# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: [PERSON]*, [PERSON], [PERSON], [PERSON]
## 1 Introduction  
Differential equations have great potential to model and understand real-world problems in science and engineering. In many cases, differential equations cannot be solved analytically, so that numerical methods are required. Therefore, numerical methods have been elaborated frequently in scientific research for solving differential equations, for example [1; 2; 3; 4]. In this connection, fuzzy approaches successfully cope with solving differential equations. One of fuzzy approaches that has been proposed in the literature is fuzzy transform (FzT). FzT is a general mathematical technique coined and developed by [PERSON] [5]. The study of FzT is rapidly expanding as a new branch of approximation method based on fuzzy sets. The main idea of FzT is usually forming a fuzzy partition of a universe into fuzzy subsets. Two shapes for the basic functions of fuzzy partition, triangular- and sinusoidal-shaped membership functions, were considered by [6]. Applications of the FzT can be used in the construction of approximate models, the approximation of functions and the solution of differential equations. FzT has been generalized from the case of constant components to the case of polynomial components [7]. Later, FzT was successfully used by [8] for a second order initial value problem. From this idea, FzT was proposed for numerical solutions of two point boundary value problems by the more efficient way in comparison with the similar ones obtained by the finite difference method [9]. Recently, in [10], a new numerical method based on FzT to solve a class of delay differential equations by means of the Picard-like numerical scheme was presented. The author demonstrated the stability of the method, and the obtained results have good agreement with existing methods. Furthermore, in some cases, a better approximation was achieved through sinusoidal-shaped basic functions, while the Bernstein basis polynomials allow better results in the other examples. On the other hand, a new approach to the fuzzy boundary value problem in the form of the fuzzy relationwas investigated by [11]. Another approach for the second order linear differential equation with constant coefficients and Dirichlet boundary conditions was introduced by [12], where the ability of FzT was demonstrated to deal with boundary value problems affected by noise, and the results were compared with the finite difference method. To confirm again the ability of FzT with respect to noisy and non-noisy source functions, FzT based on the shooting method was introduced by [13] for solving a nonlinear second-order ODE, and the obtained results were better than the classical method, namely the second order Runge-Kutta. Further, in [14], FzT to approximate the solution of boundary value problems by minimizing the integral squared error in the two-norm was considered. The trigonometric function based on higher degree FzT and the accuracy of the resulting approximation increase with the increase in the degree of FzT were presented by [15], while the weighted transform method was discussed by [16]. The conditions for quasi-consensus in a multi-agent system with sampled data based on FzT were proposed by [17]. In [18], the dynamical properties of a two-neuron system with respect to FzT and a single delay have been investigated. Multi-step FzT was first studied by [19] for solving ODEs. From this perspective, FzT was considered for solving a class of ODEs. In this regard, many approximation methods have been studied for solving ODEs, for example using neural networks [20], embedded three and two pairs of nonlinear methods [21], electrical analogy [22], multi-general purpose graphical processing units [23], the differential transform method [24] and a Galerkin finite element method [25]. A numerical method based on the trapezoidal rule for the Cauchy-Smoluchowski problem was discussed by [26]. In [27; 28], the authors studied ODEs with the initial value as the triangular fuzzy number.