# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: [PERSON]*, [PERSON], [PERSON], [PERSON]
## 2 Basic Concepts  
Throughout this section, we deal with an interval \([a,b]\subset\mathbb{R}\) of real numbers. Let \([a,b]\) be an interval on the real line \(\mathbb{R}\). Fuzzy sets on \([a,b]\) will be identified with their membership functions mapping from \([a,b]\) into \([0,1]\). We will assume an interval \([a,b]\) as a real domain. In this section, we remind about the definitions and claims that were introduced and proven by [5].  
Definition 1: (Fuzzy partition) Let \(x_{1}<\cdots<x_{n}\) be fixed nodes within \([a,b]\) such that \(x_{1}=a,\ x_{n}=b\) and \(n\geq 2\). The fuzzy sets \(A_{1},\ldots,\ A_{n}\) are often called basic functions. We say that fuzzy sets \(A_{1},\ldots,\ A_{n}\subset[a,b]\) establish a fuzzy partition of \([a,b]\) if they fulfill the following conditions for \(k=1,\ldots,n\) (for the uniformity of notation, we set \(x_{0}=a\) and \(x_{n+1}=b\)):  
1. \(A_{k}\left(x\right)\ :\ [a,\,b]\rightarrow[0,1]\) is continuous with \(A_{k}\left(x_{k}\right)=1\), \(A_{k}\left(x\right)>0\) if \(x\in\left(x_{k-1},x_{k+1}\right)\) and \(A_{k}\left(x\right)=0\) if \(x\
otin\left(x_{k-1},x_{k+1}\right)\);
2. \(A_{k}\left(x\right)\), \(k=2,\ldots,n\), strictly increases on \([x_{k-1},x_{k}]\), and \(A_{k}\left(x\right),\ k=1,\ldots,n-1\), strictly decreases on \([x_{k},x_{k+1}]\);
3. For all \(x\in[a,\ b]\), \(\sum_{k=1}^{n}A_{k}(x)=1\). This is called the Ruspini condition.  
We say that a fuzzy partition of \([a,b]\) is \(h\)-uniform if its nodes \(x_{1},\ldots,\,x_{n}\), where \(n\geq 2\) are equidistant. This means that \(x_{k}=a+h\left(k-1\right),\ k=1,\ldots,\,n\), where \(h=\frac{b-a}{n-1},\ n\geq 2\), and the two additional properties are fulfilled:  
* \(A_{k}(x_{k}-x)=A_{k}(x_{k}+x),\ k=2,\ldots,\ n-1\), for all \(x\in[0,h]\) and:
* \(A_{k}(x)=A_{k-1}(x-h)\) and \(A_{k+1}(x)=A_{k}(x-h),\ k=2,\ldots,n-1\), for all \(x\in[x_{k},\,x_{k+1}]\).  
Two uniform fuzzy partitions with triangular- and sinusoidal-shaped basic functions can be found in [5; 6]. Throughout this paper, we will write uniform fuzzy partition instead of \(h\)-uniform fuzzy partition.  
Definition 2: Let \(f\) be a continuous function on \([a,b]\) and \(A_{k}(x)\), \(k=1,\ldots,n\), be a uniform fuzzy partition of \([a,b]\), \(n\geq 2\). A vector of real numbers \(F[f]=\left(F_{1},\ F_{2},\ldots,\ F_{n}\right)\) given by (to complete this notation, we set \(x_{1}=a\) and \(x_{n+1}=b\)):  
\[F_{k}\left[f\right]=\frac{\int_{a}^{b}f\left(x\right)\ A_{k}(x)\,dx}{\int_{a}^ {b}A_{k}(x)\,dx}=\frac{\int_{x_{k-1}}^{x_{k+1}}f(x)\ A_{k}(x)dx}{\int_{x_{k-1 }}^{x_{k+1}}A_{k}(x)dx},\ k=1,\ldots,n, \tag{1}\]  
is called the direct FzT of \(f\) with respect to \(A_{1},\ldots,\ A_{n}\).  
Remark 1: The elements \(F_{1}[f],\ldots,F_{n}[f]\) are called components of the FzT. If \(A_{1},\ldots,A_{n}\) forms a uniform fuzzy partition, then the expression (1) can be simplified as follows:  
\[F_{1}[f]=\frac{2}{h}\int_{x_{1}}^{x_{2}}f(x)\ A_{1}(x)dx,\ F_{n}[f]=\frac{2} {h}\int_{x_{n-1}}^{x_{n}}f(x)\ A_{n}(x)dx,\]  
\[F_{k}[f]=\frac{1}{h}\int_{x_{k-1}}^{x_{k+1}}f(x)\ A_{k}(x)dx,\ k=2,\ldots,n-1. \tag{2}\]  
Definition 3: Let \(F[f]=\left(F_{1},\ F_{2},\ldots,\ F_{n}\right)\) be the direct FzT of \(f\in C\left[a,b\right]\) with respect to \(A_{k}(x)\), \(k=1,\ldots,n\). Then, the inverse FzT of \(f,\hat{f}\,:\,[a,b]\to R\), given by:  
\[\hat{f}(x)=\sum_{k=1}^{n}F_{k}\ A_{k}(x). \tag{3}\]  
**Lemma 1**.: _[_6_]_ _Let \(f(x)\) be continuous on \([a,b]\) and twice continuously differentiable in \((a,b)\), and let basic functions form a uniform fuzzy partition of \([a,b]\). Then, for each \(k=1,\ldots,n\):_  
\[F_{k}=f(x_{k})+\mathcal{O}(h^{2}). \tag{4}\]  
**Remark 2**.: _An important property of the direct FzT, as well as inverse FzT is their linearity, namely, given \(f\), \(g\in C\left[a,b\right]\) and \(\alpha,\ \beta\in\mathbb{R}\), if \(h=\alpha f+\beta g\), then \(F\left[h\right]=\alpha F\left[f\right]+\beta F\left[g\right]\) and \(\hat{h}=\alpha\hat{f}+\beta\hat{g}\)._