}^{m}}\left(t\right)dt=1\Rightarrow\alpha=1/\left( \int_{-1}^{1}K_{T_{1}^{m}}\left(t\right)dt\right)\). In the following, we modified the definition a triangular and raised cosine generalized uniform fuzzy partition by propose that power of the triangular and raised cosine generalized uniform fuzzy partitions can be simply using the equality \(\alpha=1/\left(\int_{-1}^{1}K(t)dt\right)\). **Definition 6**. : _Let \(m\in\mathbb{N}^{+}\). A system of fuzzy sets \(\left\{A_{k}\ |\ k\in\mathbb{Z}\right\}\) defined by_  
\[1.\ A_{k}\left(x,x_{0}\right) = \alpha K_{T_{1}^{m}}\left(\frac{x-x_{0}}{h}-k\right),\ \alpha=\frac{m+1}{2}, \tag{6}\] \[2.\ A_{k}\left(x,x_{0}\right) = \alpha K_{T_{2}^{m}}\left(\frac{x-x_{0}}{h}-k\right),\ \alpha=\frac{m+1}{2m}, \tag{7}\]_is called power of the triangular generalized uniform fuzzy partition of the real line determined by the triplet \(\left(K_{T_{i}^{m}},\,h,\,x_{0}\right),\,\,i=1,2\). Further, let \(m\) is an odd natural number. A system of fuzzy sets \(\left\{A_{k}\mid k\in\mathbb{Z}\right\}\) defined by_  
\[3.\,\,A_{k}\left(x,x_{0}\right) = \alpha K_{C^{m}}\left(\frac{x-x_{0}}{h}-k\right),\,\,\,\alpha=1, \tag{8}\]  
_is called power of the raised cosine generalized uniform fuzzy partition of the real line determined by the triplet \(\left(K_{C^{m}},h,,\,\,x_{0}\right)\). The parameter \(h\) is bandwidth of the fuzzy partition and \(x_{0}+kh=x_{k}\)._  
**Definition 7**. : _Let \(x_{1}<\ldots<x_{n}\) be fixed nodes within \(\left[a,b\right]\subset\mathbb{R}\), such that \(x_{1}=a\), \(x_{n}=b\) and \(n\geq 2\). We consider nodes \(x_{1},\ldots,\,x_{n}\) 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 power of a triangular and raised cosine generalized uniform fuzzy partitions of \(\left[a,b\right]\) if it is defined by_  
\[B_{k}\left(x\right) = \begin{cases}A_{k}(x,a),&x\in\left[a,b\right],\\ 0,&\text{otherwise.}\end{cases} \,\,\text{or equivalent}\,\,\,\,\,\,\,\,B_{k} \left(x\right) = \begin{cases}\alpha K\left(\frac{x-x_{k}}{h}\right),&x\in\left[a,b \right],\\ 0,&\text{otherwise.}\end{cases}  \tag{9}\]  
_where \(x_{k}=a+kh\). In the sequel, we denote \(K\) for a generating function determined by the Formulas (2)-(4). Further, \(\alpha,\,A_{k}(x,a),\,\,k=1,\ldots,n\), are determined by the Formulas (6)-(8)._  
**Lemma 2**. : _If \(B_{k}\left(x\right)\) determines power of the raised cosine generalized uniform fuzzy partition of \(\left[a,b\right]\), then \(B_{k}\left(x\right)\) satisfied Ruspini condition (1) when \(m\) (see (4) ) is an odd natural number._  
**Proof. ** Indeed, if \(x\in\left[a,b\right]\), there exists \(k\in\left\{1,\ldots,n-1\right\}\) such that \(x\in\left[x_{k},x_{k+1}\right]\).