# New Fuzzy Numerical Methods for Solving Cauchy Problems
## 2 Basic Concepts
### New FT Based Power of the Triangular and Raised Cosine Generalized Uniform Fuzzy Partition
: _Let \(f\in C\left[a,b\right]\). Then for any \(\varepsilon>0\) there exist \(n_{\varepsilon}\in\mathbb{N}\) and \(B_{1},\ldots,\ B_{n_{\varepsilon}}\) be basic functions form power of the triangular and raised cosine generalized uniform fuzzy partition of \(\left[a,b\right]\). Let \(F_{k},\ k=1\ldots,\ n\), be the integral FT components of \(f\) with respect to \(B_{1},\ldots,\ B_{n_{\varepsilon}}\). Then for each \(k=1\ldots,\ n_{\varepsilon}-1\) the following estimations hold: \(\left|f(t)-F_{i}\right|\leq\varepsilon\) for each \(t\in\left[a,b\right]\cap\left[t_{k},t_{k+1}\right]\) and \(i=k,\,k+1\)._  
Proof. : see [4]. **Corollary 3**. : _Let the conditions of Lemma 6 be fulfilled. Then for each \(k=1\ldots,\ n_{\varepsilon}-1\) the following estimations hold: \(\left|F_{k}-F_{k+1}\right|<\varepsilon\)._  
Proof. : According to [4, 16], let \(t\in\left[a,b\right]\cap\left[t_{k},t_{k+1}\right]\). Then by Lemma 6, for any \(k=1,\ldots,n-1\) we obtain \(\left|f\left(t\right)-F_{k}\right|<\varepsilon/2\) and \(\left|f\left(t\right)-F_{k+1}\right|<\varepsilon/2\). Thus, \(\left|F_{k}-F_{k+1}\right|\leq\left|f\left(t\right)-F_{k}\right|+\left|f\left( t\right)-F_{k+1}\right|<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon\). The following theorem estimates the difference between the original function and its direct FT with respect to power of the triangular and raised cosine generalized uniform fuzzy partition. **Theorem 1**. : _Let \(f\left(t\right)\in C^{2}\left[a,b\right]\) and the conditions of Lemma 5 be fulfilled. Then for \(k=1,\ldots,n\)_  
\[F_{k}=\alpha f\left(t_{k}\right)+\mathcal{O}\left(h^{2}\right)\,, \tag{12}\]  
_where \(\alpha>0\) or \(\alpha\) is defined by Definition 7._  
Proof. : By locality condition for Definition 1, Lemmas 3 and 5, and according to the proof of Lemma 9.3 [8], using the trapezoid formula with nodes \(t_{k-1},\ t_{k},\ t_{k+1}\) to the numerical computation of the integral, we get for \(\alpha>0\)  
\[F_{k} =\frac{1}{h}\int_{t_{k-1}}^{t_{k+1}}f\left(t\right)B_{k}(t)\,dt,\] \[=\frac{1}{h}\frac{h}{2}\left(f\left(t_{k-1}\right)B_{k}(t_{k-1})+2 f\left(t_{k}\right)B_{k}(t_{k})+f\left(t_{k+1}\right)B_{k}(t_{k+1})\right)+ \mathcal{O}\left(h^{2}\right),\] \[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}.\]0 \[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}.\]1 \[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}.\]2  
**Definition 9**. : _Let \(F[f]=\left(F_{1},\,F_{2},\ldots,\,F_{n}\right)\) be direct FT of a function \(f\in C\left[a,b\right]\) with respect to the fuzzy partition \(B_{k}(t)\), \(k=1,\ldots,n\) of \(\left[a,b\right]\). Then, the function \(\hat{f}\) defined on \(\left[a,b\right]\)_  
\[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}.\]3  
_is called the inverse FT of \(f\)._  
**Corollary 4**. : _Let the assumptions of Lemma 2 and moreover, Let \(\hat{f}\left(t\right)\) be the inverse FT of \(f\) with respect to power of the raised cosine generating function. Then, for all \(t\in\left[a,b\right]\) the following holds: \(\hat{f}\left(t\right)=\sum_{k=1}^{n}F_{k}B_{k}(t)\)._  
Proof. : This proof immediately follows from Defintion 9, Lemma 2 and then using \(\sum_{k=1}^{n}B_{k}(t)=1\). The following lemma estimates the difference between the original function and its inverse FT.  
**Lemma 7**. : _Let the assumptions of Theorem 1 and let \(\hat{f}\left(t\right)\) be the inverse FT of \(f\) with respect to the fuzzy partition of \(\left[a,b\right]\) is given by Definition 7.