# New Fuzzy Numerical Methods for Solving Cauchy Problems
## 2 Basic Concepts
### Power of the Triangular and Raised Cosine Generalized Uniform Fuzzy Partition  
Two types of basic functions, triangular and sinusoidal shaped membership functions, were proposed by [4, 8]. Later [13], the authors considered different shapes for the basic functions of fuzzy partition. Furthermore, a generalized fuzzy partition appeared in connection with the notion of a higher-degree F-transform [11]. Its even weaker version was implicitly introduced to satisfy the requirements of image compression [14]. Recently, the different conditions for generalized uniform fuzzy partitions was proposed by [10, 12]. Table 1 provides the definition two types of generating function, triangular and raised cosine generating functions [7, 10, 11, 12, 15]. In the following, we present new representations for generating function. In particular, we present three new representations, based on the triangular and raised cosine generating functions: two generating function based on the triangular generating functions and one generating function based on the raised cosine generating function. **Definition 4**. : _(natural order triangular generating function) Let \(K_{T_{i}^{m}}\,:\,\mathbb{R}\rightarrow[0,1],\,i=1,2\), be defined by_  
\[1.\,\,K_{T_{1}^{m}}\left(x\right) = \begin{cases}\left(1-\left|x\right|\right)^{m},&\left|x\right| \leq 1,\\ 0,&\text{otherwise}\end{cases}  = \min\left(\left(1-\left|x\right|\right)^{m},1\right), \tag{2}\] \[2.\,\,K_{T_{2}^{m}}\left(x\right) = \begin{cases}1-\left(\left|x\right|\right)^{m},&\left|x\right| \leq 1,\\ 0,&\text{otherwise}\end{cases}  = \min\left(1-\left(\left|x\right|\right)^{m},1\right), \tag{3}\]  
_are called power of the triangular (shaped) generating functions, when \(m\in\mathbb{N}^{+}\)._ \begin{table}
\begin{tabular}{c c} \hline \hline
**Triangular Generating Function** & **Raised Cosine Generating Function** \\ \hline \(\max\left\{1-\left|x\right|,\,0\right\}\) & \(\frac{1}{2}\left(1+\cos\left(\pi x\right)\right)_{\left|\left[-1,1\right] \right|}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Generating functions of strong uniform fuzzy partition. **Definition 5**. : _(odd natural order raised cosine generating function) Let \(K_{C^{m}}\ :\ \mathbb{R}\rightarrow[0,1]\) be defined by_  
\[K_{C^{m}}\left(x\right) = \begin{cases}\frac{1}{2}\left(1+\cos^{m}\left(\pi x\right)\right),&\left|x\right|\leq 1;\\ 0,&\text{otherwise}.\end{cases}  \tag{4}\]  
_is called power of the raised cosine generating function, when \(m\) is an odd natural number \(\left(\text{i.e., }m=2k-1,\ k\in\mathbb{N}^{+}\right)\)._  
**Remark 2**. : _Particularly, we can check the validity of Equation (4) using the following relation_  
\[K_{C^{m}}\left(x\right) = \begin{cases}\frac{1}{2}\left(1+\cos^{m}\left(\pi x\right)\right),&\left|x\right|\leq 1,\\ 0,&\text{otherwise}.\end{cases} \] \[= \begin{cases}\frac{1}{2}\left(1+\sin^{m}\left(\frac{\pi}{2}\left(2 x+1\right)\right)\right),&\left|x\right|\leq 1,\\ 0,&\text{otherwise}.\end{cases} \]  
**Lemma 1**. : _If \(K_{T_{1}^{m}}\left(x\right),\ i=1,2,\)\(\left(K_{C^{m}}\left(x\right)\right)\) determines power of the triangular (raised cosine) generating functions, then_  
\[1.\quad\int_{-1}^{1}K_{T_{1}^{m}}\left(x\right)dx= \frac{2}{n+1},\qquad 2.\quad\int_{-1}^{1}K_{T_{2}^{m}}\left(x\right)dx= \frac{2n}{n+1},\qquad 3.\quad\int_{-1}^{1}K_{C^{m}}\left(x\right)dx=1,\]  
_or equivalent_  
\[1.\quad\int_{-h}^{h}K_{T_{1}^{m}}\left(\frac{t}{h}\right)dt= \frac{2h}{n+1},\qquad 2.\quad\int_{-h}^{h}K_{T_{2}^{m}}\left(\frac{t}{h} \right)dt= \frac{2 nh}{n+1},\qquad 3.\quad\int_{-h}^{h}K_{C^{m}}\left(\frac{t}{h} \right)dt= h,\]  
_where \(0\leq\left|\frac{2}{n+1}\right|\leq 1,\ 1\leq\left|\frac{2n}{n+1}\right|\leq 2\), \(h\) be positive real numbers, \(m\) is an odd natural number and \(n\in\mathbb{N}^{+}\)._  
Proof. : The proof can be easily obtained by using integration methods within the boundaries and then substitution \(x=t/h\).