1. Introduction

This paper is an extension of work originally presented in [1]. In many physical and dynamical processes, mathematical modeling leads to the deterministic initial and boundary value problems. In practical the boundary values may be different from crisp and displays in the form of unknown parameters [2]. When the parameters or the states of the differential equations are uncertain, they can be modeled with FDE. In recent days, many methods have used FDE for modeling and control of uncertain nonlinear systems [3-5]. The basic idea of the fuzzy derivative was first introduced in [Chang]. Then it is extended in [6]. The first-order fuzzy initial value problem, as well as fuzzy partial differential equation, have been studied in [7]. By generalizing the differentiability, [6] gave an analytical solution. The Lipschitz condition, as well as the theorem for existence and uniqueness of the solution related to FDEs, are discussed in [10-12]. In [13], the analytical solutions of second order FDE are obtained. The analytical solutions of third order linear FDE are found in [14]. By the interval-valued method, [15] examined the basic solutions of nonlinear FDEs with generalized differentiability.

A novel technique in order to solve FDEs is laid down based on the Sumudu transform. Sumudu transform along with broad applications has been utilized in the area of system engineering and applied physics [16-18]. In [19], some simple and deeper fundamental theorems, as well as properties of the Sumudu Transform, were generalized. In [20], Sumudu transform is applied to the system of differential equations. In [21], Sumudu transform is used in order to find the solution of the fuzzy partial differential equation. In [22], Sumudu transform has been used to solve fractional differential equations.

In this paper, we use FST to approximate the Z-number solutions of the FDEs. The FST reduces the FDE to an algebraic equation. A very important property of the FST is that it can solve the equation without resorting to a new frequency domain. The procedure of switching FDEs to an algebraic equation is cited in [10] and is stated as an operational calculus. We extend our previous work [1] by generating more theorems for describing the properties of FST and displaying the uncertainties with Z-numbers. The Z-number is a new concept that is subjected to a higher potential to demonstrate the information of the human being as well as to utilize in information processing [23]. Z-numbers can be regarded as to answer questions and carry out the decisions [24]. There exist few structure based on the theoretical concept of Z-numbers [25]. [26] gave an inception, which results in the extension of the Z-numbers. [27] generated a theorem to convert the Z-numbers to the usual fuzzy sets.

In this paper, initially, some preliminary definitions along with properties related to FST are demonstrated. After that, solving FDEs by using the methodology of FST has been discussed. At the end, two examples along with comparisons are utilized in order to demonstrate the effectiveness of our proposed method.

2. Preliminaries

Prior to the introduction of the FST, some concepts related to the fuzzy variables and Z-numbers are laid down in this section [28, 29].

6. Conclusion

In this paper, a novel method based on the FST is proposed in order to find the solution of the first order FDEs on the basis of the Z-numbers. The new method is clarified by utilizing the concept of strongly generalized differentiability. By using the FST method, the FDE converts to an algebraic problem. Some essential theorems are laid down in order to demonstrate the properties of the FST. Two real examples are applied to demonstrate the effectiveness of the proposed technique. This work has a significant contribution in initializing a superior starting point for such extensions.

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