# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: [PERSON]*, [PERSON], [PERSON], [PERSON]
## 4 Applications
In view of Figure 1, the graphical results of Examples 1-3 show a comparison between numerical schemes (I and II) and the exact solution. Furthermore, in view of Figure 1, a comparison is given between the numerical results of Examples 1 and 2 and exact solutions for \(h=0.01\), while a comparison is given between the numerical results of Example 3 and exact solutions for \(h=0.1\). All the graphs are plotted using MATLAB software. This constitutes an important improvement to the previous fuzzy approach, which did not provide such information for SODEs. Thus, this study will be particularly important. **Remark 6**. : _We compare new results based on FzT with the conventional numerical methods. For a discussion of the conventional numerical methods, the Euler method and trapezoidal rule to solve SODEs, see for example [35; 36]._ \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**Example 1**} & \multicolumn{2}{c}{**Example 2**} & \multicolumn{2}{c}{**Example 3**} \\ \cline{2-6}  & \(x(t)\) & \(y(t)\) & \(x(t)\) & \(y(t)\) & \(x(t)\) & \(y(t)\) \\ \hline Scheme I & \(2.91443\times 10^{-1}\) & \(6.67431\times 10^{-3}\) & \(1.12399\times 10^{-1}\) & \(1.12399\times 10^{-1}\) & \(2.92534\times 10^{-4}\) & \(1.58256\times 10^{-3}\) \\ Scheme II & \(2.24139\times 10^{-2}\) & \(3.77476\times 10^{-4}\) & \(3.04846\times 10^{-4}\) & \(3.04846\times 10^{-4}\) & \(1.72082\times 10^{-5}\) & \(4.10161\times 10^{-5}\) \\ Euler & \(6.99731\times 10^{-1}\) & \(1.19826\times 10^{-2}\) & \(1.14890\times 10^{-1}\) & \(1.14890\times 10^{-1}\) & \(5.43867\times 10^{-4}\) & \(1.68059\times 10^{-3}\) \\ Trapezoidal & \(5.99915\times 10^{-1}\) & \(1.40165\times 10^{-3}\) & \(2.75574\times 10^{-2}\) & \(2.75574\times 10^{-2}\) & \(3.19103\times 10^{-5}\) & \(5.21595\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The values of MSE for Example 1–3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(t_{i}\)} & \multirow{2}{*}{**Solution \(x(t)\)**} & **Proposed** & **Proposed** & **Euler** & **Trapezoidal** \\  & & **Scheme I** & **Scheme II** & **Scheme II** & **Scheme II** \\ \hline
0.00 & 1.00000 & 1.00000 & 1.00000 & 1.00000 & 1.00000 \\
0.10 & 1.10965 & 1.10581 & 1.11259 & 1.10000 & 1.10945 \\
0.20 & 1.23706 & 1.22517 & 1.24000 & 1.21890 & 1.23671 \\
0.30 & 1.37957 & 1.36112 & 1.38257 & 1.35438 & 1.37924 \\
0.40 & 1.53406 & 1.51087 & 1.53717 & 1.50368 & 1.53402 \\
0.50 & 1.69689 & 1.67119 & 1.70015 & 1.66355 & 1.69755 \\
0.60 & 1.86386 & 1.83827 & 1.86733 & 1.83022 & 1.86575 \\
0.70 & 2.03020 & 2.00777 & 2.03392 & 1.99935 & 2.03396 \\
0.80 & 2.19055 & 2.17473 & 2.19456 & 2.16601 & 2.19692 \\
0.90 & 2.33891 & 2.33356 & 2.34322 & 2.32461 & 2.34870 \\
1.00 & 2.46869 & 2.47798 & 2.47776 & 2.46891 & 2.48270 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of numerical results of \(x(t)\) for Example 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(t_{i}\)} & \multirow{2}{*}{**Solution \(y(t)\)**} & **Proposed** & **Proposed** & **Euler** & **Trapezoidal** \\  & & **Scheme I** & **Scheme II** & **Scheme II** & **Scheme II** \\ \hline
0.00 & 0.00000 & 0.00000 & 0.00000 & 0.00000 & 0.00000 \\
0.10 & 0.10933 & 0.09948 & 0.10781 & 0.10000 & 0.10805 \\
0.20 & 0.23466 & 0.21563 & 0.23168 & 0.21610 & 0.23113 \\
0.30 & 0.37191 & 0.34407 & 0.36755 & 0.34440 & 0.36521 \\
0.40 & 0.51694 & 0.48088 & 0.51126 & 0.48098 & 0.50621 \\
0.50 & 0.66544 & 0.62209 & 0.65855 & 0.62184 & 0.64994 \\
0.60 & 0.81285 & 0.76359 & 0.80488 & 0.76288 & 0.79202 \\
0.70 & 0.95430 & 0.90109 & 0.94543 & 0.89979 & 0.92782 \\
0.80 & 1.08451 & 1.03002 & 1.07494 & 1.02801 & 1.05236 \\
0.90 & 1.19767 & 1.14549 & 1.18766 & 1.14261 & 1.16023 \\
1.00 & 1.28736 & 1.24213 & 1.28463 & 1.23823 & 1.24549 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of numerical results of \(y(t)\) for Example 3.