Ill-posedness for fuzzy Fredholm integral equations of the first kind and regularization methods

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Fuzzy Sets and Systems ••• (••••) •••–•••

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Ill-posedness for fuzzy Fredholm integral equations of the first kind and regularization methods ✩

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Hong Yang, Zengtai Gong ∗

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College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, PR China

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Received 24 November 2016; received in revised form 18 May 2018; accepted 20 May 2018

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Abstract In this study, the concept of ill-posedness for fuzzy Fredholm integral equations of the first kind was first proposed. An efficient regularization method based on classic Tikhonov regularization scheme to determine the approximate solution of the fuzzy Fredholm integral equations of the first kind is presented. To support our investigation, some examples are illustrated. © 2018 Elsevier B.V. All rights reserved.

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Keywords: Fuzzy numbers; Ill-posed problem; Fuzzy integral equation; Regularization method

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1. Introduction

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Fuzzy integral equations (FIEs) are important for studying and solving a large proportion of the problems in applied mathematics, particularly in relation to fuzzy control. The present paper proposed ill-posedness of fuzzy Fredholm integral equations of the first kind (FFIEs-1) and a regularization method for it. The FFIEs-1 (3.1) are characterized by the occurrence of the unknown function Y (s) only inside the integral sign. Because the solution of FFIEs-1 is ill-posed [25], the numerical solution researching for FFIEs-1 is more difficult than fuzzy Fredholm integral equations of the second kind (FFIEs-2) and hardly shows in literature. However, the regularization method that transforms the FFIEs-1 to the approximation FFIEs-2 is effective. There are lots of literatures are applied to handle the FFIEs-2 by the classic methods. The study of FIEs began with the investigations of Kaleva [26] and Siekkala [47] for the fuzzy Volterra integral equations which were equivalent to the initial value problem for the first order fuzzy differential equations. These concepts were developed by many researchers [12,22,35,51,52]. The Banach fixed point theorem is the main tool in studying existence and uniqueness of the solution for FFIEs-2, which is carried out in [7,8,21,33,38,39,37]. The numerical methods for solving FFIEs-2 based on the method of successive approximations and other iterative techniques are applied in [12,10,18,19,21,20, 40]. Bica proved the convergence method of successive approximations used to approximate the solution of nonlinear

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✩

This work is supported by National Natural Science Foundation of China (11461062, 61763044). * Corresponding author. E-mail addresses: [email protected] (H. Yang), [email protected], [email protected] (Z. Gong).

https://doi.org/10.1016/j.fss.2018.05.010 0165-0114/© 2018 Elsevier B.V. All rights reserved.

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Hammerstein fuzzy integral equations in [11] and the authors obtained the new error estimation of the iterative to solve nonlinear fuzzy Hammerstein–Fredholm integral equations In [53]. In [16], an iterative procedure based on the trapezoidal quadrature for solving nonlinear FFIEs-2 is proposed. Furthermore, in [6] it has presented based on the hybrid of block-pulse functions and Taylor series recently. Some noticeable methods used in the construction of the numerical techniques for FIEs are: quadrature rules and Nystorm methods [1,27], Lagrange interpolation [3], divided and finite differences [41], Bernstein polynomials [17,34], Chebyshev interpolation [9], Legendre wavelets [46], fuzzy Haar wavelets [54], Galerkin type techniques [29] and using block pulse functions [44]. Analytic-numeric methods like Adomian decomposition, homotopy analysis and homotopy perturbation are used in [1,4,5,28,32]. In [30], a numerical method to solve linear fuzzy Fredholm integral equations of the second kind by using TFs is proposed. In [45] Iterative method for numerical solution of two-dimensional nonlinear FFIEs-2 is examined. In 1923, Hadamard [25] introduced the concept of a well-posed problem, originating from the philosophy that the mathematical model of a physical problem has to have the properties of uniqueness, existence, and stability of the solution. If one of the properties fails to hold, he called the problem ill-posed. Due to the ill-posedness of the problem, numerical computation is difficult. The regularization method to obtain the numerical solution for the ill-posed problem was established independently by Tikhonov [42,43] in 1963 and Phillips [36] in 1962. In [49], the regularization method for Fredholm integral equations of the first kind is examined. In [24], a numerical method to solve initial-boundary value problems for ill-posed fuzzy partial differential equations by using regularization scheme is proposed. In [31], a hybrid of regularization method and Bernstein polynomials is used to solve the first kind fuzzy integral equations and the authors focused on the Abel integral equation that belong to Volterra type integral equation of the first kind. In this work, we will apply the regularization method that transforms a fuzzy Fredholm integral equation of the first kind to a fuzzy Fredholm integral equation of the second kind. And by converting the first kind to a second kind, then we can apply the existing techniques of the second kind to the transformed equation. The present study addresses two topics. One of our intentions is devoted to the ill-posedness for FFIEs. The other aim is using the regularization method to recover the approximation solution. The remainder of this paper is organized as follows. In Section 2, we briefly introduce the necessary notions related to fuzzy numbers. In Section 3, we define ill-posedness of FFIEs-1. The regularization method and stabilized approximation solutions of FFIEs-1 are considered in Section 4. In Section 5, we present some examples which are used to highlight the reliability of the regularization method and our conclusions are given in Section 6.

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a∈A b∈B

b∈B a∈A

where || · || denotes the usual Euclidean norm in R n [14]. Then (Pk (R n ); dH ) is a metric space.

E = {u : R → [0, 1]|u satisfies (1)–(4) below} n

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as a fuzzy number space, where

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(1) (2) (3) (4)

u is normal, i.e. there exists an x0 ∈ R n such that u(x0 ) = 1, u is fuzzy convex, i.e. u(λx + (1 − λ)y) ≥ min{u(x), u(y)} for any x, y ∈ R n and 0 ≤ λ ≤ 1, u is upper semi-continuous, [u]0 = cl{x ∈ R n |u(x) > 0} is compact.

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Definition 2.1. Denote

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Let Pk (R n ) denote the family of all nonempty compact convex subset of R n and define the addition and scalar multiplication in Pk (R n ) as usual. Let A and B be two nonempty bounded subset of R n . The distance between A and B is defined by the Hausdorff metric   dH (A, B) = max sup inf ||a − b||, sup inf ||b − a|| , (2.1)

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2. Preliminaries

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Here, cl(X) denotes the closure of set X. For 0 < α ≤ 1, the α-level set of u (or simply the α-cut) is defined by [u]α = {x ∈ R n |u(x) ≥ α}. The core of u is the set of elements of R n having membership grade 1, i.e., [u]1 = {x|x ∈ R n , u(x) = 1}. Then from above (1)–(4), it follows that the α-level set [u]α ∈ Pk (R n ) for all 0 < α ≤ 1.

