# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: [PERSON]*, [PERSON], [PERSON], [PERSON]
## Appendix A Taylor Series  
A Taylor series is given that:  
\[x(t_{k+1}) =x(t_{k})+hx^{\prime}(t_{k})+\frac{h^{2}}{2}x^{\prime\prime}(t_{k})+ \frac{h^{3}}{6}x^{\prime\prime\prime}(\varepsilon_{1k}),\] \[=x(t_{k})+hx^{\prime}(t_{k})+\frac{h^{2}}{2}\left(\frac{x^{\prime }(t_{k+1})-x^{\prime}(t_{k})}{h}-\frac{h}{2}x^{\prime\prime\prime}( \varepsilon_{2k})\right)+\frac{h^{3}}{6}x^{\prime\prime\prime}(\varepsilon_{1 k}),\] \[=x(t_{k})+\frac{h}{2}x^{\prime}(t_{k})+\frac{h}{2}x^{\prime}(t_{ k+1})+h^{3}\left[\frac{1}{6}x^{\prime\prime\prime}(\varepsilon_{1k})-\frac{1}{4}x^{ \prime\prime\prime}(\varepsilon_{2k})\right], \tag{10}\]  
where \(x^{\prime\prime}(t)=\frac{x^{\prime}(t_{k+1})-x^{\prime}(t_{k})}{h}-\frac{h}{2 }x^{\prime\prime\prime}(\varepsilon_{2k}).\) Calculus can be used to derive that:  
\[x^{\prime}(t_{k+1}) =f\left(t_{k+1},x(t_{k+1}),y(t_{k+1})\right)\] \[=f\left(t_{k+1},x(t_{k})+hf(t_{k},x(t_{k}),y(t_{k})),y(t_{k})+hg( t_{k},x(t_{k}),y(t_{k}))\right)\] \[+\frac{h^{2}}{2}f_{2}(\varepsilon_{3k},x(\varepsilon_{3k}),y( \varepsilon_{3k}))x^{\prime\prime}(\varepsilon_{3k}),\]  
where \(f_{2}(\varepsilon_{3k},x(\varepsilon_{3k}),y(\varepsilon_{3k}))=\frac{\partial }{\partial x}f(\varepsilon_{3k},x(\varepsilon_{3k}),y(\varepsilon_{3k}))+ \frac{\partial}{\partial y}f(\varepsilon_{3k},x(\varepsilon_{3k}),y( \varepsilon_{3k})).\)  
Substituting Equation (A1), it is given that:  
\[x(t_{k+1}) =x(t_{k})+\frac{h}{2}x^{\prime}(t_{k})\] \[+\frac{h}{2}f\left(t_{k+1},x(t_{k})+hf(t_{k},x(t_{k}),y(t_{k})), y(t_{k})+hg(t_{k},x(t_{k}),y(t_{k}))\right)\] \[+\frac{h}{2}\frac{h^{2}}{2}f_{2}(\varepsilon_{3k},x(\varepsilon_{ 3k}),y(\varepsilon_{3k}))x^{\prime\prime}(\varepsilon_{3k})\] \[+h^{3}\left[\frac{1}{6}x^{\prime\prime\prime}(\varepsilon_{1k})- \frac{1}{4}x^{\prime\prime\prime}(\varepsilon_{2k})\right],\] \[x(t_{k+1}) =x(t_{k})+\] \[\frac{h}{2}\left(x^{\prime}(t_{k})+f\left(t_{k+1},x(t_{k})+hf(t_{ k},x(t_{k}),y(t_{k})),y(t_{k})+hg(t_{k},x(t_{k}),y(t_{k}))\right)\right)\] \[+h^{3}\left[\frac{1}{6}x^{\prime\prime\prime}(\varepsilon_{1k})- \frac{1}{4}x^{\prime\prime\prime}(\varepsilon_{2k})+\frac{1}{4}f_{2}( \varepsilon_{3k},x(\varepsilon_{3k}),y(\varepsilon_{3k}))x^{\prime\prime}( \varepsilon_{3k})\right].\]  
It can be rewritten as:  
\[x(t_{k+1})=x(t_{k})+\frac{h}{2}\left(K_{0}+K_{f}\right)+e_{f}h^{3},\]  
where:  
\[K_{0} =x^{\prime}(t)=f(t_{k},x(t_{k}),y(t_{k})),\,K_{1}=y^{\prime}(t)=g (t_{k},x(t_{k}),y(t_{k})),K_{f}=f(t_{k+1},x(t_{k})+hK_{0},y(t_{k})+hK_{1}),\] \[e_{f} =\frac{1}{6}x^{\prime\prime\prime}(\varepsilon_{1k})-\frac{1}{4} x^{\prime\prime\prime}(\varepsilon_{2k})+\frac{1}{4}f_{2}(\varepsilon_{3k},x( \varepsilon_{3k}),y(\varepsilon_{3k}))x^{\prime\prime}(\varepsilon_{3k}),\text {and}\,\,t_{k}<\varepsilon_{1k},\,\varepsilon_{2k},\,\varepsilon_{3k}<t_{k+1}.\]  
Similarly,  
\[y(t_{k+1})=y(t_{k})+\frac{h}{2}\left(K_{1}+K_{g}\right)+e_{g}h^{3},\]where:  
\[K_{g}=g(t_{k+1},x(t_{k})+hK_{0},y(t_{k})+hK_{1}),\,e_{g}=\frac{1}{ 6}y^{\prime\prime\prime}(\xi_{1k})-\frac{1}{4}y^{\prime\prime\prime}(\xi_{2k})+ \frac{1}{4}g_{2}(\xi_{3k},x(\xi_{3k}),y(\xi_{3k}))y^{\prime\prime}(\xi_{3k}),\] \[g_{2}(\epsilon_{3k},x(\epsilon_{3k}),y(\epsilon_{3k}))=\frac{ \partial}{\partial x}g(\epsilon_{3k},x(\epsilon_{3k}),y(\epsilon_{3k}))+ \frac{\partial}{\partial y}g(\epsilon_{3k},x(\epsilon_{3k}),y(\epsilon_{3k})),\]  
and \(t_{k}<\xi_{1k},\,\xi_{2k},\,\xi_{3k}<t_{k+1}\).