Artificial bee colony algorithm for solving linear fuzzy Fredholm integral equation of the second kind
M. Hasani,a B. Asadya[1] ,M. Alavia
aDepartment of mathematics, Science and Research branch, Islamic Azad University, Hamadan,
Iran.
Abstract
In this paper, we propose a new method to solve fuzzy Fredholm integral equations(FFIE) by using artificial bee colony algorithm. In this approach, ABC algorithm is employed as the main optimizer for optimal adjustments of error control variables of the(FFIE) solution. The ability of artificial bee colony algorithm in function approximation is our main objective. Also, we offering some numerical examples to illustrate capability and robustness, of the presented method .
Keywords: Fuzzy numbers, Fuzzy Fredholm integral equations, Artificial bee colony algorithm
1 Introduction
Since many mathematical formulations of physical phenomena contain fuzzy integral equations and so these equations are very useful for solving many problems in several applied fields like mathematical physics and engineering. Also, these equations usually cannot be solved analytic, so it is required to obtain the approximate solutions. Therefore, various methods to solving these problems have been proposed. In other hand, the topic of fuzzy integral equations and particularly fuzzy control, has been rapidly developed in recent years. Before discussing fuzzy integral equations, it is essential to present a suitable brief introduction to preliminary topics same as fuzzy numbers and fuzzy calculus by zadeh and others [6, 14, 32, 37].The basic arithmetic structure for fuzzy numbers was later developed by Mizumoto and Tznaka[32]. The concept of integration of fuzzy functions was introduced by Dubois and Prade[30] for the first time and alternative approaches were later suggested by Goetschel and Voxman[16], Kaleva[18],
Matloka[29] and others. More recently, many authors have been proposed numerical methods for solving FFIE, for example, S.Abbasbandy and coworker [4] proposed a method for solving linear FFIE. As, variational iteration method proposed by X. Lan [26] and introduced Adomian decomposition method by Abbasbandy [4, 7, 3]. Also, the Homotopy analysis method (HAM) was proposed by Liao [27, 28]. Addition, nonlinear Fuzzy Feredholm integral equation has been studied by many authors, for instance, see [2, 7, 8, 9, 10, 11, 12]. In this paper, the newly proposed heuristic optimization algorithm, the ABC method, is employed to solve the fuzzy Fredholm integral equations of the second kind(FFIEs). So that, artificial bee colony algorithm (ABC) is an algorithm based on the intelligent foraging behavior of honey bee swarm, proposed by Karaboga in 2005 [20, 22, 23, 24]. Then, it has many advantages than Genetic algorithm (GA) [35], Dierential Evolution (DE) [36], Firey algorithms (FA) [25] and Particle Swarm optimization (PSO) [13] in the solving numerical optimization problems ( for more explain see [1, 31, 33]). Therefore, we propose a method for solving fuzzy Fredholm integral equations of the second kind(FFIEs) by using ABC method. Addition, the method is illustrated by numerical examples and compared with other methods.
2 PRELIMINARIES
This section brifley deals with the foundation of fuzzy numbers and integral equations which are used in the next sections. We started by defining the
fuzzy number.
Definition 2.1: A fuzzy number is a fuzzy set u:R1→I=[0,1] which satisfies
1. u is upper semicontinuous,
2. u=0 outside some interval [a,b],
3. There are real numbers b,c:a≤b≤c≤d for which:
(3.a) u(x) is monotonically increasing on [a,b],
(3.b) u(x) is monotonically decreasing on[c,d],
(3.c) ux=1, b≤x≤c.
The set of all fuzzy numbers (as given by Definition 2.1) is denoted by E1 [14].
Definition 2.2: A fuzzy number u is a pair (
[1] Corresponding author, ,