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The distance between two fuzzy numbers u and v is defined by

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D(u, v) = sup dH ([u]α , [v]α ).

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Definition 2.2. According to Zadeh’s extension principle, we have addition and scalar multiplication in fuzzy number space E 1 as follows:

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[u ⊕ v]α = [u]α + [v]α = {x + y|x ∈ [u]α , y ∈ [v]α },

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[k  u]α = k[u]α = {kx|x ∈ [u]α }, [0]α = {0},

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where u, v ∈ E 1 and 0 < α ≤ 1. [u]α + [v]α means the usual addition of two intervals (as subset of R) and k[u]α means the usual product between a scalar and a subset of R. Also, according to[12,51], the following algebraic properties for any u, v, w ∈ E 1 hold: (1) (2) (3) (4) (5) (6)

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u ⊕ (v ⊕ w) = (u ⊕ v) ⊕ w, u ⊕ 0 = 0 ⊕ u, 0 has opposite in (E 1 , ⊕), with respect to  0, none u ∈ (E 1 − R), u =  (a + b)  u = a  u + b  u, ∀a, b ∈ R with ab ≥ 0, a  (u ⊕ v) = a  u ⊕ a  v, ∀a ∈ R, a  (b  u) = (ab)  u, ∀a, b ∈ R and 1  u = u.

Definition 2.3. (See [51]) For arbitrary fuzzy numbers u, v ∈ E 1 , u = [u(α), u(α)], v = [v(α), v(α)], the quantity D(u, v) = supα∈[0,1] max{|u(α) − v(α)|, |u(α) − v(α)|} is the distance between u and v and also the following properties hold:

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(2.2)

(1) (2) (3) (4) (5) (6)

(E 1 , D) is a complete metric space, D(u ⊕ w, v ⊕ w) = D(u, v), ∀u, v, w ∈ E 1 , D(u ⊕ v, w ⊕ e) ≤ D(u, w) + D(v, e), ∀u, v, w, e ∈ E 1 , D(u ⊕ v, 0) ≤ D(u, 0) + D(v, 0), ∀u, v, ∈ E 1 , D(k  u, k  v) = |k|D(u, v), ∀u, v, ∈ E 1 , k ∈ R, 0), ∀u ∈ E 1 , k1 , k2 ∈ R with k1 · k2 ≥ 0. D(k1  u, k2  u) = |k1 − k2 |D(u,

Let us recall the definition of the Hukuhara difference (H-difference) [13]. Let u, v ∈ E 1 . The Hukuhara Hdifference has been introduced as a set w for which u H v = w ⇐⇒ u = v ⊕ w. The H-difference is unique, but it does not always exist (a necessary condition for u H v to exist is that u contains a translate {c} ⊕ v of v). A generalization of the Hukuhara difference aims to overcome this situation.

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Definition 2.4. (B. Bede and L. Stefanini [13]) The generalized Hukuhara difference between two fuzzy numbers u, v ∈ E 1 is defined as follows  (i) u = v ⊕ w; u gH v = w ⇐⇒ (2.3) or (ii) v = u ⊕ (−w). In terms of the α-levels, we have [u gH v]α = [min{u(α) − v(α), u(α) − v(α)}, max{u(α) − v(α), u(α) − v(α)}] and if the H-difference exists, then u  v = u gH v; the conditions for the existence of w = u gH v ∈ E 1 are  w(α) = u(α) − v(α) and w(α) = u(α) − v(α), ∀α ∈ [0, 1], (2.4) case (i) with w(α) increasing, w(α) decreasing, w(α) ≤ w(α).  w(α) = u(α) − v(α) and w(α) = u(α) − v(α), ∀α ∈ [0, 1], (2.5) case (ii) with w(α) increasing, w(α) decreasing, w(α) ≤ w(α).

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It is easy to show that (i) and (ii) are both valid if and only if w is a crisp number. In the fuzzy case, it is possible that the gH-difference of two fuzzy numbers does not exist. To address this shortcoming, a new difference between fuzzy numbers was proposed in [13].

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Definition 2.5. (See [13]) The generalized difference (g-difference) of two fuzzy numbers u, v ∈ level sets as  [u g v]α = cl ([u]β gH [v]α ), ∀α ∈ [0, 1],

En

is given by its

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β≥α

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where the gH-difference gH is with interval operands [u]β and [v]α .

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Theorem 2.1 (Decomposition theorem [26]). If u ∈ E n , then  u= (α · [u]α ).

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(2.7)

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The following well-known characterization theorem makes the connection between a fuzzy interval and its LUrepresentation.

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∈ E1

Definition 2.9. (See [22]) Let f f is Fuzzy–Riemann integrable to I if for any ε > 0, there exists δ > 0 such that for any division P = {[u, v]; ξ } of [a, b] with the norms (P ) < δ, we have ∗  D( (v − u)  f (ξ ); I ) < ε, P

where

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denotes the fuzzy summation. We choose to write

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b I := (F R)

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f (x)dx. a

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: [a, b] → E 1 . We say that

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Definition 2.8. (See [50]) If f, g : [a, b] → E 1 are fuzzy continuous functions, then the function F : [a, b] → R+ defined by F (x) = D(f (x), g(x)) is continuous on [a, b]. Also D(f (x),  0) ≤ M, ∀x ∈ [a, b], M > 0, that is f is fuzzy bounded.

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Definition 2.7. (See [2]) A fuzzy-number-valued function f : [a, b] → E 1 is said to bounded iff there is M > 0 such that D(f (t), 0) = f (u) ≤ M for all t ∈ [a, b]. Equivalently we get χ−M ≤ f (x) ≤ χM , ∀x ∈ [a, b].

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Definition 2.6. (See [2]) A fuzzy-number-valued function f : [a, b] → E 1 is said to be continuous at t0 ∈ [a, b] if for each ε > 0 there is δ > 0 such that D(f (t), f (t0 )) < ε whenever |t − t0 | < δ. If f is continuous for each t ∈ [a, b] then we say that f is fuzzy continuous on [a, b].

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Reciprocally, given two functions that satisfy the above conditions, they uniquely determine a fuzzy number.

