# New Approximation Methods Based on Fuzzy Transform for Solving SODEs: II
## 1 Introduction  
Differential equation is particularly useful for different areas of applied sciences and engineering. Many differential equations have no closed form solutions. Thus, many researchers are developing approximation methods for solving differential equations, for example [1; 2; 3]. In this paper, we continue the study of approximation methods based on FzT to solutions of differential equations. The core idea of FzT is a fuzzy partition of a universe into fuzzy subsets. The first fuzzy partition of FzT with the Ruspini condition was introduced by [4] and was extensively investigated by [5]. This condition implies normality of the fuzzy partition. In addition, the fuzzy partition with the generalized Ruspini condition (fuzzy \(r\)-partition) was introduced by [6]. This fuzzy partition was achieved by replacing the partition of unity by fuzzy \(r\)-partition. This type of partition was used by [6; 7] for smoothing or filtering data based on the inverse FzT. Further, a generalized fuzzy partition appeared in connection with the notion of the FzT, where FzT components are polynomials of degree \(m\)[8]. By [9], different types of fuzzy partitions are taken into consideration such as B-splines, Shepard kernels, Bernstein basis polynomials and Favard-Szasz-Mirakjan operators. Later, the higher degree FzT based on B-splines was proposed [10] to improve the quality of the function approximation of two variables. A generalized fuzzy partition was implicitly introduced by [11] with the purpose of meeting the requirements of image compression. In addition, a generalized fuzzy partition can also be considered in connection with radial membership functions [12]. Further, necessary and sufficient conditionsfor modeling the generalized fuzzy partition was provided by [13]. Recently, a new representation formula for basic functions of FzT and a new fuzzy numerical method based on block pulse functions for numerical solution of integral equations were presented by [14]. The approximation method based on the FzT with Shepard-type basic functions for linear Fredholm integral equations was discussed by [15]. New representations of the generalized uniform fuzzy partitions with the normal case to obtain better approximation solutions for solving Cauchy problems were presented by [16]. FzT is a soft computing method developed by [PERSON] [5] that has many applications, for example, in differential and integral equations. FzT for solving ordinary Cauchy problems with one variable was initiated by [4]. The generalization of the Euler method has been discussed by [17] for solving ordinary Cauchy problems. The author has applied this technique to reef growth and sea level variations models. Further, FzT has been generalized from the case of constant components to the case of polynomial components by [8]. Later, the first and second degree FzT based mid-point rule for solving the Cauchy problem and the uncertain initial value problem have been proposed by [18]. Furthermore, an algorithm to obtain the approximate solutions of second order initial value problems was constructed by [19]. From this idea, FzT for numerical solutions of two point boundary value problems was proposed by [20]. FzT of two variables based on finite differences method was used by [21] for solving a type of partial differential equations with Dirichlet boundary conditions and initial conditions. In addition, the first degree FzT of two variables was introduced by [22]. By [23], the partial derivatives using the first FzT were approximated and modification of the Canny edge detector was proposed. Furthermore, the uniform stability result for the vibrations of a telegraph equation using FzT of two variables was proposed by [24]. The composition of inverse and direct discrete FzT method was extended to numerical solution of Fredholm integral equations and Volterra Fredholm integral equations [25]. The general form of the higher order FzT was constructed by [26] for solving differential and integral equations using any arbitrary basis functions.