# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: [PERSON]*, [PERSON], [PERSON], [PERSON]
## 4 Applications  
One of the main problems of mathematics appears with variable coefficients when \(\alpha\left(t\right),\,\beta\left(t\right),\,\delta\left(t\right),\,\gamma \left(t\right)\) are analytic functions and added to the model. The new differential equations are represented by non-autonomous SODEs. In this model, time varying values for the growth rate of the prey, the efficiency of the predator, being the ability to capture prey, the death rate of the predator and the growth rate of the predator are considered. It is important to remark that since in this problem, the coefficients are time varying, careful attention must be paid in order to obtain the correct recurrence equation system of the model. The model, incorporating the above functions, is as follows [30; 33; 34]:  
\[\frac{dx}{dt}= \alpha\left(t\right)x\left(t\right)-\beta\left(t\right)x\left(t \right)y\left(t\right), x\left(0\right)=x_{1}\] \[\frac{dy}{dt}= \delta\left(t\right)x\left(t\right)y\left(t\right)-\gamma\left(t \right)y\left(t\right), y\left(0\right)=y_{1} \tag{35}\]Three examples are discussed in order to prove the results obtained by Scheme I (14)-(15) and Scheme II (29)-(30), two examples for the numerical solution of the model (35) and one example for the linear case. **Example 1**. : _Consider the problem of the Lotka-Volterra prey-predator model (35). We take_  
\[ \begin{array}{l}\alpha\left(t\right)=4+\tan\left(t\right),\,\beta\left(t \right)=\exp(2t),\,\gamma\left(t\right)=-2,\,\delta\left(t\right)=\cos(t),\,x (0)=-4\text{ and }y(0)=4.\end{array} \]  
_The exact solution for these coefficients is \(x(t)=\frac{-4}{\cos(t)}\), \(y(t)=4\exp(-2t)\), as proposed by [30; 33; 34]._  
**Example 2**. : _Consider the problem of the Lotka-Volterra prey-predator model (35) with_  
\[ \begin{array}{l}\alpha\left(t\right)=-t,\,\beta\left(t\right)=-t,\,\gamma \left(t\right)=t,\,\delta\left(t\right)=t,\,x(0)=2\text{ and }y(0)=2.\end{array} \]  
_The exact solution for these coefficients is \(x(t)=\frac{2}{2-\exp(t^{2}/2)}\), \(y(t)=\frac{2}{2-\exp(t^{2}/2)}\), as proposed by [30; 33]._  
**Example 3**. : _Consider the following non-autonomous SODEs with initial values (5):_  
\[ \begin{cases}x^{\prime}(t)=x(t)-y(t)+2t-t^{2}-t^{3}&,\,x(0)=1\,\,\,,t\in[0,1] \\ y^{\prime}(t)=x(t)+y(t)-4t^{2}+t^{3}&,\,y(0)=0.\end{cases}  \tag{36}\]  
_The exact solution of (36) is given by \(x(t)=e^{t}\cos(t)+t^{2}\) and \(y(t)=e^{t}\sin(t)-t^{3}\)._  
The results are listed in Tables 1-7 by the proposed fuzzy approximation methods with respect to the raised cosine generating function and Table 8 by the proposed fuzzy approximation methods with respect to the triangular generating function and raised cosine generating function. The proposed fuzzy approximation methods are generated by Algorithms A1 and A2 (please see Appendix B). The mean square error (MSE) is 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 2-7, a comparison is made between the two new proposed schemes (15), (29), the Euler method and the trapezoidal rule based on the Euler method for Examples 1-3. 2. Moreover, a comparison of MSE for Examples 1-3 is shown in Table 1. It is observed that the new fuzzy approximation methods yield more accurate results in comparison with the classical Euler and classical trapezoidal rule (one-step). The best result (in comparison with the Schemes I and II) is obtained by Scheme II. 3. In Table 8, a comparison is given between the errors for two proposed schemes based on the FzT with respect to fuzzy partitions determined by [6; 19]. The better results (in comparison with the non-linear case) are obtained by the linear case and non-autonomous SODEs in Example 3. Further, the results obtained using proposed fuzzy approximation methods for Examples 1-3 are shown in Figure 1 by using the raised cosine generating function.