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(i) u is a bounded, non-decreasing, left-continuous function in (0, 1] and it is right-continuous at 0. (ii) u is a bounded, non-increasing, left-continuous function in (0, 1] and it is right-continuous at 0. (iii) u(1) ≤ u(1).

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Theorem 2.2. (See [22]) Let u ∈ E 1 be a fuzzy number. Then the functions u, u: [0, 1] → R, defining the endpoints of the α-level sets of u, satisfy the following conditions:

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We also call an f as above (F R)-integrable.

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Theorem 2.3. (See [22]) If f, g : [a, b] → E 1 are (F R)-integrable fuzzy functions, and α, β are real numbers, then

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b α  f (x) ⊕ β  g(x)dx = α  (F R)

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b f (x)dx ⊕ β  (F R)

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g(x)dx.

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Theorem 2.4. (See [23]) Let f : [a, b] → Then f is (F R)-integrable if and only if f α and f integrable for any α ∈ [0, 1]. Furthermore, for any α ∈ [0, 1], E1.

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[(F R)

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b f (x)dx]α = [(R)

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are Riemann

[f (x)]α dx, (R) a

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[f (x)] dx].

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In what follows, we assume that the necessary conditions to guarantee solutions of these equations hold. Such as, the data function F (t) must contain components which are matched by the corresponding t components of the arbitrary kernel K(s, t), F (t) is fuzzy continuous on [c, d], and K(s, t) is a continuous bounded function on [a, b] × [c, d] and so on.

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F (t) = λ  (F R)

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k(s, t)  Y (s)ds,

t ∈ [c, d],

(3.1)

where a and b are constants, λ is a parameter, F (t) is the known fuzzy-number-valued function, k(s, t) is the kernel of the integral equation, and Y (s) is the unknown fuzzy-number-valued function that will be determined. Equation (3.1) is called the fuzzy Fredholm integral equations of the first kind characterized by the occurrence of the unknown function Y (s) only inside the integral sign. The existence of Y (s) inside the integral sign causes special difficulties.

b For any α ∈ [0, 1], the α-cut sets of F (t), Y (s) and (F R) a k(s, t)  Y (s)ds are shown as follows respectively: (3.2)

[Y (s)]α = [Y (s, α), Y (s, α)],

(3.3)

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b k(s, t)  Y (s)ds]α = [(R)

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[F (t)]α = [F (t, α), F (t, α)],

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b [k(s, t)  Y (s)]α ds, (R)

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[k(s, t)  Y (s)]α ds],

(3.4)

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Now, we consider a fuzzy Fredholm integral equation of the first kind as follows.

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3. Ill-posedness of FFIEs-1

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where

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[k(s, t)  Y (s)]α = min{k(s, t)  Y (s, α), k(s, t)  Y (s, α)},

(3.5)

[k(s, t)  Y (s)]α = max{k(s, t)  Y (s, α), k(s, t)  Y (s, α)}.

(3.6)

The following definition, we discuss the ill-posedness of FFIEs-1. The classical Fredholm integral equations of the first kind are often ill-posed problems, that may have no solution, or if a solution exists it is not unique and may not depend continuously on the data F (t). The Fredholm integral equations of the first kind appear in many physical models such as radiography, spectroscopy, cosmic radiation, image processing and in the theory of signal processing. For any α ∈ [0, 1], in order to investigate equation (3.1), we consider the parametric form of FFIE-1 (3.1) is as follows.

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F (t, α) =

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F (t, α) =

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⎧ b ⎪ ⎪ ⎪ ⎪ ⎪ λ(R) v1 (s, t, Y (s, α), Y (s, α))ds, t ∈ [c, d], λ > 0, ⎪ ⎪ ⎨ a

⎪ ⎪ ⎪ ⎪ ⎪ λ(R) ⎪ ⎪ ⎩

(3.7)

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⎧ b ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ(R) v2 (s, t, Y (s, α), Y (s, α))ds, t ∈ [c, d], λ > 0, ⎪ ⎨

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⎪ ⎪ ⎪ ⎪ ⎪ λ(R) ⎪ ⎪ ⎩

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(3.8)

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Where

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v1 (s, t, Y (s, α), Y (s, α)) = v2 (s, t, Y (s, α), Y (s, α)) =

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k(s, t) · Y (s, α), t ∈ [c, d], k(s, t) ≥ 0,

k(s, t) · Y (s, α), t ∈ [c, d], k(s, t) < 0.  k(s, t) · Y (s, α), t ∈ [c, d], k(s, t) ≥ 0, k(s, t) · Y (s, α), t ∈ [c, d], k(s, t) < 0.

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(3.9)

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(3.10)

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v1 (s, t, Y (s, α), Y (s, α))ds, t ∈ [c, d], λ < 0.

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v2 (s, t, Y (s, α), Y (s, α))ds, t ∈ [c, d], λ < 0.

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Definition 3.1. (Hadamard’s definition of well-posedness [25]) If a mathematical model for a physical problem (can be considered as equations or optimization) has to be well-posed in the sense that it has the following properties (3.11)–(3.13):

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1. There exists a solution of the problem (existence).

(3.11)

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2. There is at most one solution of the problem (uniqueness).

(3.12)

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3. The solution depends continuously on the data (stability).

(3.13)

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Conversely, problems for which (at least) one of these properties does not hold are called ill-posed.

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Theorem 3.1. If equation (3.7) or equation (3.8) has no solution, then equation (3.1) has no solution.

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Proof. If equation (3.7) or equation (3.8) has no solution, for any α ∈ [0, 1], from Theorem 2.2 we obtain there is no F (t, α) or F (t, α) to F (t), then equation (3.1) has no solution. 2

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Proof. Assume λ > 0, k(s, t) > 0, from equation (3.7) and equation (3.8), we obtain the following two equation, respectively.

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b

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Theorem 3.2. If the solution of equation (3.7) or equation (3.8) exists and it is not unique, then a solution of equation (3.1) exists and it is not unique.

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F (t, α) = λ(R)

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k(s, t) · Y (s, α)ds,

(3.14)

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b

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F (t, α) = λ(R)

k(s, t) · Y (s, α)ds.

(3.15)

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Assume Y1 (s, α) and Y2 (s, α) is respectively the solutions of equation (3.14) that correspond to the data F (t, α), and Y1 (s, α) = Y2 (s, α), we have the following two equations.

