# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: [PERSON]*, [PERSON], [PERSON], [PERSON]
## 3 FzT for Solving SODEs
### Numerical Scheme II for SODEs  
In this subsection, we will construct numerical Scheme II, a more advanced method than that of Scheme I. The components of EzT of \(x\) and \(y\) can be approximated by the average of the two methods, Scheme I (14)-(15) and the backward Scheme I (or implicit Scheme I), then the EzT is given by:  
\[ \begin{array}{ccl}X_{p}=&X_{k}+h\hat{F}_{k},\qquad\quad X_{1}=\alpha,\\ X_{c}=&X_{k}+h\hat{F}_{k+1},\qquad X_{1}=\alpha,\\ X_{k+1}=&\frac{1}{2}\left(X_{p}+X_{c}\right),\end{array} \qquad \begin{array}{ ccl}Y_{p}=&Y_{k}+h\hat{G}_{k},\qquad Y_{1}=\beta,\\ Y_{c}=&Y_{k}+h\hat{G}_{k+1},\qquad Y_{1}=\beta,\\ Y_{k+1}=&\frac{1}{2}\left(Y_{p}+Y_{c}\right),\end{array}  \tag{28}\]  
for \(k=1,\ldots,n-1\). The problem with the previous scheme (28) is that the unknown quantities \(\hat{F}_{k+1}\) and \(\hat{G}_{k+1}\) appear on both sides (an implicit method). Therefore, one solution to this problem would be to use an explicit method such as another fuzzy approach. The following Scheme II for \(k=1,\ldots,n-1\), \(X_{1}=\alpha\), \(Y_{1}=\beta\):  
\[\left. \begin{array}{ccl}X_{k+1}^{*}&=&X_{k}+h\hat{F}_{k},\\ X_{k+1}&=&X_{k}+\frac{h}{2}\left(\hat{F}_{k}+\hat{F}_{k+1}^{*}\right),\end{array}  \right| \begin{array}{ccl}Y_{k+1}^{*}&=&Y_{k}+h\hat{G}_{k},\\ Y_{k+1}&=&Y_{k}+\frac{h}{2}\left(\hat{G}_{k}+\hat{G}_{k+1}^{*}\right),\end{array}  \tag{29}\]  
where:  
\[ \begin{array}{ccl}\hat{F}_{k}[f]&=&\frac{\int_{t_{1}}^{t_{n}}f(t,X_{k},Y_{k} )A_{k}(t)dt}{\int_{t_{1}}^{t_{n}}A_{k}(x)dx},\\ \hat{F}_{k+1}^{*}[f]&=&\frac{\int_{t_{1}}^{t_{n}}f(t,X_{k+1}^{*},Y_{k+1}^{*})A _{k+1}(t)dt}{\int_{t_{1}}^{t_{n}}A_{k+1}(x)dx},\end{array} \qquad\hat{G}_{k}[g] = \frac{\int_{t_{1}}^{t_{n}}g(t,X_{k},Y_{k})A_{k}(t)dt}{\int_{t_{1}} ^{t_{n}}A_{k}(x)dx}, \tag{30}\]  
This method computes the approximate coordinates \([X_{1},\ldots,X_{n}]\) and \([Y_{1},\ldots,Y_{n}]\) of the direct EzT of the functions \(x(t)\) and \(y(t)\), respectively. In the sequel, the inverse EzT (16) approximates the solution \(x(t)\ \left(y(t)\right)\) of the SODEs (5). In the next theorem, we obtain an error estimate in the context of a fuzzy partition and error analysis of approximation Scheme II. **Theorem 5**. : _Suppose that \(f(t,x_{1},x_{2})\), \(g(t,x_{1},x_{2})\in C^{2}\left[t_{1},t_{2}\right]\). Let \(\left|\frac{\partial f}{\partial x_{i}}\right|\leq L_{f}\left(\left|\frac{ \partial g}{\partial x_{i}}\right|\leq L_{g}\right),\)\(i=1,2\), and \(\left|f(t,x_{1},x_{2})\right|\leq M_{1}\)\((\left|g(t,x_{1},x_{2})\right|\leq M_{2})\). Consider Scheme II (29)-(30) for some positive integer \(k\) and \(\left\{A_{k}\ |\ k=1,\ldots,n-1\right\},\)\(n\geq 2\), to be a uniform fuzzy partition of \([t_{1},t_{n}]\), then the following hold true:_  
1. _for a value of_ \(k\) _in the range_ \(1\leq k\leq n-1\)_:_ \[\left|\hat{F}_{k+1}^{*}-\hat{F}_{k}^{*}\right|\leq Lh\left(1+2 Lh^{2}\right)U_{k,k-1},\qquad\quad\left| \hat{G}_{k+1}^{*}-\hat{G}_{k}^{*}\right|\leq Lh\left(1+2 Lh^{2}\right)U_{k,k-1},\] _where_ \(U_{k,k-1}=|X_{k}-X_{k-1}|+|Y_{k}-Y_{k-1}|\)_._
2. _for all_ \(k=1,\ldots,n-1\)__ \[|X_{n+1}-X_{n}|\leq\frac{Mh}{2}e^{n}(2 Lh^{2}),\ \ |Y_{n+1}-Y_{n}|\leq\frac{Mh}{2}e^{n}(2 Lh^{2}),\] _where_ \(L=L_{f}+L_{g}\) _and_ \(M=\sum_{i}^{2}M_{i}\)_._Proof. : The proof is similar to the proof of Theorem 3, so we just write out the procedure. The proofs of Part 1 is as follows.