Abstract
Fuzzy Inf. Eng. (2011) 2: 147-156 DOI 10.1007/s12543-011-0073-x ORIGINAL ARTICLE A Computational Method for Fuzzy Volterra-Fredholm Integral Equations Hossein Attari · Allahbakhsh Yazdani Received: 30 January 2011/ Revised: 20 April 2011/ Accepted: 22 May 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, we will study the application of homotopy perturbation method for solving fuzzy nonlinear Volterra-Fredholm integral equations of the sec- ond kind. Some examples are proposed to exhibit the eﬃciency of the method. Keywords Fuzzy arithmetic · Fuzzy nonlinear · Volterra-Fredholm integral equa- tion· Homotopy perturbation method 1. Introduction In recent years, there has been growing interest in study of fuzzy integral equations, in particular in relation to fuzzy control. Prior to discussing fuzzy integral equations and their associated numerical algorithms, it is necessary to present an appropriate brief introduction to preliminary topics such as fuzzy numbers and fuzzy calculus. The concept of fuzzy sets, was originally introduced by Zadeh [29], led to the deﬁni- tion of fuzzy numbers and its implementation in fuzzy control [20] and approximate reasoning problems [30]. The basic arithmetic structure for fuzzy numbers was later developed by Mizumoto and Tanaka [21], Nahmias [22], Dubois and Prade [8]. All of them observed fuzzy numbers as a collection of α-levels, 0 ≤ α ≤ 1. Additional related material can be found in [3]. Goetschel and Voxman [11] suggested a new approach. They represented fuzzy number in a parameterized form (see Section 2) and then embedded the set of fuzzy numbers into a topological vector space. This enabled them to design the basis of a fuzzy calculus. The subject of embedding fuzzy numbers in either a topological or a Banach space was investigated also by Kaleva [13] and Ouyang [25]. The establishment of the embedding Banach space and its in- duced metric over its subset of fuzzy numbers led to immediate applications such as Hossein Attari ()· Allahbakhsh Yazdani () Department of Mathematics, Mazandaran University, Babolsar, Iran email: h.attari@umz.ac.ir yazdani@umz.ac.ir 148 Hossein Attari · Allahbakhsh Yazdani (2011) fuzzy least squares [7], fuzzy linear systems [5] and characterization of compact sets in fuzzy number space [19]. Further applications such as solving integral equations required appropriate and applicable deﬁnitions of fuzzy function and fuzzy integral of a fuzzy function. The fuzzy mapping function was introduced by Chang and Zadeh [6]. Later, Dubois and Prade [9] presented an elementary fuzzy calculus based on the extension principle [29]. The concept of integration of fuzzy functions was ﬁrst introduced by Dubois and Prade [9]. Alternative approaches were later suggested by Goetschel and Voxman [11], Kaleva [13], Nanda [23] and others. While Goetschel and Voxman [11] preferred a Riemann integral type approach, Kaleva [13] chose to deﬁne the integral of fuzzy function, using the Lebesgue-type concept for integration. Recently various powerful mathematical methods such as Adomian decomposi- tion method [2], Nystrom or quadrature method [1] have been proposed to obtain numerical solutions for linear fuzzy integral equation problems. The application of homotopy perturbation method in linear and nonlinear crisp problems has been de- voted by scientists and engineers, because this method is to continuously deform a simple problem which is easy to solve into the under study which is diﬃcult to solve. This paper, applies the homotopy perturbation method to