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F (t, α) = λ(R)

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k(s, t) · Y1 (s, α)ds,

(3.16)

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F (t, α) = λ(R)

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k(s, t) · Y2 (s, α)ds,

(3.18)

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(3.19)

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Where for any α ∈ [0, 1], from the properties of Riemann Integration there exists Y1 (s, α) = Y2 (s, α), such that equation (3.19) hold, and we have

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We will give the following an example to explain Theorem 3.4.

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Theorem 3.3. (See [15,48]) For any α ∈ [0, 1], a solution of equation (3.7) and equation (3.8) not depend continuously on the data F (t, α) and F (t, α), respectively.

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Theorem 3.4. A solution of equation (3.1) not depend continuously on the data F (t).

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Proof. Assume Z : [a, d] → E 1 be a fuzzy-number-valued function. Let Y1 (s) and Y2 (s) = Y1 (s) ⊕ N sin(ωs)  Z(s) (where N is a nonzero constant) be the solutions of equation (3.1) that correspond to the data F1 (t) and F2 (t), respectively, then we have the following two equations

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Remark 3.1. If λ > 0, k(s, t) > 0 or λ < 0, k(s, t) > 0 or λ < 0, k(s, t) < 0, then the methods of proof for Theorem 3.2 are similar to the situation of λ > 0, k(s, t) > 0.

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From equation (3.2), equation (3.3) and equation (3.20), we obtain Y1 (s) and Y2 (s) are two different solutions of the equation (3.1). Hence, the solution of equation (3.7) or equation (3.8) exists and it may not be unique, then a solution of equation (3.1) exists and it may not be unique. 2

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17

19

30 31

16

(3.20)

[Y1 (s)]α = [Y1 (s, α), Y (s, α)], [Y2 (s)]α = [Y2 (s, α), Y (s, α)].

28 29

14 15

a

25 26

11 12

k(s, t) · [Y1 (s, α) − Y2 (s, α)]ds = 0.

(R)

6

8

b k(s, t) · Y1 (s, α)ds = λ(R)

λ(R)

5

7

From the equation (3.16) and equation (3.17), we have

9

11

(3.17)

a

7 8

k(s, t) · Y2 (s, α)ds.

37 38 39

b F1 (t) = λ  (F R)

36

k(s, t)  Y1 (s)ds,

40

t ∈ [c, d].

(3.21)

41

42

a

42

43

b

43

44 45 46 47

F2 (t) = λ  (F R)

k(s, t)  Y2 (s)ds,

50 51 52

(3.22)

44 45

a

46

From equation (3.21), (3.22) and Theorem 2.3, we have

48 49

t ∈ [c, d].

47 48

b F2 (t) = F1 (t) ⊕ λ  (F R)

k(s, t) · N sin(ωs)  Z(s)ds, a

From equation (3.23) and Definition 2.4, we obtain

t ∈ [c, d].

(3.23)

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8

b

1 2

F2 (t) gH F1 (t) = λ  (F R)

3 4 5 6 7 8 9 10 11 12

(F2 gH F1 )(t, α) =

18 19

⎧ b ⎪ ⎪ ⎪ ⎪ ⎪ λ(R) w2 (s, t, N, sin(ωs), Z(s, α), Z(s, α))ds, t ∈ [c, d], λ > 0, ⎪ ⎪ ⎨ a

⎪ b ⎪ ⎪ ⎪ ⎪ λ(R) w1 (s, t, N, sin(ωs), Z(s, α), Z(s, α))ds, t ∈ [c, d], λ < 0. ⎪ ⎪ ⎩

22 23 24 25 26 27 28 29 30 31 32

35

w1 (s, t, N, sin(ωs), Z(s, α), Z(s, α))  k(s, t) · N sin(ωs) · Z(s, α), t ∈ [c, d], (k(s, t) · N · sin(ωs)) ≥ 0, = k(s, t) · N sin(ωs) · Z(s, α), t ∈ [c, d], (k(s, t) · N · sin(ωs)) < 0. w2 (s, t, N, sin(ωs), Z(s, α), Z(s, α))  k(s, t) · N sin(ωs) · Z(s, α), t ∈ [c, d], (k(s, t) · N · sin(ωs)) ≥ 0, = k(s, t) · N sin(ωs) · Z(s, α), t ∈ [c, d], (k(s, t) · N · sin(ωs)) < 0.

(F2 gH F1 )(t, α) = λ(R)

(F2 gH F1 )(t, α) = λ(R)

39

45 46 47

50 51 52

10 11 12

16 17

19 20

22 23

(3.27)

26 27

(3.28)

28 29 30

(3.29)

31 32

34 35 36 37

k(s, t) · N · sin(ωs) · Z(s, α)ds.

(3.30)

38 39 40

From equation (3.29), equation (3.30), we obtain

41

(F2 gH F1 )(t, α) − (F2 gH F1 )(t, α)

42 43

b k(s, t) · N · sin(ωs) · [Z(s, α) − Z(s, α)]ds.

= λ(R)

24 25

(3.31)

44 45

a

46

From equation Y2 (s) = Y1 (s) ⊕ N sin(ωs)  Z(s), we obtain

47

48 49

9

18

a

43 44

8

33

k(s, t) · N · sin(ωs) · Z(s, α)ds, b

42

(3.26)

b

37

41

7

15

Assume λ > 0, k(s, t) > 0, N > 0, sin(ωs) > 0, from equation (3.25) and equation (3.26), we obtain the following two equation, respectively.

a

40

6

21

36

38

5

14

Where

33 34

4

13

a

20

2 3

a

15

21

(3.24)

For any α ∈ [0, 1], in order to investigate equation (3.24), we consider the parametric form of FFIE-1 (3.24) as follows. ⎧ b ⎪ ⎪ ⎪ ⎪ ⎪ λ(R) w1 (s, t, N, sin(ωs), Z(s, α), Z(s, α))ds, t ∈ [c, d], λ > 0, ⎪ ⎪ ⎨ a (F2 gH F1 )(t, α) = (3.25) ⎪ b ⎪ ⎪ ⎪ ⎪ λ(R) w2 (s, t, N, sin(ωs), Z(s, α), Z(s, α))ds, t ∈ [c, d], λ < 0. ⎪ ⎪ ⎩

14

17

t ∈ [c, d].

a

13

16

1

k(s, t) · N sin(ωs)  Z(s)ds,

48

Y2 (s) gH Y1 (s) = N sin(ωs)  Z(s).

