# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: [PERSON]*, [PERSON], [PERSON], [PERSON]
## 3 FzT for Solving SODEs
### Numerical Scheme I for SODEs  
In this subsection, we present a modified scheme to solve SODEs using the FzT. Suppose that the functions \(f\) and \(g\) on \([t_{1},t_{n}]\) are sufficiently smooth in (5). For solving SODEs (5) on \([t_{1},t_{n}]\), the interval is divided into \(n-1\) subintervals. Let us choose some uniform fuzzy partition of interval \([t_{1},t_{n}]\) with parameter \(h=(t_{n}-t_{1})/(n-1)\), \(n\geq 2\), and basic functions \(A_{1},\dots,A_{n}\). In view of the methodological remarks in Subsection 3.1, we describe the complete sequence of steps, which leads to the approximation solution of SODEs (5) (see [6] for technical details). Before we apply the direct FzT to both parts of the differential equation, we will use the Taylor expansion and replace the first derivatives of the left-hand sides in (5) by their approximations, i.e.,  
\[\begin{cases}x(t+h)=x(t)+hx^{\prime}(t)+\mathcal{O}(h^{2}),\\ y(t+h)=y(t)+hy^{\prime}(t)+\mathcal{O}(h  
Thus, according to (5), the equality (8) can be rewritten (up to \(\mathcal{O}\left(h^{2}\right)\)) as matrix equality:  
\[ \begin{cases}F_{n}[x^{\prime}]=DF_{n}[x],\\ G_{n}[y^{\prime}]=DG_{n}[y],\end{cases}  \tag{9}\]  
where \( \begin{cases}F_{n}[x^{\prime}]=[X_{1}^{\prime},\dots,X_{n-1}^{\prime}]^{T},\\ G_{n}[y^{\prime}]=[Y_{1}^{\prime},\dots,Y_{n-1}^{\prime}]^{T},\end{cases} \) and \( \begin{cases}F_{n}[x]=[X_{1},\dots,X_{n-1}]^{T},\\ G_{n}[y]=[Y_{1},\dots,Y_{n-1}]^{T}.\end{cases} \)  
Now, let us return to the problem (5) and apply the FzT to both sides of the differential equation. Based on the linearity of FzT and Formula (9), we obtain the following system with respect to the unknown \(F_{n}[x]\) and \(G_{n}[y]\):  
\[ \begin{cases}DF_{n}[x]=F_{n}[f],\\ DG_{n}[y]=G_{n}[g],\end{cases}  \tag{10}\]  
where \( \begin{cases}F_{n}[f]=[F_{1},\dots,F_{n-1}]^{T},\\ G_{n}[g]=[G_{1},\dots,G_{n-1}]^{T},\end{cases} \) are the FzT of \(f(t,x(t),y(t))\) (\(g(t,x(t),y(t))\)) as a function of \(t\) w.r.t. the chosen basic functions \(A_{1},\dots,A_{n}\). Note that the last components \(F_{n}\) and \(G_{n}\) are not presented in \(F_{n}[f]\) and \(G_{n}[g]\), respectively, due to the dimensionality limitation, and (10) does not include two initial conditions of (5). Thus, we complete the matrix \(D\) by adding the first row as the initial value as follows:  
\[D^{*}=\frac{1}{h}\left( \begin{array}{ccccc}1&0&0&\cdots&0\\ -1&1&0&\cdots&0\\ 0&-1&1&\cdots&0\\ \vdots&&&\ddots&\vdots\\ 0&0&\cdots&-1&1\end{array} \right),\]  
so that \(D^{*}\) is an \(n\times n\) nonsingular matrix. Based on the initial conditions and the matrix \(D^{*}\), we also expand \(F_{n}[f]\) and \(G_{n}[g]\) by adding the first element, as follows:  
\[ \begin{cases}F_{n}^{*}[f]=[\frac{x_{1}}{h},F_{1},\dots,F_{n-1}]^{T},\\ G_{n}^{*}[g]=[\frac{y_{1}}{h},G_{1},\dots,G_{n-1}]^{T}.\end{cases} \]  
Then, the problem (5) can be completely represented by the following expression with respect to the unknown \(F_{n}[x]\) and \(G_{n}[y]\):  
\[ \begin{cases}D^{*}.F_{n}[x]=F_{n}^{*}[f],\\ D^{*}.G_{n}[y]=G_{n}^{*}[g].\end{cases}  \tag{11}\]  
The solution of (11) can be obtained by the following formula:  
\[ \begin{cases}F_{n}[x]=(D^{*})^{-1}.F_{n}^{*}[f],\\ G_{n}[y]=(D^{*})^{-1}.G_{n}^{*}[g].\end{cases}  \tag{12}\]In fact, to obtain the solution of (12), we should compute the inverse matrix of \(D^{*}\). Therefore, we have:  
\[(D^{*})^{-1}=h \begin{pmatrix}1&0&0&\cdots&0\\ 1&1&0&\cdots&0\\ 0&1&1&\cdots&0\\ \vdots&&&\ddots&\vdots\\ 1&1&\cdots&1&1\end{pmatrix} ,\]  
and by (12), we get:  
\[X_{k+1}=X_{k}+hF_{k},\qquad X_{1}=\alpha,\ \big{|}\ Y_{k+1}=Y_{k}+hG_{k}, \qquad Y_{1}=\beta,\ k=1,\ldots,n-1, \tag{13}\]  
where \(F_{k}\ \ (G_{k})\) is given by Formula (6). Formula (13) can be applied to the computation of \(X_{2},\ldots,X_{n}\) and \(Y_{2},\ldots,Y_{n}\). However, it cannot be applied directly by using the function \(f(t,x,y)\) or \(g(t,x,y)\), because it uses unknown functions \(x\) and \(y\).