# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II
## 4 New Fuzzy Numerical Methods for Solving SODEs  
Consider the initial value problem (IVP) for the SODEs:  
\[ \begin{cases}x^{\prime}(t)=f\left(t,x,y\right),\ \ x\left(t_{1}\right)=x_{1},\ a=t_{1} \leq t\leq t_{n}=b\\ y^{\prime}(t)=g\left(t,x,y\right),\ \ y\left(t_{1}\right)=y_{1},\end{cases}  \tag{16}\]  
where \(x_{1},\ y_{1}\in R\) and \(f,\ g\) are continuous function on \(D=\left[a,b\right]\times R\times R\). If \(f\left(g\right)\) satisfies a Lipschitz condition on \(D\) in the variable \(x\left(y\right)\), then the initial-value problem (16) has a unique solution \(x(t)\left(y(t)\right)\) for \(a\leq t\leq b\). In many cases, the problem (16) cannot be solved analytically so that numerical solutions are required. In [16], new representations of basic function based on the FzT are constructed for solving generalized Cauchy problems with help of NIM, FzT and classical methods (one-step, two-step and three-step) have presented while Euler method and Mid-point rule, based on FzT to solve Cauchy problem proposed by [17; 18]. Further, NIM has been proposed for solving ODEs and delay differential equations [3]. Moreover, Adams-Bashforth methods and Adams-Moulton methods are noted as two families of multistep methods in literature. Multistep methods refer to using several previous values from the previous steps. The Adams-Bashforth methods were presented by [PERSON] to solve a differential equation modelling capillary action due to [PERSON] and it follows that the Adams-Moulton method was developed improved multistep methods for solving ballistic equations by [PERSON]. In particular, the Adams-Moulton method is similar to the Adams-Bashforth method and the Adams-Moulton method was used [PERSON]'s method to solve the implicit equation. Clearly, Adams-Bashforth methods are explicit methods and the Adams-Moulton methods are implicit methods, for example, see ([40], p. 111). Necessary steps of construction of the generalized uniform fuzzy partitions can be summarized as follows. 1. Specify the number \(n\) of components and compute the step \(h=\left(t_{n}-t_{1}\right)/\left(n-1\right)\). If we want to obtain as best approximation of \(f\) as possible, then \(n\) should be large. 2. Construct the nodes \(t_{1}<\ldots<t_{n}\), where \(t_{k}=t_{1}+h(k-1)\). 3. Select the shape of basic functions. This is achieved by selecting the shape of generating function. 4. Construct a \(h\)-uniform generalized fuzzy partition of \(\left[t_{1},t_{n}\right]\) by new representations of basic functions are defined by Definition 4. To begin the derivation of a modified Trapezoidal rule (1-step) and Adams Moulton method (2 and 3-step), integrate (16) on the interval \(\left[t_{k},t_{k+1}\right]\), \(k=1,\ldots,n-1\) to obtain=  
\[x(t_{k+1}) =x\left(t_{k}\right)+\int_{t_{k}}^{t_{k+1}}f\left(s,x(s),y\left(s \right)\right)ds,\] \[y(t_{k+1}) =y\left(t_{k}\right)+\int_{t_{k}}^{t_{k+1}}g\left(s,x(s),y\left(s \right)\right)ds. \tag{17}\]  
Consider the following integral  
\[I_{f} =\int_{t_{k}}^{t_{k+1}}f\left(s,x(s),y\left(s\right)\right)ds,\] \[I_{g} =\int_{t_{k}}^{t_{k+1}}g\left(s,x(s),y\left(s\right)\right)ds. \tag{18}\]  
However, we cannot integrate \(f\left(s,x(s),y\left(s\right)\right)\) and \(g\left(s,x(s),y\left(s\right)\right)\) without knowing \(x(s)\) and \(y(s)\). So, the above integral (18) can be approximated by the following approach \[I_{f} \approx\int_{t_{k}}^{t_{k+1}}f_{1}\left(s,x(s),y\left(s\right) \right)ds,\] \[I_{g} \approx\int_{t_{k}}^{t_{k+1}}g_{1}\left(s,x(s),y\left(s\right) \right)ds, \tag{19}\]  
where \(f_{1}\) and \(g_{1}\) are the approximation of \(f\) and \(g\) on the interval \([t_{k},t_{k+1}]\). Choosing different \(f_{1}\) (\(g_{1}\)) leads to different schemes. In particular, we choose \(f_{1}\) (\(g_{1}\)) which contributes to the one, two and three-step methods based on FzT. Later, modification of these methods based on FzT and NIM.