(3.32)

50

For any α ∈ [0, 1], we consider the parametric form of FFIE-1 (3.32) as follows. (Y2 gH Y1 )(s, α) = γ1 (N, sin(ωs), Z(s, α), Z(s, α)).

49

51

(3.33)

52

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(Y2 gH Y1 )(s, α) = γ2 (N, sin(ωs), Z(s, α), Z(s, α)).

9

(3.34)

2

Where

3

γ1 (s, N, sin(ωs), Z(s, α), Z(s, α))  N sin(ωs) · Z(s, α), s ∈ [a, b], N · sin(ωs)) ≥ 0, = N sin(ωs) · Z(s, α), s ∈ [a, b], N · sin(ωs)) < 0. γ2 (s, N, sin(ωs), Z(s, α), Z(s, α))  N sin(ωs) · Z(s, α), s ∈ [a, b], N · sin(ωs)) ≥ 0, = N sin(ωs) · Z(s, α), s ∈ [a, b], N · sin(ωs)) < 0.

4

(3.35)

14 15 16 17 18 19 20 21 22 23 24 25 26 27

30 31 32 33 34 35 36 37 38 39

9

(3.36)

42 43 44 45 46 47 48 49 50 51 52

10 11 12 13 14

(Y2 gH Y1 )(s, α) = N · sin(ωs) · Z(s, α),

(3.37)

15

(Y2 gH Y1 )(s, α) = N · sin(ωs) · Z(s, α).

(3.38)

17

(Y2 gH Y1 )(s, α) − (Y2 gH Y1 )(s, α) = N · sin(ωs) · [Z(s, α) − Z(s, α)].

19

(3.39)

22 23 24

L

c

(3.40)

⎧ ⎨

25 26 27

a

However, as ω −→ ∞, equation     (Y2 gH Y1 )(s, α) − (Y2 gH Y1 )(s, α)

20 21

From Riemann–Lebesgue Lemma [42,43,15,36], as ω −→ ∞, we have equation     (F2 gH F1 )(t, α) − (F2 gH F1 )(t, α) 2 ⎧ ⎫ 12 d b ⎨ ⎬ = |λ · N | (R) [ k(s, t) · sin(ωs) · [Z(s, α) − Z(s, α)]ds]2 dt −→ 0. ⎩ ⎭

16

18

From equation (3.37), equation (3.38), we obtain

28 29 30 31

L2

⎫ 12 b ⎬ = |N | (R)[ sin2 (ωs) · [Z(s, α) − Z(s, α)]2 ds]  0, ⎩ ⎭

32

(3.41)

33 34 35

a

where  ·  denotes the L2 -norm. Hence, from equation (3.40) and equation (3.41), we infer that the solution of equation (3.1) does not depend continuously on the data F(t). 2

40 41

6

8

Assume N > 0, sin(ωs) > 0, from equation (3.34) and equation (3.35), we obtain the following two equations, respectively.

28 29

5

7

12 13

1

36 37 38 39 40

We will give the following example to explain Theorem 3.4. = v ∈ E 1 is given by a triangular fuzzy number Example 3.1. Let Z v  s : [0, 1] → E 1 . Where  ⎧ ⎪ x ∈ (−1, 0), ⎨x + 1,  v (x) = −x + 1, x ∈ (0, 1), ⎪ ⎩ 0, x ∈ (−∞, −1] ∪ [1, +∞).

41 42 43 44 45

(3.42)

47 48

 is given by The α-cut set of Z(s)

49

 α = [sv(t, α), sv(t, α)] [Z(s)] = [s(α − 1), s(1 − α)].

46

50 51

(3.43)

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10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

From equation (3.40) (3.41), we have equation     (F2 gH F1 )(t, α) − (F2 gH F1 )(t, α)

1 2 3

L2

⎧ ⎫ 1 1 ⎨ ⎬ = |λ · N | (R) [ k(s, t) · sin(ωs) · [2s(1 − α)]ds]2 dt ⎩ ⎭ 0

0

⎧ ⎨

4

1 2

5 6

(3.44)

⎫ 12 ⎬

8

1 1 = 2(1 − α)|λ · N | (R) [ sk(s, t) · sin(ωs)ds]2 dt −→ 0, ⎩ ⎭ 0

11

13 14 15

L2

16

1 2

⎧ ⎨

⎫ 12 1 ⎬ = 2(1 − α)|N | (R)[ sin2 (ωs) · s 2 ds] ⎩ ⎭ 

10

12

⎧ ⎫ 1 ⎨ ⎬ = |N | (R)[ sin2 (ωs) · [2s(1 − α)]2 ds] ⎩ ⎭ 0

9

0

as ω −→ ∞. However, as ω −→ ∞, equation     (Y2 gH Y1 )(s, α) − (Y2 gH Y1 )(s, α)

17 18 19

(3.45)

22 23

0

24 25 26

L2 -norm.

27

28 29 30

28

Remark 3.2. Under other circumstance, the methods of proof for Theorem 3.4 are similar to the situation of λ > 0, k(s, t) > 0, N > 0, sin(ωs) > 0.

31 32 33

36 37 38

43 44 45 46 47 48 49

52

33

35 36 37 38

40 41

In what follows we will present a brief summary of the regularization method. Details can be found in [15,42,43, 36]. To simplify the analysis, the problems (3.1), (3.7) and (3.8), are written in their linear operator forms, respectively. Tx =y

42 43 44 45

(4.1)

where T is a bounded linear operator between Hilbert spaces X and Y . We will call y attainable if y ∈ R(T ) holds. Then, (3.11) is equivalent to every y ∈ Y being attainable. (3.12) holds if and only if N (T ) = {0}, and if (3.11) and (3.12) hold. So that T −1 exists, and (3.13) is equivalent to the continuity (or boundedness) of T −1 .

50 51

32

39

4. Regularization solution of FFIEs-1

41 42

30

34

It is important to note that the fuzzy Fredholm integral equation of the first kind is ill-posed problem. The solution for an ill-posed problem may not exist, and if it exists it may not be unique, or it does not depend continuously on the given Cauchy data and any small perturbation in the given data may cause large change to the solution. In general terms, regularization is the approximation of an ill-posed problem by a family neighboring well-posed problems.

39 40

29

31

Definition 3.2. The fuzzy Fredholm integral equation of the first kind (3.1) is said to be ill-posed if the Fredholm integral equations of the first kind (3.7) or the Fredholm integral equations of the first kind (3.8) is ill-posed.