the fuzzy nonlinear Volterra-Fredholm integral equations. This paper is organized as follows: Section 2 present some preliminaries of fuzzy calculus and homotopy perturbation method. Section 3 is concerned with apply- ing the homotopy perturbation method for solving the system of nonlinear Volterra- Fredholm integral equations. Section 4 discusses the method in order to solve fuzzy nonlinear Volterra-Fredholm integral equations. To demonstrate the above idea, some numerical examples are given in Section 5. Section 6 ceases with conclusion. 2. Preliminaries 2.1. Fuzzy Integral In this section, some basic notations used in fuzzy calculus and fuzzy integral equa- tion are introduced. Deﬁnition 1 An arbitrary fuzzy number is an ordered pair of functions (u(r), u(r)), 0 ≤ r ≤ 1 which satisfy the following requirements [11,20]: a. u(r) is a bounded left continuous nondecreasing function over [0, 1]; b. u(r) is a bounded left continuous nonincreasing function over [0, 1]; c. u(r) ≤ u(r), 0 ≤ r ≤ 1. A crisp number a is simply represented by u(r) = u(r) = a, 0 ≤ r ≤ 1. For arbitrary u = (u, u), v = (v, v), we deﬁne addition (u+ v) and multiplication by real k as (u+ v)(r) = u(r)+ v(r), (u+ v)(r) = u(r)+ v(r), (1) (kx, kx) k ≥ 0, kx = (kx, kx) k ≤ 0. Fuzzy Inf. Eng. (2011) 2: 147-156 149 The set of all the fuzzy numbers with addition and multiplication as deﬁned by Equation (1) is denoted by E and is a convex cone. It can be shown that Equation (1) are equivalent to the addition and multiplication as deﬁned by using the α-cut approach [11] and the extension principles [24]. Deﬁnition 2 For arbitrary fuzzy numbers u = (u, u) and v = (v, v), the quantity D(u, v) = max sup |u(r)− v(r)|, sup |u(r)− v(r)| (2) 0≤r≤1 0≤r≤1 is distance between u and v. It is shown [21] that (E , D) is a complete metric space. Following [11], we deﬁne the integral of a fuzzy function using the Riemann inte- gral concept. Let f :[a, b] → E . For each partition P = t , t ,··· , t of [a, b] with 0 1 n h = max|t − t | and for arbitraryζ : t ≤ ζ ≤ t, 1 ≤ i ≤ n, let i i−1 i i−1 i i R = f (ζ )(t − t ). P i i i−1 i=1 The deﬁnite integral of f (t)over[a, b]is f (t)dt = lim R . (3) h→0 Provided that this limit exists in the metric D. If the fuzzy function f (t) is contin- uous in the metric D, its deﬁnite integral exists [11]. Furthermore, ⎛ ⎞ b b ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ f (t, r)dt = f (t, r)dt, ⎜ ⎟ ⎝ ⎠ a a ⎛ ⎞ (4) ⎜ b ⎟ b ⎜
⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ f (t, r)dt⎟ = f (t, r)dt. ⎜ ⎟ ⎝ ⎠ a a More details about the properties of the fuzzy integral are given in [11,13]. 2.2. Homotopy Perturbation Method The application of homotopy perturbation method in linear and nonlinear problems has been devoted by scientists and engineers. The fundamental work was done by Liao and He. Liao proposed the Homotopy Analysis Method [17,18]. He introduced and further developed the Homotopy perturbation Method [14-16], He’s technique in particular, eliminated some of the traditional limitations of perturbation methods and was successfully applied to solve many problems in various ﬁelds including ﬂuid mechanics, heat transfer and so on (see [26,28]). The method, which is a coupling of the traditional perturbation method and ho- motopy in topology, deforms the original problem diﬃcult to solve continuously to an easily solved problem. This method, which does not require a small parameter 150 Hossein Attari · Allahbakhsh Yazdani (2011) in an equation, has a signiﬁcant advantage in that it provides an analytical approxi- mate