# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II
## 4 New Fuzzy Numerical Methods for Solving SODEs
### Numerical Scheme II: Modified 2-Step Adams Moulton Method Based on FzT and NIM for SODEs  
The Scheme I uses 1-step method for solving SODEs (16). In this subsection, we improve 2-step Adams Moulton method using FzT and NIM for solving SODEs (16). Let us recall that the modified 2-step Adams Moulton method proposed by [16]. From this idea, the modified 2-step Adams Moulton method can be extended to approximate the solution of (16) by necessary steps of construction of the generalized uniform fuzzy partitions in Section 4. It is worth noting that three terms of NIM were used in [16], while four terms of NIM are used in this study. Let 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 (5) and let \(X_{1}=x_{1},Y_{1}=y_{1}\), \(X_{2}=x_{2}\) and \(Y_{2}=y_{2}\) if possible; otherwise, we can compute FzT components \(X_{2}\) and \(Y_{2}\) from the numerical Scheme I. In the following, we apply the F-transform and NIM to the SODEs (16) for obtaining the numerical Scheme II, where \(k=2,\ldots,n-1\). First, if \(f_{1}\) in the Equation (19) is approximated by  
\[f_{1}=\left(p_{0}+p_{1}\right)F_{k+1}+p_{2}F_{k}+p_{3}F_{k-1}, \tag{26}\]  
where  
\[F_{k}=\frac{\int_{a}^{b}f(t,X_{k},Y_{k})\ B_{k}(t)\ dt}{\int_{a}^{b}B_{k}(t)\ dt},\]  
\(p_{k}=\left(-1\right)^{k}\int_{0}^{1}\binom{-s+1}{k}ds\). Substituting (26) into (19), then for \(k=1,\ldots,n-1\)  
\[I_{f}\approx\frac{h}{12}\left(5F_{k+1}+8F_{k}-F_{k-1}\right).\]  
Similarity,  
\[I_{g}\approx\frac{h}{12}\left(5G_{k+1}+8G_{k}-G_{k-1}\right).\]  
Thus, the two step method based FzT for (17) is given for \(k=1,\ldots,n-1\) as  
\[X_{k+1} = X_{k}+h\left(8F_{k}-F_{k-1}+5F_{k+1}\right)/12,\ \big{|}\ Y_{k+1} = Y_{k}+h\left(8G_{k}-G_{k-1}+5G_{k+1}\right)/12, \tag{27}\]  
where  
\[F_{k} = \frac{\int_{a}^{b}f(t,X_{k},Y_{k})B_{k}(t)dt}{\int_{a}^{b}A_{k}(t) dt},\ \bigg{|}\ G_{k} = \frac{\int_{a}^{b}g(t,X_{k},Y_{k})B_{k}(t)dt}{\int_{a}^{b}B_{k}(t) dt},\]  
The problem with the previous scheme (27) is that the unknown quantities \(F_{k+1}\) and \(G_{k+1}\). Therefore, one solution to this problem would be to use an explicit method. For this purpose, the scheme (27) is of the form  
\[X_{k+1} = f_{x}+N(X_{k+1}),\ \big{|}\ Y_{k+1} = f_{y}+N(Y_{k+1}),\]  
and can be solved by NIM (2), where  
\[f_{x}=X_{k}+\frac{h}{12}\left(8F_{k}-F_{k-1}\right),\ \text{and}\ N(X_{k+1})=\frac{5h}{12}F_{k+1}.\]  
\[f_{y}=Y_{k}+\frac{h}{12}\left(8G_{k}-G_{k-1}\right),\ \text{and}\ N(Y_{k+1})=\frac{5h}{12}G_{k+1}.\]  
The four term approximation of the NIM (3) gives the following formulas for solving SODEs (16):\[u_{x0} = X_{k}+\frac{h}{12}\left(8F_{k}-F_{k-1}\right),\] \[F_{k}=\frac{\int_{a}^{b}f(t,X_{k},Y_{k})\ B_{k}(t)\ dt}{\int_{a}^{b}B_{k}(t)\ dt},\]0 \[F_{k}=\frac{\int_{a}^{b}f(t,X_{k},Y_{k})\ B_{k}(t)\ dt}{\int_{a}^{b}B_{k}(t)\ dt},\]1 \[F_{k}=\frac{\int_{a}^{b}f(t,X_{k},Y_{k})\ B_{k}(t)\ dt}{\int_{a}^{b}B_{k}(t)\ dt},\]2  
Hence, the four term approximate solution is  
\[F_{k}=\frac{\int_{a}^{b}f(t,X_{k},Y_{k})\ B_{k}(t)\ dt}{\int_{a}^{b}B_{k}(t)\ dt},\]3  
and  
\[F_{k}=\frac{\int_{a}^{b}f(t,X_{k},Y_{k})\ B_{k}(t)\ dt}{\int_{a}^{b}B_{k}(t)\ dt},\]4  
which leads to the following formulas. \[\left. \begin{array}{rcl}X_{k+1}^{*}&=&X_{k}+h\left(8F_{k}-F_{k-1}\right)/12,\\ \\ X_{k+1}^{**}&=&X_{k+1}^{*}+5 hF_{k+1}^{*}/12,\\ \\ X_{k+1}^{**}&=&X_{k+1}^{*}+5 hF_{k+1}^{**}/12,\\ \\ X_{k+1}&=&X_{k}+h\left(8F_{k}-F_{k-1}+5F_{k+1}^{***}\right)/12,\end{array}  \right| \begin{array}{rcl}Y_{k+1}^{*}&=&Y_{k}+h\left(8G_{k}-G_{k-1}\right)/12,\\ \\ Y_{k+1}^{**}&=&Y_{k+1}^{*}+5 hG_{k+1}^{*}/12,\\ \\ Y_{k+1}&=&Y_{k}+h\left(8G_{k}-G_{k-1}+5G_{k+1}^{***}\right)/12,\end{array}