34 35

20 21

1 1 cos(2ω) 2 1−α 1 2ω2 − 1 = 2(1 − α)|N | sin(2ω) − − [ ] → 0, = 6 4ω 2ω2 ω 3 where  ·  denotes the

7

46 47 48 49 50

Definition 4.1. [15] Let T : X → Y be a bounded linear operator between Hilbert spaces X and Y , θ0 ∈ (0, +∞]. For every θ ∈ (0, θ0 ), let

51 52

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Rθ : Y → X

1 2 3 4

lim sup{Rθ(δ,y δ ) y − T δ

−1

δ→0

6

y : y ∈ Y, y − y ≤ δ} = 0 δ

δ

(4.2)

(4.4)

δ→0

20

1 Yθ (t) = [F (t) gH λ  (F R) θ

21 22

k(s, t)  Yθ (s)ds],

t ∈ [c, d],

28 29 30

24

Yθ (t) −→ Y (t),

as θ −→ 0.

25 26

Proof. For any α ∈ [0, 1], the α-cut set of Yθ (t) is shown as follows:

37 38 39 40 41 42 43

27 28

[Yθ (t)]α = [Yθ (t, α), Yθ (t, α)].

(4.6)

31

b [F (t) gH λ  (F R)

32

k(s, t)  Yθ (s)ds]α = [m1 , m2 ],

(4.7)

46 47 48 49

35

where

⎧ ⎫ ⎪ ⎪ b b ⎨ ⎬ m1 = min F − [λ  (F R) k(s, t)  Yθ (s)ds], F − [λ  (F R) k(s, t)  Yθ (s)ds] , ⎪ ⎪ ⎩ ⎭ a a ⎧ ⎫ ⎪ ⎪ b b ⎨ ⎬ m2 = max F − [λ  (F R) k(s, t)  Yθ (s)ds], F − [λ  (F R) k(s, t)  Yθ (s)ds] . ⎪ ⎪ ⎩ ⎭ a

36 37

(4.8)

52

38 39 40 41

(4.9)

42 43

a

44

Hence, from the equation (4.5) and equation (4.6) we have 1 1 [Yθ (t, α), Yθ (t, α)] = [ m1 , m2 ]. θ θ Assume λ > 0, k(s, t) > 0, from equation (4.8)–(4.10) and equation (3.14)–(3.15), we have

45 46

(4.10)

Yθ (t, α) = F (t, α) − λ(R)

49 50

k(s, t) · Yθ (s, α)ds, a

47 48

b

50 51

33 34

a

44 45

29 30

From the Definition 2.4, in terms of α-cut we have

34

36

21

23

then

32

35

18

22

a

31

33

17

20

(4.5)

26 27

15

19

b

24 25

14

16

Theorem 4.1. Assume that Y (t) is a solution of problem (3.1). The regularization method transforms the problem (3.1) to the approximation fuzzy Fredholm integral equation of the second kind

19

23

12 13

For a specific y ∈ D(T −1 ), a pair (Rθ , θ ) is called a regularization method (for solving T x = y) if (4.2) and (4.4) hold.

16

18

9

11

lim sup{θ (δ, y δ )| y δ ∈ Y, y δ − y ≤ δ} = 0.

13

17

5

10

is such that

12

15

4

8

(4.3)

11

14

3

7

θ : R + × Y → (0, θ0 )

9

2

6

holds. Here,

8

10

1

be a continuous (not necessarily linear) operator. The family {Rθ } is called a regularization or a regularization operator (for T −1 ), if for all y ∈ D(T −1 ), there exists parameter choice rule θ = θ (δ, y δ ) such that

5

7

11

(4.11)

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12

b

1 2

Yθ (t, α) = F (t, α) − λ(R)

3 4

1

k(s, t) · Yθ (s, α)ds.

(4.12)

3

a

4

Where assume

5

5

b

6 7

F − [λ  (F R)

8 9 10 11 12 13 14 15

b k(s, t)  Yθ (s)ds] < F − [λ  (F R)

a

6

k(s, t)  Yθ (s)ds].

7 8

a

Moreover, it was proved by Tikhonov [42,43] and Phillips [36] that the solution Yθ (t, α) of Eq. (4.11) and the solution Yθ (t, α) of Eq. (4.12) converges to the solution Y (t, α) of (3.14) and to the solution Y (t, α) of (3.15) as θ −→ 0, respectively. Hence

18 19 20

Under other circumstance, the methods of proof are similar to the above. The proof is complete. 2

25 26

As stated before, we will apply the regularization method to transform the fuzzy Fredholm integral equation of the first kind to the fuzzy integral equation of the second kind. The resulting fuzzy integral equation of the second kind will be then solved by the well-known existing techniques mentioned in section 1 but that will not be presented in this work.

31

34 35 36 37 38 39 40

43

19 20

23

In what follows, we will apply the regularization method combined with one appropriate technique to illustrate the analysis presented before. Notice first that the data function F (x) must contain components which are matched by the corresponding x components of the kernel K(x, t). This is a necessary condition to guarantee a solution.

24 25 26 27

Example 5.1. Consider the following fuzzy Fredholm integral equation of the first kind

28 29

1 2t   v (x) = λ  (F R)

st  Y (s)ds,

30

t ∈ [0, 1].

(5.1)

31

0

32

is given by a triangular fuzzy number ⎧ ⎪ x ∈ (−1, 0), ⎨x + 1,  v (x) = −x + 1, x ∈ (0, 1), ⎪ ⎩ 0, x ∈ (−∞, −1] ∪ [1, +∞).

33

Where  v ∈ E1

34 35

(5.2)

38 39 40 41

1 Yθ (t) = 2t   v (x) gH λ  (F R)

36 37

Assume λ > 0, from k(s, t) = st > 0, and Theorem 4.1, the regularization method carries Eq. (5.1) to the following fuzzy Fredholm integral equation of the second kind

41 42

18

22

32 33

17

21

5. Examples

29 30

12

16

27 28

11

15

23 24

10

14

21 22

9

13

lim Yθ (t, α) = Y (t, α).

θ→0

16 17

2

st  Yθ (s)ds,

t ∈ [0, 1].

(5.3)

42 43

44

0

44

45

From equation (3.14)–(3.15), Eq. (5.1) carries to the following two equation.

45

46 47 48

46

1 2t (α − 1) = λ(R)

st · Y (s, α)ds,

t, α ∈ [0, 1],

47

(5.4)

48

49

0

49

50

1

50

51 52

2t (1 − α) = λ(R)

st · Y (s, α)ds, 0

t, α ∈ [0, 1].

