# New Fuzzy Numerical Methods for Solving Cauchy Problems
## 1 Introduction  
In fact, most mathematical models in engineering and science requires the solution of ordinary differential equations (ODEs). Generally, it is difficult to obtain the closed form solutions for ODEs, especially, for nonlinear and nonhomogeneous cases. Many models often lead to ordinary differential equations which consist of Cauchy problems are an important branch of modern mathematics that arises naturally in different areas of applied sciences, physics, and engineering. Thus, many researchers start developing methods for solving Cauchy problems are of particular importance [1; 2; 3].  
FT was coined by [PERSON] as a new mathematical method was developed [4]. The core idea of FT is a fuzzy partition of a universe into fuzzy subsets. The technique of FT has been successfully applied into other mathematical problems as well including image processing, analysis of time series and elsewhere [5; 6; 7]. This idea has been applied to Cauchy problems was first published as well as other numerical classical methods [8], by proposing generalized Euler and Euler- Cauchy methods, so that the Mid-point FT method was demonstrated in [9]. The success of these applications is due in part to the fact that FT is capable to accurately approximate any continuous function. Thus, we will propose new fuzzy numerical methods for Cauchy problems with help of the FT and new iterative method.  
The motivation of the proposed study comes from the papers [3; 8; 10]. Numeric Solution to the Cauchy problem was considered and the authors showed that the error can be reduced by using FT with uniform fuzzy partitions [8; 9]. At the same time, [10; 11], the concept of generalized fuzzy partition was proposed. Besides others, a necessary and sufficient condition making it possible to design easilythe generalized fuzzy partition was provided [12]. This is important for various practical applications of FT. Further [3], the authors have proposed modifications trapezoidal rule and Adams-Moulton methods (2 and 3-step) to solve ODEs based on the new iterative method was introduced [2].  
In this paper, we discuss the problem that considered in [8; 9]. The triangular and raised cosine generating function was replaced by new representations formulas for generalized uniform fuzzy partition of FT such as power of the triangular and raised cosine generating function. We study approximation properties of the FT based on powers of triangular and raised cosine generalized uniform fuzzy partition can be constructed in such way that the FT can reduce error. Also, we propose modifications in the FT introduced by [PERSON] [4] with respect to new representations formulas for generalized uniform fuzzy partition of FT and then the technique of FT is combined with traditional methods based on the new iterative method [2; 3] to solve Cauchy problems. It is observed that the new methods proposed are more accurate results than the fuzzy approximation method [8; 9].  
This paper is organized as follows. In Section 2, we introduce the basic concepts and results of the FT with respect to the generalized uniform fuzzy partition needed throughout this paper. The main part of this paper is Sections 3 and 4, new representations for basic functions of FT, followed by the modified one step, 2-step, and 3-step based on new representations formulas for generalized uniform fuzzy partition of FT. In Section 5, numeric examples are discussed. Concluding remarks are presented in Section 6.  
Throughout the paper, we denote by \(\mathbb{N}\), \(\mathbb{N}^{+}\), \(\mathbb{Z}\), \(\mathbb{R}\), and \(\mathbb{R}^{+}\) the sets of natural (including zero), positive natural, integer, real, and positive real numbers, respectively.