# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II
## 4 New Fuzzy Numerical Methods for Solving SODEs
### Numerical Scheme I: Modified Trapezoidal Rule Based on FzT and NIM for SODEs  
According to necessary steps of construction of the generalized uniform fuzzy partitions in Section 4, we contributed to approximation methods of SODEs (16) by scheme provides formulas for the FzT components, \(X_{k}\) (\(Y_{k}\)), \(k=2,\ldots,n-1\), of the unknown function \(x\left(t\right)\) (\(y\left(t\right)\)) with respect to choose some of the \(h\)-uniform generalized fuzzy partition, \(B_{1},\ldots,B_{n}\), of interval \([a,b]\) with parameter \(h\) to approximate solution of SODEs (16). As initial step, choose the number \(n\geq 2\) and compute \(h=\left(b-a\right)/\left(n-1\right)\), then construct the \(h\)-uniform generalized fuzzy partition of \([a,b]\) using Definition 4. Let \(X_{1}=x_{1}\) and \(Y_{1}=y_{1}\). In the following, we apply the FzT and NIM to the SODEs (16) for obtaining the numerical Scheme I, where \(k=1,\ldots,n-1\). First, let \(f_{1}\) (\(g_{1}\)) in the Equation (19) is chosen as  
\[f_{1}= B_{k}F_{k}+B_{k+1}F_{k+1},\] \[g_{1}= B_{k}G_{k}+B_{k+1}G_{k+1}, \tag{20}\]  
where  
\[F_{k}=\frac{\int_{a}^{b}f(t,X_{k},Y_{k})\ B_{k}(t)\ dt}{\int_{a}^{b}B_{k}(t)\ dt},\ G_{k}=\frac{\int_{a}^{b}g(t,X_{k},Y_{k})\ B_{k}(t)\ dt}{\int_{a}^{b}B_{k}(t)\ dt}, \tag{21}\]  
and \(B_{k}\) represents the generalized uniform fuzzy partitions that are defined by Definition 4. Then, substituting (20) into (19) for \(k=1,\ldots,n-1\)  
\[I_{f}\approx\int_{t_{k}}^{t_{k+1}}B_{k}F_{k}ds+\int_{t_{k}}^{t_{k+1}}B_{k+1}F _{k+1}ds,\]  
\[I_{g}\approx\int_{t_{k}}^{t_{k+1}}B_{k}G_{k}ds+\int_{t_{k}}^{t_{k+1}}B_{k+1}G _{k+1}ds.\]  
By Remark 4 in the interval \([t_{k},t_{k+1}]\), we have  
\[I_{f}\approx\frac{h}{2}\left(F_{k}+F_{k+1}\right),\ I_{g}\approx\frac{h}{2} \left(G_{k}+G_{k+1}\right).\]  
Hence, the one step method based on FzT for (17) is derived as follows, where \(k=1,\ldots,n-1\). \[X_{k+1} =X_{k}+\frac{h}{2}\left(F_{k}+F_{k+1}\right),\] \[Y_{k+1} =Y_{k}+\frac{h}{2}\left(G_{k}+G_{k+1}\right), \tag{22}\]  
where \(F_{k}\) and \(G_{k}\) are defined by (21). This method computes the approximate coordinates \(\left[X_{1},\ldots,X_{n}\right]\) and \(\left[Y_{1},\ldots,Y_{n}\right]\) of the FzT for the functions \(x(t)\) and \(y(t)\). The problem with the previous scheme (22) is that the unknown quantities \(F_{k+1}\) and \(G_{k+1}\) which means that \(X_{k+1}(Y_{k+1})\) appears on both sides and an implicit method. Therefore, one solution to this problem would be to use an explicit method such as another fuzzy approach, namely Scheme I. For this purpose, the scheme (22) is of the form  
\[X_{k+1} = f_{x}+N(X_{k+1}),\;\left|\;Y_{k+1}\right. = f_{y}+N(Y_{k+1}),\]  
and can be solved by NIM (2), where  
\[f_{x}=X_{k}+\frac{h}{2}F_{k},\;\text{and}\;N(X_{k+1})=\frac{h}{2}F_{k+1}.\]  
\[g_{1}= B_{k}G_{k}+B_{k+1}G_{k+1}, \tag{20}\]0  
The three term approximation of the NIM (3) gives the following formulas for solving SODEs (16):  
\[\left. \begin{array}{rcl}u_{x0}&=&X_{k}+\frac{h}{2}F_{k},\\ u_{x1}&=&N\left(u_{x0}\right),\\ u_{x2}&=&N\left(u_{x0}+u_{x1}\right)-N\left(u_{x0}\right),\end{array} \right| \left. \begin{array}{rcl}u_{y0}&=&Y_{k}+\frac{h}{2}G_{k},\\ u_{y1}&=&N\left(u_{y0}\right),\\ u_{y2}&=&N\left(u_{y0}+u_{y1}\right)-N\left(u_{y0}\right),\end{array} \right\}\]  
Hence, the three term approximate solution is  
\[u_{x}=u_{x0}+u_{x1}+u_{x2}=u_{x0}+N\left(u_{x0}+u_{x1}\right)\]  
and  
\[u_{y}=u_{y0}+N\left(u_{y0}+u_{y1}\right),\]  
which leads to the following formulas. \[\left. \begin{array}{rcl}X_{k+1}^{*}&=&X_{k}+hF_{k}/2,\\ X_{k+1}^{**}&=&X_{k+1}^{*}+hF_{k+1}^{*}/2,\\ X_{k+1}^{**}&=&Y_{k+1}^{*}+hG_{k+1}^{*}/2,\\ X_{k+1}&=&X_{k}+h\left(F_{k}+F_{k+1}^{**}\right)/2,\end{array} \right|\left. \begin{array}{rcl}Y_{k+1}^{*}&=&Y_{k}+hG_{k}/2,\\ Y_{k+1}^{**}&=&Y_{k+1}^{*}+hG_{k+1}^{*}/2,\\ Y_{k+1}&=&Y_{k}+h\left(G_{k}+G_{k+1}^{**}\right)/2,\end{array} \right\} \tag{23}\]  
where  
\[\left.