(5.5)

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Using the regularization method, Eq. (5.4) and (5.5) can be transformed respectively to

2 3 4

13

2 1 Yθ (t, α) = t (α − 1) − λ(R) θ θ

1 2

1 st · Yθ (s, α)ds,

t, α ∈ [0, 1],

3

(5.6)

4

5

0

5

6

1

6

7 8

2 1 Yθ (t, α) = t (1 − α) − λ(R) θ θ

11 12 13

16

Yθ0 (t, α) = 0,

19 20 21

24 25 26 27 28 29 30 31 32 33 34 35

10 11 12 13

1

14

st · Yθn−1 (s, α)ds,

t, α ∈ [0, 1],

(5.8)

17

we obtain the following approximations

18

Yθ0 (t, α) = 0, Yθ1 (t, α) = Yθ2 (t, α) = Yθ3 (t, α) = Yθ4 (t, α) =

2 t (α − 1), θ 2 t (α − 1) − θ 2 t (α − 1) − θ 2 t (α − 1) − θ

19 20 21 22

2λ t (α − 1), 3θ 2 2λ t (α − 1) + 3θ 2 2λ t (α − 1) + 3θ 2

23

(5.9) 2λ2 t (α − 1), 9θ 3 2λ2 2λ3 t (α − 1) − t (α − 1), 3 9θ 27θ 4

38 39 40 41 42 43 44 45 46 47 48 49

26 27 28

and so on. Furthermore, in the equation (5.8), θn → θ, θn−1 → θ, as n → ∞. Based on this we obtain the approximate solution     2 λ3 2 λ λ2 3θ Yθ (t, α) = t (α − 1) 1 − + · · · = + 2− t (α − 1). (5.10) θ 3θ θ 3θ + λ 9θ 27θ 3

52

29 30 31 32 33 34 35

The exact solution Y (t, α) of Eq. (5.6) can be obtained by

36

6 Y (t, α) = lim Yθ (t, α) = t (α − 1). θ→0 λ

(5.11)

6 Y (t, α) = lim Yθ (t, α) = t (1 − α). θ→0 λ From Theorem 4.1, we can obtain the exact solution Y (t) of Eq. (5.1)

39 40

(5.12)

42 43

(5.13)

47 48 49

1

50

e 0

45 46

Example 5.2. Consider the following fuzzy Fredholm integral equation of the first kind (e − 1)e   v (x) = λ  (F R)

41

44

6t  v (x), t ∈ [0, 1], λ where  v is given by Eq. (5.2). Y (t) =

3t

37 38

Similarly, we have the exact solution Y (t, α) of Eq. (5.7) can be obtained by

50 51

24 25

36 37

15 16

0

22 23

9

Consequently, from Eq. (5.6), based on iteration procedure 2 1 Yθn (t, α) = t (α − 1) − λ(R) θ θ

7 8

Yθ0 (t, α) = 0.

17 18

(5.7)

To use the successive approximations method, we first select

14 15

t, α ∈ [0, 1].

0

9 10

st · Yθ (s, α)ds,

3t−4s

 Y (s)ds,

t ∈ [0, 1].

(5.14)

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14

1 2 3

Where  v ∈ E 1 is given by Eq. (5.2). Assume λ > 0, from k(s, t) = e3t−4s > 0, and Theorem 4.1, the regularization method carries Eq. (5.14) to the following fuzzy Fredholm integral equation of the second kind

4

7

Yθ (t) = (e − 1)e   v (x) gH λ  (F R) 3t

e

 Yθ (s)ds,

t ∈ [0, 1].

(5.15)

8 9

From equation (3.14)–(3.15), Eq. (5.14) carries to the following two equation.

10

1

11

13

(e − 1)e (α − 1) = λ(R) 3t

15

17

20

e

· Y (s, α)ds,

23

(e − 1)e3t (1 − α) = λ(R)

e3t−4s · Y (s, α)ds,

28

t, α ∈ [0, 1].

(5.17)

0

1 1 Yθ (t, α) = (e − 1)e3t (α − 1) − λ(R) θ θ

31

21

1 e3t−4s · Yθ (s, α)ds,

t, α ∈ [0, 1],

22

(5.18)

34

25

1 e3t−4s · Yθ (s, α)ds,

t, α ∈ [0, 1].

26

(5.19)

29

Yθ (t, α) =

∞ 

Yθ (t, α) =

∞ 

Yθn (t, α),

(5.20)

41 42 43 44 45 46

49 50 51 52

33 34

36

Yθn (t, α).

(5.21)

37 38

n=0

39

For equation (5.18)–(5.19), we have the following recurrence relation 1 Yθ0 (t, α) = (e − 1)e3t (α − 1), θ 1 1 Yθk+1 (t, α) = − λ(R) e3t−4s · Yθk (s, α)ds, k ≥ 0. θ

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(5.22)

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0

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n=0

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38

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The resulting Fredholm integral equation (5.18)–(5.19) of the second kind will be solved by the Adomian decomposition method. The Adomian decomposition method admits the use of

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1 1 Yθ (t, α) = (e − 1)e3t (1 − α) − λ(R) θ θ

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Using the regularization method, Eq. (5.16) and (5.17) can be transformed respectively to

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(5.16)

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t, α ∈ [0, 1],

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3t−4s

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1 Yθ0 (t, α) = (e − 1)e3t (1 − α), θ 1 1 Yθk+1 (t, α) = − λ(R) e3t−4s · Yθk (s, α)ds, k ≥ 0. θ 0

48 49

(5.23)

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For equation (5.22), which in turn gives the components

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1 Yθ0 (t, α) = (e − 1)e3t (α − 1), θ λ Yθ1 (t, α) = − 2 (e − 1)2 e3t (α − 1), eθ λ2 Yθ2 (t, α) = 2 3 (e − 1)3 e3t (α − 1), e θ λ3 Yθ3 (t, α) = − 3 4 (e − 1)4 e3t (α − 1), e θ

3 4 5

(5.24)

13 14 15 16 17 18

9 10 11

and so on. Substituting this result into equation (5.20) gives the approximate solution   1 (e − 1)λ (e − 1)2 λ2 (e − 1)3 λ3 Yθ (t, α) = (e − 1)e3t (α − 1) 1 − − + · · · + θ eθ e2 θ 2 e3 θ 3   1 eθ = (e − 1)e3t (α − 1). θ eθ + (e − 1)λ

12 13 14

(5.25)

21 22 23 24 25

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The exact solution Y (t, α) of Eq. (5.16) can be obtained by

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1 Y (t, α) = lim Yθ (t, α) = e3t+1 (α − 1). θ→0 λ

(5.26)

1 Y (t, α) = lim Yθ (t, α) = e3t+1 (1 − α). θ→0 λ

28 29 30 31 32 33 34 35

24

(5.27)

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25 26

From Theorem 4.1, we can obtain the exact solution Y (t) of Eq. (5.14) 1 Y (t) = e3t+1   v (x), λ where  v is given by Eq. (5.2).

