Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 273830, 9 pages

http://dx.doi.org/10.1155/2015/273830

## Approximate Method for Solving the Linear Fuzzy Delay Differential Equations

Department of Applied Mathematics, Bharathiar University, Coimbatore 641 046, India

Received 7 September 2015; Accepted 8 October 2015

Academic Editor: Carlo Bianca

Copyright © 2015 S. Narayanamoorthy and T. L. Yookesh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose an algorithm of the approximate method to solve linear fuzzy delay differential equations using Adomian decomposition method. The detailed algorithm of the approach is provided. The approximate solution is compared with the exact solution to confirm the validity and efficiency of the method to handle linear fuzzy delay differential equation. To show this proper features of this proposed method, numerical example is illustrated.

#### 1. Introduction

Fuzzy set theory is a powerful tool for modeling uncertainty and for processing vague or subjective information in mathematical models, which has been applied to a wide variety of real problems. Thus the theory of fuzzy differential equation has attracted widespread attention and has been rapidly growing. It is massively studied by many researchers [1, 2]. The theory of fuzzy differential equations has undergone a rapid evolution in the last three decades because the idea of fuzzy set theory is simple and natural. The concept of fuzzy was first introduced by Zadeh [3]. Chang and Zadeh [4] had introduced many concepts in fuzzy derivatives. Fuzzy delay differential equations have a wide range of applications in real time applications of control theory, physics, ecology, economics, population study, inventory control, and the theory of nuclear reactors. It is difficult to obtain exact solutions of fuzzy differential equations and hence several numerical methods were proposed [5–8]. Abbasbandy and Allahviranloo [9] developed numerical algorithms for solving fuzzy differential equations based on Seikkala’s derivative [10]. Jafari et al. [11] worked on the solving th order fuzzy differential equations by the variational iteration method. Allahviranloo et al. [12] used the predictor-corrector method to find the numerical solution of fuzzy differential equations. Delay differential equations are the type of differential equations in which the derivative of the unknown function at a certain time is given in terms of the values of the function at a previous time.

Delay differential equations and ordinary differential equations are used to describe physical phenomena, but they are different. While in ordinary differential equations the derivatives of unknown functions are dependent only on the current value of the independent variable, in delay differential equations the derivatives of unknown functions are dependent on the values of the functions at previous time. This implies that the solution of delay differential equations requires the knowledge of the current state and also the certain previous time state. For example, the present state of change of unknown functions depends upon the past values of the same functions in both physical and biological systems.

Linear differential equations with retarded argument concept proposed by Myškis (1951) [13] and later many researchers continued to work on these concepts [14, 15]. Driver [15] wrote a book about ordinary differential equations and delay differential equations which explains the equations clearly. Bellen and Zennaro [16] presented the numerical solutions to delay differential equations. Khastan et al. [17] discussed the concepts of fuzzy delay differential equations under the generalized differentiability. Barzinji et al. [18] studied the linear fuzzy delay differential systems to analyze the stability of steady state. For solving the delay differential equations Ibrahim et al. [19] used the step spline method. Evans and Raslan [20] solved delay differential equation by the domain decomposition method. Shakeri and Dehghan [21] found the solution to delay differential equation by the homotopy perturbation method. Mirzaee and Latifi [22] used the differential transform method to the delay differential equation. For the delay differential equation Sadehgi Hafshejani et al. [23] discussed the Legendre Wavelet method. Numerical and theoretical treatment for both linear and nonlinear delay differential equations using the variational iteration method was proposed by Khader [24]. Guerrini worked on delay differential AK model with nonpositive population growth rate [25] and the time delay effects on the qualitative behavior of an economic growth model [26]. Akgül and Kiliçman [27] solved delay differential equations by an accurate method with interpolation. Abdul Aziz et al. [28] derived delay differential equations of small and vanishing lag using the multistep block method. The general form of the delay differential equation is given as follows:where is constant for each .

Let us consider the fuzzy delay differential equation as follows: where is constant for each .

The Adomian decomposition method is named after Adomian [29]. The Adomian decomposition method is well suited to solve Cauchy problems. The advantage of the method is that it can be applied directly for all types of differential and integral equations, linear or nonlinear, homogeneous or inhomogeneous, and with constant coefficients or with variable coefficients. Another important advantage is that the method is capable of greatly reducing the size of computational work while still maintaining high accuracy of the numerical solution. This method is based on the decomposition of a solution of a nonlinear operator equation in a series of functions. Each of the series is obtained from an expansion of an analytic function into an abstract formulation, but the difficulty arises in calculating the polynomial and in proving the convergence in the series of functions. The Adomian decomposition method is used to solve a wide range of physical problems in various engineering fields such as wave and heat and mass transfer equations. Thus, many researchers have applied and solved their problems by using Adomian decomposition method.

Motivated by the work of Evans and Raslan [20], in this paper, the author proposes an approximate method to solve the linear fuzzy delay differential equations using Adomian decomposition method. The purpose of this paper is to find how this technique works on delay differential equations under fuzzy environment. The Adomian decomposition method converges rapidly to form the exact solution. This method does not involve any discretization of variables and hence it is free from rounding off errors. We illustrate a numerical example to explain the proposed method.

#### 2. Preliminaries

In this section, we present the most basic notations and definitions, which are used throughout this work. We start with defining a membership function.

*Definition 1 (see [30]). *Let be a nonempty set. A fuzzy set is characterized by its membership function and is interpreted as the degree of membership of element in fuzzy set for each . It is clear that is determined by the set of tuples

*Definition 2. *A fuzzy number is a map which satisfies the following three conditions: (i)is upper semicontinuous.(ii) outside some interval .(iii)There exist real numbers such that ,where (1) is monotonic increasing on ,(2) is monotonic decreasing on ,(3) The set of all such fuzzy numbers is represented by .

*Definition 3. *Given a fuzzy set defined on and a number , the -cut, , and the strong -cut, , are the crisp sets Unlike in the conventional set theory, convexity of fuzzy sets refers to properties of the membership function rather than to the support of a fuzzy set.

*Definition 4. *An arbitrary fuzzy number is parametric form represented by an ordered pair of functions , which satisfy the following requirements: (1) is a bounded left continuous nondecreasing function over ,(2) is a bounded left continuous nonincreasing function over ,(3). For arbitrary , and scalar , we define addition , subtracting by as(1)Addition:(2)Subtraction:(3)Scalar multiplication:

*Definition 5. *A fuzzy function is a fuzzy set on with membership function Its inverse is a fuzzy set on with

*Definition 6. *Suppose that . For each partition of and for arbitrary points , let where Then the definite integral of over is defined by provide the limits exist.

*Definition 7. *A fuzzy set is the triangular fuzzy number with peak (or center) , left width and right (Figure 1) if its membership function has the following form: