# New Fuzzy Numerical Methods for Solving Cauchy Problems
## 5 Numerical Examples  
In this section, we present examples of the Cauchy problem (15). **Example 1**. : _Consider the following initial value problem with initial conditions \(y\left(0\right)=1\) and with a smooth right-hand function_  
\[y^{\prime}(t)=t^{2}-y,\ \ \ t\in\left[0,2\right]. \tag{27}\]  
**Example 2**. : _Consider the Cauchy problem (15) with oscillating right-hand function. We take \(f(t,y)=1+2y\cos\left(t^{2}\right)+\sin\left(2t^{2}\right)\), \(t(\frac{\pi}{2})=2.1951\), \(a=\frac{\pi}{2}\) and \(b=\frac{3\pi}{2}\)._  
The results are listed in Tables 2-4 by fuzzy numerical methods proposed in this paper with respect to case \(K_{T_{1}^{1}}\) and Table 5 by fuzzy numerical methods proposed in this paper with respect to case \(K_{T_{1}^{1}}\), \(K_{T_{1}^{3}}\), \(K_{T_{1}^{201}}\), \(K_{C1}\). The Euclidean distance is given by Norm \(\ell_{2}\) defined as \(\left\|Y-y(t)\right\|_{2}=\sqrt{\sum_{k}\left(Y_{k}-y(t_{k})\right)^{2}}\) and mean square error (MSE) defined as \(\text{MSE }=\frac{1}{n}\left(\left\|Y_{k}-y(t_{k})\right\|_{2}\right)^{2}\). This is an easily computable quantity for a particular sample. Concluding remarks are summarized as follows:  
* In view of Table 2, a comparison between the Euler method (Euler-FT) [8], the Mid-point rule (Mid-FT), Scheme I and II [9] and three new schemes (16), (19) and (20) in this paper for Example 1. We can easily observe from Table 2, the better results (in comparison with the Euler-FT method [8]) are obtained by the three new schemes in this paper and the best result (in comparison with the Scheme I, II and II) is obtained by the Scheme III. Also, the better results (in comparison with the Mid-point rule (Mid-FT), Scheme I and II [9]) are obtained by the Scheme II (19) and Scheme III (20) in this paper where all fuzzy numerical methods used the FT components and the best approximation is shown by the Scheme III (20) with FT components. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(t_{i}\) & **Solution \(y(t)\)** & **Proposed** & **Proposed** & **Proposed** & **Euler-FT** & **Mid-FT** & **Scheme I** & **Scheme II** \\  & **Scheme I** & **Scheme II** & **Scheme III** & **in[8]** & **in[9]** & **in[9]** & **in[9]** \\ \hline
0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\  
& 1.864665 & 1.863636 & 1.863899 & 1.864097 & 1.779378 & 1.865684 & 1.852585 & 1.874993 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of numeric results for Example 1. The columns contain the exact and seven approximate solutions of the Cauchy problem (27) with a smooth right-hand function: the first three approximate solution is obtained by the three new schemes ((16), (19) and (20)), the fourth approximate solution by the Euler-FT [8] with FT components and the last three by the schemes are proposed in [9]. The best approximation is shown by the Scheme III proposed above (20) with FT components. * In Tabel 3, a comparison of MSE and a comparison of Norm \(\ell_{2}\) for Examples 1 and 2. We can easily observe, the best results are obtained by the three new schemes in this paper and the better results (in comparison with the other numerical classical methods) are obtained by all fuzzy numerical methods used the FT components except Euler-FT [8] for these examples. * In view of Table 4, a comparison between the three new schemes (16), (19) and (20) in this paper and the Trapezoidal Rule, 2-Step Adams Moulton Method and 3-Step Adams Moulton Method based on Euler method for Example 2. We can easily observe from Table 4, the better results are obtained by the three new schemes in this paper and the best result (in comparison with the Scheme I, II and II) is obtained by the Scheme III.