# 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  
In this subsection, we present the main principles of F-transform detailed in [8,10,11] that are modified with respect to power of the triangular and raised cosine generalized uniform fuzzy partition. Further, we will show that FT components with respect to power of the triangular and raised cosine generalized uniform fuzzy partition can be simplified and approximated of an original function, say \(f\). **Definition 8**. : _Let \(f\) be a continuous function on \([a,b]\) and \(B_{k}(t),\)\(k=1,\ldots,n,\) be power of the triangular and raised cosine generalized uniform fuzzy partition of \([a,b]\)\(,\)\(n\geq 2\). A vector of real numbers \(F[f]=(F_{1},\ F_{2},\ldots,\ F_{n})\) given by_  
\[F_{k}=\frac{\int_{a}^{b}f\left(t\right)B_{k}(t)\,dt}{\int_{a}^{b}B_{k}(t)\,dt}, \tag{10}\]  
_for \(k=1,\ldots,n\) is called the direct FT of \(f\) with respect to power of the triangular and raised cosine generalized uniform fuzzy partition \(B_{k}\)._  
In the following, we assume a generating function \(K\) in the Formulas (2)-(4). We will simplify the representation (10). **Lemma 4**. : _Let \(f\in C\left([a,b]\right)\) and according to Definition 7, fuzzy sets \(B_{k},\ k=1,\ldots,n,\ n\geq 2,\) be power of a triangular and raised cosine generalized uniform fuzzy partition of \([a,b]\) with a generating function \(K\), then representation (10) of direct FT can be simplified as follows for \(k=1,\ldots,n\)_  
\[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}.\]  
Proof. : In this proof, we will write a generating function \(K\) instead of (2)-(4). By Definition 7, we get  
\[B_{k}\left(t\right)=\alpha K\left(\frac{t-t_{k}}{h}\right),\ \ t\in[t_{k}-h,t_{k}+h]\,\]  
for \(k=1,\ldots,n\)\(,\)\(t_{0}=t_{1},\ t_{n+1}=t_{n}\), and substituting \(u=\frac{t-t_{k}}{h}\) and then substituting \(t=s/h\). Thus, we get  
\[\int_{t_{k-1}}^{t_{k+1}}f\left(t\right)B_{k}(t)\,dt=\alpha h\int_{-1}^{1}f \left(th+t_{k}\right)K(t)\,dt=\alpha\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{ t}{h})\,dt\]  
\[\int_{t_{k-1}}^{t_{k+1}}B_{k}(t)\,dt=\alpha h\int_{-1}^{1}K(t)\,dt=\alpha\int_ {-h}^{h}K(\frac{t}{h})\,dt\]  
and its corresponding results with representation (10). Indeed, the previous lemma holds for every fuzzy partition generated by a kernel. Now, we will simplify the above given expressions for the coefficients \(F[f]=(F_{1},\ F_{2},\ldots,\ F_{n})\) in the representation (10) even more. This fact is very important for applications which are more flexible and consequently easier to use. **Lemma 5**. : _Let the assumptions of Lemma 4 be fulfilled. Then, the coefficients \(F[f]=\left(F_{1},\ F_{2},\ldots,\ F_{n}\right)\) in the expression (10) of the FT component \(F_{k}\) of \(f\) as follows:_  
\[F_{k}=\frac{1}{h}\int_{a}^{b}f\left(t\right)B_{k}(t)\,dt=\frac{\alpha}{h}\int_{ a}^{b}f\left(t\right)K\left(\frac{t-t_{k}}{h}\right)\ dt, \tag{11}\]  
_for \(k=1,\ldots,\ n\), where interval \(\left[a,b\right]\) is partitioned by power of the triangular and raised cosine generalized uniform fuzzy partition \(B_{1},\ldots,\ B_{n}\) and \(\alpha\) is defined by Definition 7._  
Proof. : Let \(k\in\left\{1,\ldots,\ n\right\}\) and consider set of fuzzy sets \(B_{k}(x)\) from power of the triangular and raised cosine generalized uniform fuzzy partition of \(\left[a,b\right]\) in (9). We will prove the equality \(\int_{t_{k-1}}^{t_{k+1}}B_{k}(t)\,dt=h\). We get by virtue of Lemmas 1 and 4, and (6):  
\[\int_{t_{k-1}}^{t_{k+1}}B_{k}(t)\,dt=\int_{t_{k-1}}^{t_{k+1}}A_{k}(t,a),dt=\int _{t_{k}-h}^{t_{k}+h}\left(\frac{m+1}{2}\right)K_{T_{1}^{n}}\left(\frac{t-t_{k }}{h}\right)\,dt=h\int_{-1}^{1}\left(\frac{m+1}{2}\right)K_{T_{1}^{n}}\left(t \right)\,dt=h,\]  
where \(h\) is the bandwidth of the fuzzy partition and \(t_{k}=a+kh\). Similarly, the other Formulas (7) and (8) will be proved and then its corresponding in the expression (10). **Lemma 6**.