# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II
## 5 Applications  
A general model for the dynamical system may be written as \(\frac{dx}{dt}=xg(x,y)\), \(\frac{dy}{dt}=yh(x,y)\), where \(g\) and \(h\) are arbitrary functions of the prey and predator species whose populations are \(x(t)\) and \(y(t)\) at time \(t\). However, the following problem of Lotka-Volterra equation with derivatives and with variable coefficients \(\alpha\left(t\right),\ \beta\left(t\right),\ \delta\left(t\right),\ \gamma\left(t\right)\) as functions of time \(t\) have not yet been solved by any fuzzy numerical method. The new differential equations are represented by a non-autonomous ordinary differential equation system. The model, incorporating the above functions is as follows [41; 42; 43]:  
\[\frac{dx}{dt}= \alpha\left(t\right)x\left(t\right)-\beta\left(t\right)x\left(t \right)y\left(t\right), x\left(0\right)=x_{1}\] \[\frac{dy}{dt}= \delta\left(t\right)x\left(t\right)y\left(t\right)-\gamma\left(t \right)y\left(t\right), y\left(0\right)=y_{1} \tag{36}\]  
Two examples are discussed in order to prove that the results obtained by Scheme I (23), II (28) and III (29) for the numerical solution of the model (36). **Example 2**. : _Consider the problem of Lotka-Volterra-prey- predator model (36). We take \(\alpha\left(t\right)=4+\tan\left(t\right),\ \beta\left(t\right)=\exp(2t)\), \(\gamma\left(t\right)=-2\), \(\delta\left(t\right)=\cos(t)\), \(x(0)=-4\) and \(y(0)=4\)._  
_The exact solution for these coefficients is \(x(t)=\frac{-4}{\cos(t)}\), \(y(t)=4\exp(-2t)\) proposed by [41; 42; 43]._  
**Example 3**. : _Consider the problem of Lotka-Volterra-prey-predator model (36) with \(\alpha\left(t\right)=-t\), \(\beta\left(t\right)=-t\), \(\gamma\left(t\right)=t\), \(\delta\left(t\right)=t\), \(x(0)=2\) and \(y(0)=2\)._  
_The exact solution for these coefficients is \(x(t)=\frac{2}{2-\exp(t^{2}/2)}\), \(y(t)=\frac{2}{2-\exp(t^{2}/2)}\) proposed by [41; 42]._  
The results are listed in Tables 2-6 by the proposed fuzzy approximation methods with respect to case \(K_{\mathrm{C}_{2}^{200}}\) is defined by Example 1. The proposed fuzzy approximation methods are generated by Algorithms A1-A3 (Appendix A). The mean square error (MSE) defined as \(\text{MSE}\ =\frac{1}{n}\left(\left\|Y_{k}-y(t_{k})\right\|_{2}\right)^{2}\). This is an easily computable quantity for a particular sample. From the numerical tests, the results are summarized as follows:  
1. In view of Tables 2-5, a comparison between the three new proposed schemes ((23), (28) and (29)) and the classical Trapezoidal rule (1-step), classical 2-step Adams Moulton Method and classical 3-step Adams Moulton Method based on Euler method for Examples 2 and 3. 2. Moreover, comparison of MSE for Examples 2 and 3 shown in Table 6. It is observed that the new fuzzy approximation methods yield more accurate results in comparison with the classical Trapezoidal rule (one step) and classical Adams Moulton method (two and three steps). Hence, the new fuzzy approximation methods provide alternative techniques for solving SODEs with better results. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(t_{i}\)} & \multirow{2}{*}{**Solution \(x(t)\)**} & \multirow{2}{*}{**Proposed**} & \multirow{2}{*}{**Proposed**} & \multirow{2}{*}{**Proposed**} & \multirow{2}{*}{**Trap 1**} & \multirow{2}{*}{**2-Step Adams 2**} & \multirow{2}{*}{**3-Step Adams 3**} \\  & & &  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(t_{i}\)} & \multirow{2}{*}{**Solution \(x(t)\)**} & \multicolumn{1}{c}{**Proposed**} & \multicolumn{1}{c}{**Proposed**} & \multicolumn{1}{c}{**Proposed**} & \multirow{2}{*}{**Trap \({}^{1}\)**} & \multirow{2}{*}{**2-Step Adams \({}^{2}\)**}  
Further, the results obtained using proposed fuzzy approximation methods for Examples 2 and 3 are shown in Figures 1-3 by using \(K_{\mathrm{C}_{2}^{201}}\).