# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: [PERSON]*, [PERSON], [PERSON], [PERSON]
## 3 FzT for Solving SODEs
### Numerical Scheme I for SODEs
Then, for each \(k=1,\ldots,n\):_  
\[X_{k+1}=X_{k}+h\hat{F}_{k},\qquad X_{1}=\alpha,\quad|\quad Y_{k+1}=Y_{k}+h \hat{G}_{k},\qquad Y_{1}=\beta,\ k=1,\ldots,n-1. \tag{15}\]1  
_where \(F_{k}\), \(G_{k}\) are defined by (6) and \(\hat{F}_{k}\), \(\hat{G}_{k}\) are defined by (14)._  
**Proof. ** By Lemma 1, \(x_{k}-X_{k}=\mathcal{O}(h^{2})\) and \(y_{k}-Y_{k}=\mathcal{O}(h^{2})\) and (20), we get:  
\[X_{k+1}=X_{k}+h\hat{F}_{k},\qquad X_{1}=\alpha,\quad|\quad Y_{k+1}=Y_{k}+h \hat{G}_{k},\qquad Y_{1}=\beta,\ k=1,\ldots,n-1. \tag{15}\]2  
and using the trapezoid rule and Remark 1, we obtain:  
\[X_{k+1}=X_{k}+h\hat{F}_{k},\qquad X_{1}=\alpha,\quad|\quad Y_{k+1}=Y_{k}+h \hat{G}_{k},\qquad Y_{1}=\beta,\ k=1,\ldots,n-1. \tag{15}\]3 \[X_{k+1}=X_{k}+h\hat{F}_{k},\qquad X_{1}=\alpha,\quad|\quad Y_{k+1}=Y_{k}+h \hat{G}_{k},\qquad Y_{1}=\beta,\ k=1,\ldots,n-1. \tag{15}\]4 \[X_{k+1}=X_{k}+h\hat{F}_{k},\qquad X_{1}=\alpha,\quad|\quad Y_{k+1}=Y_{k}+h \hat{G}_{k},\qquad Y_{1}=\beta,\ k=1,\ldots,n-1. \tag{15}\]5  
which together with the previous estimation proves that:  
\[X_{k+1}=X_{k}+h\hat{F}_{k},\qquad X_{1}=\alpha,\quad|\quad Y_{k+1}=Y_{k}+h \hat{G}_{k},\qquad Y_{1}=\beta,\ k=1,\ldots,n-1. \tag{15}\]6  
Similarly,  
\[X_{k+1}=X_{k}+h\hat{F}_{k},\qquad X_{1}=\alpha,\quad|\quad Y_{k+1}=Y_{k}+h \hat{G}_{k},\qquad Y_{1}=\beta,\ k=1,\ldots,n-1. \tag{15}\]7  
**Corollary 2**. : _The approximation method for (5) is given by Scheme I (14)-(15) with the local error \(\mathcal{O}(h^{2})\). The approximate solution to (5) can be found by taking the inverse FzT (16) where \(A_{1},\ldots,A_{n}\) are fixed basic functions._  
**Theorem 2**. : _Let \(f\), \(g\in C^{2}\left[t_{1},t_{n}\right]\) and bounded on \(I=\left[t_{1},t_{n}\right]\). Let, moreover, basic functions \(\left\{A_{k}\,|\,k=1,\ldots,n\right\},\ n\geq 2\), form a uniform fuzzy partition of \(I\). Assume that \(F_{k}\left(G_{k}\right),\,k=1\ldots,\,n,\) and \(F_{k}^{\prime}\left(G_{k}^{\prime}\right),\,k=1\ldots,\,n,\) are the FzT components of \(f\left(g\right)\) and \(f^{\prime}\left(g^{\prime}\right)\) with respect to \(\left\{A_{k}\,|\,k=1,\ldots,n\right\}\), respectively. Then, for \(k=1,\ldots,n-1\):_  
\[X_{k+1}=X_{k}+h\hat{F}_{k},\qquad X_{1}=\alpha,\quad|\quad Y_{k+1}=Y_{k}+h \hat{G}_{k},\qquad Y_{1}=\beta,\ k=1,\ldots,n-1. \tag{15}\]8  
_where \(M_{f}=\underset{t\in I}{\max}\left|f^{\prime\prime}(t,x(t),y(t))\right|\) and \(M_{g}=\underset{t\in I}{\max}\left|g^{\prime\prime}(t,x(t),y(t))\right|\)._  
**Proof. ** Let us write the following result from [PERSON]'s theorem:  
\[X_{k+1}=X_{k}+h\hat{F}_{k},\qquad X_{1}=\alpha,\quad|\quad Y_{k+1}=Y_{k}+h \hat{G}_{k},\qquad Y_{1}=\beta,\ k=1,\ldots,n-1. \tag{15}\]9 \[x_{n}(t)=\sum_{k=1}^{n}X_{k}\ A_{k}(t),\ \ y_{n}(t)=\sum_{k=1}^{n}Y_{k}\ A_{k}(t), \tag{16}\]0  
where \(t_{k}<\varepsilon,\ \xi<t_{k+1}\). Using the linearity of the FzT by Remark 2 and the properties of the uniform fuzzy partition by Definition 1, we get:  
\[x_{n}(t)=\sum_{k=1}^{n}X_{k}\ A_{k}(t),\ \ y_{n}(t)=\sum_{k=1}^{n}Y_{k}\ A_{k}(t), \tag{16}\]1 \[x_{n}(t)=\sum_{k=1}^{n}X_{k}\ A_{k}(t),\ \ y_{n}(t)=\sum_{k=1}^{n}Y_{k}\ A_{k}(t), \tag{16}\]2 \[x_{n}(t)=\sum_{k=1}^{n}X_{k}\ A_{k}(t),\ \ y_{n}(t)=\sum_{k=1}^{n}Y_{k}\ A_{k}(t), \tag{16}\]3and:  
\[x_{n}(t)=\sum_{k=1}^{n}X_{k}\ A_{k}(t),\ \ y_{n}(t)=\sum_{k=1}^{n}Y_{k}\ A_{k}(t), \tag{16}\]4 \[x_{n}(t)=\sum_{k=1}^{n}X_{k}\ A_{k}(t),\ \ y_{n}(t)=\sum_{k=1}^{n}Y_{k}\ A_{k}(t), \tag{16}\]5  
which easily leads to (23). **Lemma 2**. : _Consider Scheme I (14)-(15). If the set of fuzzy sets \(\left\{A_{k}\ \mid k=1,\ldots,n-1\right\},\ n\geq 2,\) is a uniform fuzzy partition of \([t_{1},t_{n}]\), then we have for fixed \(k=1,\ldots,n-1\):_  
\[X_{k+1}-X_{k}=\left\{