# 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
Then, for all \(t\in\left[a,b\right]\) the following estimation holds:_  
\[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}.\]4  
Proof. : Let \(t\in\left[a,b\right]\) so that \(x\in\left[t_{k},t_{k+1}\right]\) for some \(k=1,\ldots,\,n\). By Theorem 1,  
\[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}.\]5 \[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}.\]6  
**Corollary 5**. : _Let the assumptions of Lemma 7, then \(\left|\hat{f}\left(t\right)-f\left(t\right)\right|<\varepsilon\)._  
Proof. : The proof easily follows from the proof of Lemma 7 and then using Lemma 6 as follows:  
\[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}.\]7  
**Remark 3**. : _According to the Definitions 1 and 2, if the normality is considered to be an additional condition for generating function (i.e., \(K(0)=1\)) and generalized uniform fuzzy partition of \(\left[a,b\right]\) satisfies \(A_{k}(x_{k})=\alpha,\,\alpha>0\), then it is easy to see that the inverse FT \(\hat{f}\left(t_{k}\right)=F_{k}\) for all \(k=1,\ldots,\,n\). This is true for Definition 7. Moreover, if orthogonality condition (Ruspini condition (1)) is replaced by covering condition in Definition 1 and generalized uniform fuzzy partition of \(\left[a,b\right]\) satisfies \(A_{k}(x_{k})=\alpha=1\), then it is easy to also see that the inverse FT \(\hat{f}\left(t_{k}\right)=F_{k}\) for all \(k=1,\ldots,\,n\). This is true for Formula (8) only._  
Important property of the direct FT as well as inverse FT is their linearity, namely, given \(f\), \(g\in C\left[a,b\right]\) and \(\alpha,\,\beta\in R\), if \(h=\alpha f+\beta g\), then \(F\left[h\right]=\alpha F\left[f\right]+\beta F\left[g\right]\) and \(\hat{h}=\alpha\hat{f}+\beta\hat{g}\). In the next section, we present new fuzzy numerical methods based on the FT and a new iterative method to numeric solution of the Cauchy problem. **4. New Fuzzy Numerical Methods for Cauchy Problem**  
Consider the initial value problem (IVP) for the Cauchy problem:  
\[F_{k}=\frac{\int_{-1}^{1}f\left(th+t_{k}\right)K(t)\,dt}{\int_{-1}^{1}K(t)\, dt}=\frac{\int_{-h}^{h}f\left(t+t_{k}\right)K(\frac{t}{h})\,dt}{\int_{-h}^{h}K( \frac{t}{h})\,dt}.\]8where \(y_{1}\in\mathbb{R}\) and \(f\) is continuous function on \([a,b]\times\mathbb{R}\) and satisfies Lipschitz condition. In fact, the analytical solution of problem (15) is often difficult and sometimes impossible to obtain. Instead, numerical analysis is interested with obtaining approximate solutions with errors within reasonable bounds. Thus, a usage of fuzzy numerical methods seems to be suitable. In [8; 9], the authors have presented Euler method and Mid-point rule, based on FT to numeric solution of Cauchy problem (15). A new iterative method (NIM) has been proposed for solving linear (nonlinear) functional equations, ordinary differential equations and delay differential equations [2; 3]. In this section, we present three new schemes to solve Cauchy problem (15), that use the FT and NIM. Our motivation stems from the classical approach, trapezoidal rule (1-step) and Adams Moulton methods (2 and 3-step). For the rest of this paper, suppose that we are given the Cauchy problem (15), where the function \(f\) on \([a,b]\) are sufficiently smooth and we assume that all necessary requirements for constructing the FT of the solution of Cauchy problem (15) are fulfilled. Now, we present numerical Scheme I, II, and III. The first scheme uses 1-step method, while the second one uses 2-step method, and the third uses 3-step method.