# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II
## 1 Introduction
The FzT has investigated for solving the Volterra population growth model using the approximation for the Caputo derivative [27]. A new numerical method based FzT was demonstrated to solve a class of delay differential equations by means of the Picard-like numerical scheme [28]. FzT was considered to approximate the solution of boundary value problems by minimizing the integral squared error in 2-norm [29]. In [30], the dynamical properties of a two neuron system with respect to FzT and a single delay have been investigated. The conditions under which quasi-consensus in a multi-agent system with sampled data based on FzT were proposed by [31]. NIM was proposed to solve nonlinear functional equations and the existence of solution for nonlinear Volterra integral equations [2]. At the same time, NIM was introduced for solving nonlinear equations by using a different decomposition technique [32]. From this conception, NIM was considered in terms up to fourth-order in Taylor series for solving nonlinear equations [33]. Sufficiency conditions have been presented for convergence of the NIM [34]. A new predictor-corrector approach was developed based on NIM for fractional differential equations [35]. Classical methods are modified by [3] to derive numerous formulas for solving the differential equations. The motivation of the proposed study comes from [16; 36; 37]. In [16], new fuzzy numerical methods to solve the Cauchy problem was considered and the authors showed that the error can be reduced by FzT and NIM with respect to new generalized uniform fuzzy partitions, namely power of the triangular and raised cosine generalized uniform fuzzy partitions, where generating functions are normal (see also [37] for another approach). In addition, two basic approximation methods, modified Euler method and Trapezoidal rule, with help from FzT for solving SODEs are analyzed in detail by [36]. For this purpose, more generally, new generalized uniform fuzzy partitions are proposed in this study, where a generating function is not normal. The membership functions in underlying fuzzy partitions are often called basic functions. There has been a growing interest in investigating the properties of fuzzy partitions. However, the problem arises on how one can effectively construct the basic function of fuzzy partitions. In this paper, new representations of basic functions are proposed. This is achieved by introducing new generalized uniform fuzzy partitions, where a generating function is not normal. Further, new fuzzy numerical methods based on NIM and FzT for solving SODEs are introduced and discussed. In particular, we consider functions of two variables with initial conditions. In accordance with the existing methods, Trapezoidal rule and Adams Moulton are improved using FzT and NIM. The methods are combined with one-step, two-step and three-step. As an application, all these methods are used to solve a general model of the dynamical system, i.e., Lotka-Volterra equation with derivatives and with variable coefficients. Furthermore, numerical examples are presented. It is observed that the new fuzzy numerical methods yield more accurate results than classical Trapezoidal rule and classical Adams Moulton methods (2 and 3-step). The paper is organized as follows. The main part of the paper is Sections 3 and 4, which provides new representations for basic functions of FzT, followed by the modified one step, 2-step and 3-step based on NIM and FzT method with respect to new representations formulas for generalized uniform fuzzy partition of FzT. In Section 5, numerical examples are discussed. Finally, conclusions are given in Section 6.