Journal of Fuzzy Set Valued Analysis 2013 (2013) 1-9
Available online at www.ispacs.com/jfsva Volume 2013, Year 2013 Article ID jfsva-00146, 9 Pages doi:10.5899/2013/jfsva-00146 Research Article
Kernel iterative method for solving two-dimensional fuzzy Fredholm integral equations of the second kind Azim Rivaz1∗ , Farzaneh Yousefi 1 (1) Department of Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran. c Azim Rivaz and Farzaneh Yousefi. This is an open access article distributed under the Creative Commons Attribution License, Copyright 2013 ⃝ which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract Using parametric form of fuzzy functions, we convert a linear two-dimensional fuzzy Fredholm integral equation of the second kind (2D-FFIE-2) to a linear system of Fredholm integral equations of the second kind with three variables in crisp case. In this paper an iterative method is presented to find the approximate solution of 2D-FFIE-2. Also a proof of convergence of this method is discussed in detail. Finally, the proposed method is illustrated by two numerical example. Keywords: Two-dimensional fuzzy Fredholm integral equation; Parametric form of fuzzy integral equation; Kernel iterative method.
1
Introduction
The concept of integration of fuzzy functions was first introduced by Dubois and Prade. Alternative approaches were later suggested by Goetschel and Voxman [4], Kaleva [6], Matloka [7], Nanda [8] and others. One of the first applications of fuzzy integration was given by Wu and Ma (Wu et al. 1990), who investigated the fuzzy Fredholm integral equation of the second kind (FFIE-2). In recent years, some numerical methods have been introduced to solve (FFIE-2). These methods can be found in [5, 9, 13]. Also, some numerical methods are presented to solve two-dimensional fuzzy Fredholm integral equations of the second kind; for example, see in [10, 11]. In this work, we apply kernel iterative method for solving 2D-FFIE-2. This paper is organized as follows: in section 2, we briefly mention the basic notations of fuzzy number, fuzzy function and fuzzy integral. 2D-FFIE-2 and it’s parametric form are discussed in section 3. In section 4, the proposed algorithm is illustrated, also a proof of convergence of the method is discused in this section. We apply the method for two examples in section 5. 2 preliminaries Definition 2.1. [4] The parametric form of a fuzzy number u is a pair of functions (u(r), u(r)), 0 ≤ r ≤ 1, which satisfies in the following requirements: 1. u(r) is a bounded, continuous, monotonic increasing function over [0, 1]. 2. u(r) is a bounded, continuous, monotonic decreasing function over [0, 1]. 3. u(r) ≤ u(r), 0 ≤ r ≤ 1. (u(r), u(r)), 0 ≤ r ≤ 1, are called the r-cut sets of u. ∗ Corresponding
author. Email address:
[email protected], Tel:+98 3413202484
Journal of Fuzzy Set Valued Analysis http://www.ispacs.com/journals/jfsva/2013/jfsva-00146/
Page 2 of 9
The set of all fuzzy numbers is denoted by E 1 . Definition 2.2. [12] For arbitrary fuzzy numbers u = (u(r), u(r)), v = (v(r), v(r)) and real number k, we have (u + v)(r) = (u(r) + v(r), u(r) + v(r)), (u − v)(r) = (u(r) − v(r), u(r) − v(r)), { (ku(r), ku(r)) k ≥ 0, (ku)(r) = (ku(r), ku(r)) k < 0. Lemma 2.1. [1] Let (u(r), u(r)), 0 ≤ r ≤ 1, be a given family of non-empty intervals. If 1) (u(r1 ), u(r1 )) ⊇ (u(r2 ), u(r2 )) for 0 ≤ r1 ≤ r2 ≤ 1, 2) (limk→∞ u(rk ), limk→∞ u(rk )) = (u(r), u(r)), whenever (rk ) is a non-decreasing converging sequence converges to r ∈ [0, 1], then the family (u(r), u(r)), 0 ≤ r ≤ 1, represent the r-cut sets of a fuzzy number u ∈ E 1 . Conversely, if (u(r), u(r)), 0 ≤ r ≤ 1, are the r-cut sets of a fuzzy number u ∈ E 1 , then the conditions (1) and (2) hold. Definition 2.3. [4] For arbitrary fuzzy numbers u = (u, u) and v = (v, v) the quantity { } sup |u(r) − v(r)| , sup |u(r) − v(r)| D(u, v) = max , 0≤r≤1 0≤r≤1 is called the distance between u and v. It is shown that (E 1 , D) is a complete metric space [7]. Definition 2.4. [2] A function f : R2 → E 1 is called a fuzzy function in two-dimensional space. f is said to be continuous, if for arbitrary fixed t0 ∈ R2 and ε > 0 a δ > 0 exists such that ∥t − t0 ∥ < δ ⇒ D ( f (t), f (t0 )) < ε ,
t = (x, y), t0 = (x0 , y0 ).
Definition 2.5. [2] Let f : [a, b] × [c, d] → E 1 . For each partition p = {x1 , x2 , . . . , xm } of [a, b] and q = {y1 , y2 , . . . , yn } of [c, d] and for arbitrary ξi : xi−1 ≤ ξi ≤ xi , 2 ≤ i ≤ m, and for arbitrary η j : y j−1 ≤ η j ≤ y j , 2 ≤ j ≤ n, let m
n
R p = ∑ ∑ f (ξi , η j )(xi − xi−1 )(y j − y j−1 ), i=2 j=2
the definite integral of f (x, y) over [a, b] × [c, d] is ∫ d∫ b c
a
f (x, y)dxdy = lim R p ,
(max |xi − xi−1 |, max y j − y j−1 ) → (0, 0), 2≤i≤m
2≤ j≤n
provided that this limit exists in metric D. If the function f (x, y) is continuous in the metric D, it’s definite integral exists. Furthermore, (∫ ∫ ) ∫ ∫ d b = cd ab f (x, y, r)dxdy, f (x, y, r)dxdy c a (∫ ∫ ) ∫ ∫ d b f (x, y, r)dxdy = cd ab f (x, y, r)dxdy. c a
International Scientific Publications and Consulting Services
Journal of Fuzzy Set Valued Analysis http://www.ispacs.com/journals/jfsva/2013/jfsva-00146/
Page 3 of 9
Definition 2.6. The linear fuzzy Fredholm integral equation of the second kind (FFIE-2) is u(x) = f (x) + λ
∫ b a
k(x,t)u(t)dt, x ∈ D,
(2.1)
where u(x) and f (x) are fuzzy functions on D = [a, b] and k(x,t) is an arbitrary kernel function over T = [a, b] × [a, b], and u is unknown on D. Theorem 2.1. [3] Let k(x,t) be a continuous function over T and f (x) be a fuzzy continuous function on D. If 1 , M(b − a)
|λ |
