# New Fuzzy Numerical Methods for Solving Cauchy Problems
## 6 Conclusions  
We extended applicability of fuzzy numeric methods to the initial value problem (the Cauchy problem). We proposed three new numeric methods based on the FT and NIM and then analyzed their suitability. We considered in the case of the generalized uniform fuzzy partition is power of the triangular (raised cosine) generalized uniform fuzzy partition and showed that the newly proposed schemes outperform the Euler-FT [8] and Mid-FT, Scheme I, and Scheme II [9] especially on examples where the generalized uniform fuzzy partition is power of the triangular generalized uniform fuzzy partition by using generating function (2). Alos, the newly proposed schemes in this paper outperform the Trapezoidal Rule, 2-Step Adams Moulton Method and 3-Step Adams Moulton Method. Moreover, we proved that the Scheme I determines an approximate solution which converges to the exact solution. This constitutes an important improvement to previous results were coined by [PERSON] [8]. To conclude previous sections, the proposed schemes are more accurate and stable. In particular, these schemes can be used for solving initial value problem. [PERSON] and [PERSON] proposed and designed the numerical methods. [PERSON] performed the numerical experiments. [PERSON] evaluated the results and supported this work. [PERSON] project administration. [PERSON] provided software and data curation. This work of [PERSON] has been supported by the project \"LQ1602 IT4 Innovations excellence in science\" and by the Grant Agency of the Czech Republic (project No. 16-09541S). Also, many thanks given to Universiti Malaysia Perlis for providing all facilities until this work completed successfully. The authors declare no conflicts of interest. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Proposed Scheme**} & \multirow{2}{*}{**Case**} & \multicolumn{2}{c}{**Ex.1**} & \multicolumn{2}{c}{**Ex.2**} \\ \cline{3-6}  & & **Norm \(\boldsymbol{\ell_{2}}\)** & **MSE** & **Norm \(\boldsymbol{\ell_{2}}\)** & **MSE** \\ \hline \multirow{4}{*}{I} & \(K_{T_{1}^{1}}\) & \(7.81857\times 10^{-03}\) & \(2.91095\times 10^{-06}\) & \(6.38151\times 10^{-01}\) & \(1.93922\times 10^{-02}\) \\  & \(K_{T_{1}^{1}}\) & \(5.06528\times 10^{-03}\) & \(1.22176\times 10^{-06}\) & \(4.65538\times 10^{-01}\) & \(1.03203\times 10^{-02}\) \\  & \(K_{T_{1}^{2m}}\) & \(2.21945\times 10^{-03}\) & \(2.34569\times 10^{-07}\) & \(3.42892\times 10^{-01}\) & \(5.59882\times 10^{-03}\) \\  & \(K_{C^{1}}\) & \(6.96371\times 10^{-03}\) & \(2.30920\times 10^{-06}\) & \(5.79002\times 10^{-01}\) & \(1.59640\times 10^{-02}\) \\ \hline \multirow{4}{*}{II} & \(K_{T_{1}^{1}}\) & \(5.92425\times 10^{-03}\) & \(1.67127\times 10^{-06}\) & \(5.45616\times 10^{-01}\) & \(1.41761\times 10^{-02}\) \\  & \(K_{T_{1}^{2m}}\) & \(3.58895\times 10^{-03}\) & \(6.13360\times 10^{-07}\) & \(4.29959\times 10^{-01}\) & \(8.80307\times 10^{-03}\) \\  & \(K_{T_{1}^{2m}}\) & \(1.28684\times 10^{-03}\) & \(7.88551\times 10^{-08}\) & \(3.76033\times 10^{-01}\) & \(6.73336\times 10^{-03}\) \\  & \(K_{C^{1}}\) & \(5.18129\times 10^{-03}\) & \(1.27837\times 10^{-06}\) & \(5.01710\times 10^{-01}\) & \(1.19864\times 10^{-02}\) \\ \hline \multirow{4}{*}{III} & \(K_{T_{1}^{1}}\) & \(5.31047\times 10^{-03}\) & \(1.34291\times 10^{-06}\) & \(4.42442\times 10^{-01}\) & \(9.32167\times 10^{-03}\) \\  & \(K_{T_{1}^{2m}}\) & \(3.09350\times 10^{-03}\) & \(4.55702\times 10^{-07}\) & \(2.88083\times 10^{-01}\) & \(3.95199\times 10^{-03}\) \\ \cline{1-1}  & \(K_{T_{1}^{2m}}\) & \(9.28253\times 10^{-04}\) & \(4.10311\times 10^{-08}\) & \(2.15401\times 10^{-01}\) & \(2.20942\times 10^{-03}\) \\ \cline{1-1}  & \(K_{C^{1}}\) & \(4.59684\times 10^{-03}\) & \(1.00624\times 10^{-06}\) & \(3.84860\times 10^{-01}\) & \(7.05320\times 10^{-03}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The values of MSE and Norm \(\ell_{2}\) for Examples 1 and 2 by the three schemes with respect to the power of the triangular and raised cosine generalized uniform fuzzy partition are proposed in this paper.