27 28

t ∈ [0, 1],

(5.28)

Remark 5.1. Notice also that Y (t) = λ1 e5t   v (x), t ∈ [0, 1] is a solution of equation (5.14). As stated before, the Fredholm integral equation of the first kind is an ill-posed problem. For ill-posed problems, the solution might not exist, and if it exists, the solution may not be unique.

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6. Conclusions

38 39

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22

Similarly, we have the exact solution Y (t, α) of Eq. (5.17) can be obtained by

26 27

15 16

19 20

7 8

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6

37 38

In this paper, we defined the ill-posedness for the fuzzy Fredholm integral equation of the first kind using the Zadeh’s decomposition theorem of fuzzy number. We employed the regularization method combined with some of the proper well-known techniques to handle the fuzzy Fredholm integral equations of the first kind. Where the regularization scheme consists of transforming first kind fuzzy integral equations to second kind equations. The method showed reliability in handling these ill-posed problems. Finally, two examples show that our proposed regularization method is effective. Unfortunately, the error analysis of the proposed regularization method has not been discussed deeply and the main reasons are the following. In general, for the error analysis of regularization methods of ill-posed problems, we always suppose that (3.11) and (3.12) hold, whereas (3.13) does not hold. That is, an ill-posed problem means that the solution does not depend continuously on the given Cauchy data. However, since there is generalized Hukuhara difference “gH ” between two fuzzy numbers on the right-hand side of the equation (4.5) and the equation (3.1) is a fuzzy Fredholm integral equations of the first kind characterized by the occurrence of the unknown function Y (s) only inside the integral sign, the classic iterative method based on the Hausdorff metric “D(u, v)” between two fuzzy numbers u and v could not be used directly in its error analysis.

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Nevertheless, we could discuss error analysis for the method proposed in this paper based on the parametric representation of the fuzzy-number-valued functions follows these ideas. Note that Y (t) is an exact solution of equation (3.1) and Yθ (t) is its regularization solution, [Y (t)]α = [Y (t, α), Y (t, α)], [Yθ (t)]α = [Yθ (t, α), Yθ (t, α)] for any α ∈ [0, 1]. This way, for any given α ∈ [0, 1], the error analysis of Yθ → Y , as θ → 0 and the error analysis of Yθ → Y , as θ → 0 can be discussed by classical methods respectively. However, it is not as smooth as it seems, since the apriori estimates of the quantities Rθ  and Rθ (T Y (t, α)) − Y (t, α) are very difficult. Thus, there is a lot of work to do in the course of processing error analysis. For the sake of simplicity, we shall explain these by linear operator equation (4.1) as follows. We use the problems (3.7) to support our argument. Now we assume λ > 0, k(s, t) > 0, for any given α ∈ [0, 1], let F (t, α) be the exact left-hand side of the problems (3.7) and F δ (t, α) be the measured date with F δ (t, α) − F (t, α) ≤ δ. From the Definition 4.1, we define Yθ δ (t, α) := Rθ F δ (t, α) as an approximation of the solution Y (t, α) of T Y (t, α) = F (t, α). Then the error splits into two parts by the following obvious application of the triangle inequality:

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Yθδ (t, α) − Y (t, α) ≤ Rθ F δ (t, α) − Yθ (t, α) + Yθ (t, α) − Y (t, α) ≤ Rθ F (t, α) − Rθ F (t, α) + Yθ (t, α) − Y (t, α) δ

≤ Rθ F δ (t, α) − F (t, α) + Yθ (t, α) − Y (t, α) ≤ δRθ  + Rθ F (t, α) − Y (t, α) ≤ δRθ  + Rθ (T Y (t, α)) − Y (t, α) ≤ δRθ  + Rθ (F (t, α)) − T −1 (F (t, α)) = δRθ  + (Rθ − T −1 )F (t, α). We observe that the error between the exact and computed solutions consists of two parts: The first term on the righthand side describes the error in the data multiplied by the “condition number” Rθ  of the regularized problem. The second term denotes the approximation error (Rθ − T −1 )F (t, α) at the exact right-hand side F (t, α) = T (Y (t, α)). We need a strategy to choose θ = θ (δ) dependent on δ in order to keep the total error as small as possible. This means that we would like to minimize δRθ  + Rθ (T Y (t, α)) − Y (t, α). The procedure is the same in every concrete situation: One has to estimate the quantities Rθ  and Rθ (T Y (t, α)) − Y (t, α) (i.e. Yθ (t, α) − Y (t, α)) in term of θ and then minimize this upper with respect to θ . Thus, the a-priori estimates of the quantities Rθ  and Rθ (T Y (t, α)) − Y (t, α) are difficult problems and there is a lot of work to do in the course of processing error analysis. Under these circumstances, the problems remains to be discussed in a separate paper. Acknowledgements The authors would like to thank the reviewers for their constructive comments and valuable suggestions, which improved the quality of this paper. References [1] S. Abbasbandy, T. Allahviranloo, The Adomian decomposition method applied to the fuzzy system of Fredholm integral equations of the second kind, Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 14 (2006) 101–110. [2] G.A. Anastassiou, Fuzzy Mathematics: Approximation Theory, Springer-Verlag, Berlin, Heidelberg, 2010. [3] M.A.F. Araghi, N. Parandin, Numerical solution of fuzzy Fredhom integral equation by the Lagrange interpolation based on the extension principle, Soft Comput. 15 (2011) 2449–2456. [4] H. Attari, Y. Yazdani, A computational method for fuzzy Volterra–Fredholm integral equations, Fuzzy Inf. Eng. 2 (2011) 147–156. [5] E. Babolian, H. Sadeghi Goghary, S. Abbasbandy, Numerical solution of linear Fredholm fuzzy integral equations of the second kind by Adomian method, Appl. Math. Comput. 161 (2005) 733–744.

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