# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: [PERSON]*, [PERSON], [PERSON], [PERSON]
## 3 FzT for Solving SODEs
### Methodological Remarks to Applications of the FzT  
Consider the Initial Value Problem (IVP) for the SODEs:  
\[\begin{cases}x^{\prime}(t)=f\left(t,x\left(t\right),y\left(t\right)\right), \quad x\left(t_{1}\right)=\alpha,t_{1}\leq t\leq t_{n},\\ y^{\prime}(t)=g\left(t,x\left(t\right),y\left(t\right)\right),\quad y\left(t_ {1}\right)=\beta,\end{cases} \tag{5}\]  
where \(\alpha,\ \beta\in\mathbb{R}\), \(f\) and \(g\) are continuous functions on \([t_{1},\ t_{n}]\times\mathbb{R}\times\mathbb{R}\) and satisfy the Lipschitz condition. Unfortunately, the analytical solution \((x(t),\ y(t))\) of Problem (5) is often difficult and sometimes impossible to obtain. Thus, approximate solutions by means of FzT are extremely important for solving (5). A numerical method for (5) is an algorithm that computes FzT components \(X_{k}\approx x(t_{k})\) and \(Y_{k}\approx y(t_{k})\), for each \(k\ =\ 2,\ldots,n\) (to complete this notation, we set \(\alpha=X_{1}=x(t_{1})\) and \(\beta=Y_{1}=y(t_{1})\)).  
Below, we extend the main principles of FzT detailed in Formulas (6) that are needed later.  
**Definition 4**.: _Let \(f\left(g\right)\) be a continuous function on \([t_{1},\ t_{n}]\) and \(A_{1},\ldots,A_{n}\) be the fuzzy partition of \([t_{1},\ t_{n}]\). A vector of real numbers \(F_{k}[f]=\left(F_{1}[f],\ldots,F_{n}[f]\right)\)\(\left(G_{k}[g]=\left(G_{1}[g],\ldots,G_{n}[g]\right)\right)\) given by:_  
\[F_{k}[f]=\frac{\int_{t_{1}}^{t_{n}}f\left(t,x\left(t\right),y\left(t\right) \right)\,A_{k}(t)\;dt}{\int_{t_{1}}^{t_{n}}A_{k}(t)\;dt}\ \ \left(G_{k}[g]=\frac{\int_{t_{1}}^{t_{n}}g\left(t,x\left(t\right),y\left(t \right)\right)\,A_{k}(t)\;dt}{\int_{t_{1}}^{t_{n}}A_{k}(t)\;dt}\right), \tag{6}\]  
_is called the direct FzT of \(f\left(g\right)\) that is extended with independent variable \(t\) and two dependent variables \(x\) and \(y\)._  
**Remark 3**.: _We need a way to approximate the direct FzT components \(6\). This is discussed in Corollary 1._  
In the following, the necessary steps of the FzT are given.  
1. Construction of the fuzzy partition: 1. Specify the number \(n\) of components, and compute the step \(h=\left(t_{n}-t_{1}\right)/\left(n-1\right)\). 2. Construct the nodes \(t_{1}<\ldots<t_{n}\), where \(t_{k}=t_{1}+h(k-1)\). 3. Select the shape of basic functions. We mostly use triangular- or sinusoidal-shaped basic functions. Recall that the shape of the basic functions determines the course of \(\hat{f}\), that is whether it is piecewise linear or nonlinear. 4. Construct a uniform fuzzy partition of \([t_{1},t_{n}]\) by triangular- or sinusoidal-shaped basic functions [5].
2. Computation of FzT: We replace \(x^{\prime}\left(t\right)\) and \(y^{\prime}\left(t\right)\) by their approximations based on the Taylor expansion as new functions with respect to the fuzzy partition \(A_{1},\ldots,A_{n}\) by Step 1. In this way, similarly to [6], we transfer the original SODEs to the space of fuzzy units, solve them in the newspace and then transfer them back by the inverse FzT. Compute the approximation for \(x\) and \(y\) by the inverse FzT applied to \([X_{1},\dots,X_{n}]\) and \([Y_{1},\dots,Y_{n}]\). In the next subsections, the schemes provide formulas for the computation of components of FzT.