# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II
## 4 New Fuzzy Numerical Methods for Solving SODEs
### Error Analysis of Numerical Scheme I for SODEs  
In this subsection, we present error analysis for numerical Scheme I and consider the Formula (23). If \(x(t_{k})=x_{k}\) and \(y(t_{k})=y_{k}\) denote the exact solution and \(X_{k}\), \(Y_{k}\) denote the numerical solution. Then, substituting the exact solution in the Formula (23), we get  
\[\left.\begin{array}{rcl}x_{k+1}^{*}&=&x_{k}+hF_{k}^{e}/2,\\ x_{k+1}^{**}&=&x_{k+1}^{*}+hF_{k+1}^{e*}/2,\\ x_{k+1}&=&x_{k}+h\left(F_{k}^{e}+F_{k+1}^{**}\right)/2,\end{array}\right|\left. \begin{array}{rcl}y_{k+1}^{*}&=&y_{k}+hG_{k}^{e}/2,\\ y_{k+1}^{**}&=&y_{k+1}^{*}+hG_{k+1}^{e*}/2,\\ y_{k+1}&=&y_{k}+h\left(G_{k}^{e}+G_{k+1}^{**}\right)/2,\end{array}\right\} \tag{30}\]  
where  
\[\left.\begin{array}{rcl}F_{k}^{e}&=&\frac{\int_{a}^{b}f(t,x_{k},y_{k})B_{k} \left(t\right)dt}{\int_{a}^{b}B_{k}\left(t\right)dt},\\ &&\\ F_{k+1}^{e*}&=&\frac{\int_{a}^{b}f(t,x_{k+1}^{*},y_{k+1}^{*})B_{k+1} \left(t\right)dt}{\int_{a}^{b}B_{k+1}\left(t\right)dt},\\ &&\\ F_{k+1}^{e**}&=&\frac{\int_{a}^{b}f(t,x_{k+1}^{**},y_{k+1}^{**})B_{k+1} \left(t\right)dt}{\int_{a}^{b}B_{k+1}\left(t\right)dt},\\ &&\\ F_{k+1}^{e**}&=&\frac{\int_{a}^{b}g(t,x_{k+1}^{**},y_{k+1}^{**})B_{k+1} \left(t\right)dt}{\int_{a}^{b}B_{k+1}\left(t\right)dt}.\end{array}\right\} \tag{31}\]  
and the truncation error \(Tx_{k}\) and \(Ty_{k}\) of the scheme I are given by  
\[Tx_{k} = \frac{x_{k+1}-x_{k}}{h}-\frac{1}{2}\left(F_{k}^{e}+F_{k+1}^{**} \right),\;\Big{|}\;Ty_{k} = \frac{y_{k+1}-y_{k}}{h}-\frac{1}{2}\left(G_{k}^{e}+G_{k+1}^{**} \right). \tag{32}\]  
Rearranging (23), we get  
\[0 = \frac{X_{k+1}-X_{k}}{h}-\frac{1}{2}\left(F_{k}+F_{k+1}^{**}\right), \;\Big{|}\;0 = \frac{Y_{k+1}-Y_{k}}{h}-\frac{1}{2}\left(G_{k}+G_{k+1}^{**}\right). \tag{33}\]  
Let \(e_{k+1}=X_{k+1}-x_{k+1}\) and \(d_{k+1}=Y_{k+1}-y_{k+1}\), then subtracting (33) from (32), we get  
\[Tx_{k}h =e_{k+1}-e_{k}-\frac{h}{2}\left(F_{k}-F_{k}^{e}\right)-\frac{h}{2} \left(F_{k+1}^{**}-F_{k+1}^{e**}\right),\] \[Ty_{k}h =d_{k+1}-d_{k}-\frac{h}{2}\left(G_{k}-G_{k}^{e}\right)-\frac{h}{2} \left(G_{k+1}^{**}-G_{k+1}^{e**}\right), \tag{34}\]  
Similarly to Lemma 8 and Theorem 2 by [16], we have the following results.  
**Lemma 5**.: _Let \(f\), \(g\) are assumed to be sufficiently smooth functions of its arguments on \([a,b]\) and be Lipschitz continuous with respect to \(x\) and \(y\), i.e., there exists a constant \(L\in\mathbb{R}\), such that for all \(t\in[a,b]\) and \(x,x^{\prime},y,y^{\prime}\in\mathbb{R}\),_  
\[|f(t,x,y)-f(t,x^{\prime},y^{\prime})| \leq L(|x-x^{\prime}|+|y-y^{\prime}|),\] \[|g(t,x,y)-g(t,x^{\prime},y^{\prime})| \leq L(|x-x^{\prime}|+|y-y^{\prime}|). \tag{35}\]  
_Assume that \(\{B_{k}\mid k=1,\ldots,n\}\), \(n\geq 2\), is a \(h\)-uniform generalized fuzzy partition of \([a,b]\). Then we get for \(k=1,\ldots,n\),_  
\[|e_{k+1}|\leq |e_{k}|\left(1+c\right)+Th\quad\text{and}\quad\left|F_{k}^{e}-F_{k+1 }^{e**}\right|\leq LhM_{2},\] \[|d_{k+1}|\leq |d_{k}|\left(1+c\right)+Th\quad\text{and}\quad\left|G_{k}^{e}-G_{k+1 }^{e**}\right|\leq LhM_{3},\]_where \(c=hL+\frac{h^{2}L^{2}}{2}+\frac{h^{3}L^{3}}{8}\), \(T=\max\limits_{1\leq k\leq n}|Tx_{k},Ty_{k}|\), \(F_{k}^{c}\), \(F_{k+1}^{e*},\ G_{k}^{e},\ G_{k+1}^{e**}\) are determined by Formula (31) and \(M_{2}\), \(M_{3}\) are upper bound for \(f\) and \(g\) respectively on \([a,b]\)._  
**Theorem 3**.: _Let \(f\), \(g:[a,b]\rightarrow\mathbb{R}\) be twice continuously differentiable on \([a,b]\). Let moreover, \(f\), \(g:[a,b]\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) be Lipschitz continuous with respect to \(x\) and \(y\). Assume that \(\{B_{k}\mid k=1,\ldots,n\}\), \(n\geq 2\), is a \(h\)-uniform generalized fuzzy partition of \([a,b]\). Then the scheme I (23) is convergent._  
The technique of error analysis for rest schemes can be obtained analogously to numerical Scheme I.