solution to a wide range of nonlinear problems in applied sciences. The homo- topy perturbation method is applied to nonlinear oscillators [14] and to other ﬁelds [25,26,28]. To illustrate the basic idea of this method [14], we consider the following nonlinear diﬀerential equation: A(u)− f (x) = 0, x ∈ Ω, (5) with boundary conditions ∂u B u, , x ∈ Γ, (6) ∂n where A and B are a general diﬀerential operator and a boundary operator, respect- fully, f (x) is a known analytical function andΓ is the boundary of domainΩ. The operator A can be divided into two parts, L and N, where L is a linear, but N is nonlinear. Equation (5) can be therefore, rewritten as follows: L(u)+ N(u)− f (x) = 0. By the homotopy technique we construct a homotopy U(x, p): Ω×[0, 1] → R, which satisﬁes: H(u, p)= (1− p)[L(u)− L(u )]+ p[A(u)− f (x)] = 0, (7) p ∈ [0, 1], x ∈ Ω or H(u, p) = L(u)− L(u )+ pL(u )+ p[N(u)− f (x)] = 0, (8) 0 0 where p ∈ [0, 1] is an embedding parameter, u is an initial approximation of Equa- tion (5), satisfying the boundary conditions. Obviously, from Equation (7) and (8) we will have H(u, 0) = L(u)− L(u ) = 0, H(u, 1) = A(u)− f (x) = 0. The changing process of p from zero to unity is just that of u(x, p) from u (x)to u(x). In topology, this is called homotopy. According to the Homotopy perturbation Method, we can ﬁrst use the embedding parameter p as a small parameter, and assume that the solution of Equation (7) and (8) can be written as a power series in p: u = u + pu + p u +··· . 0 1 2 Setting p = 1, results in the approximate solution of Equation (5) u = lim u = u + u + u +··· . (9) 0 1 2 p→1 The series (9) is convergent for most cases [14]. 3. System of Nonlinear Volterra-Fredholm Integral Equations In this section, we illustrate applying the method for a system of nonlinear Volterra- Fredholm integral equations given by: x b 1 r 2 s U(x) = F(x)+ K (x, t)[U(t)] dt+ K (x, t)[U(t)] dt, (10) a a Fuzzy Inf. Eng. (2011) 2: 147-156 151 where U(x) = (u (x),··· , u (x)) , 1 n F(x) = ( f (x),··· , f (x)) , 1 n 1 1 (x, t) = k (x, t) , i, j = 1,··· , n, ij 2 2 K (x, t) = k (x, t) , i, j = 1,··· , n, ij 1 2 r, s are positive integers and f (x), k (x, t) and k (x, t) are functions having n-th ij ij derivatives on an interval a ≤ x, t ≤ b such that a, b are constants [12]. We assume that Equation (10) has a unique solution. Now, to explain Homotopy perturbation Method, we consider Equation (10) as ⎛ ⎞ ⎜ A (U)⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ A(U) = ⎜ . ⎟ = 0, (11) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ A (U) where 1 r A (U(x))= u (x)− f (x)− K (x, t)[u(t)] dt (12) i i i j=1 2 s − K (x, t)[u(t)] dt j=1 for i = 1, 2,··· , n. We construct a homotopy as follows: H(U, 0) = L(U), H(U, 1) = A(U), where L(U(x)) = (u (x)− f (x),··· , u (x)− f (x)) = 0, 1 1 n n H(U, p) = (1− p)L(U)+ pA(U) = 0. (13) Now consider the i-th Equation (13), we can rewrite it in the following form: 1 r H (U(x)) = u (x)− f (x)− p K (x, t)[u(t)] dt i i i j=1 (14) 2 s −p K (x, t)[u(t)] dt j=1 for i = 1, 2,··· , n. The solution of Equation (14) may be written as a power series in p: u (x) = u (x)+ pu (x)+ p u (x)+··· , i = 1,··· , n. (15) i i0 i1 i2 Substituting Equation (15) into Equation (14) and equating coeﬃcients of the terms 152 Hossein Attari · Allahbakhsh Yazdani (2011) with same power of p in both side, we have (e.g., r = 3, s = 4): p → u − f = 0, i0 i x b n
n 1 3 1 4 2 p → u − (u ) K dt− (u ) K dt = 0, i1 j0 j0 ij ij j=1 j=1 a a x b n
n 2 2 1 3 2 p → u − (3u u )K dt− (4u u )K dt = 0, i2 j1 j1 j0 ij j0 ij j=1 j=1 a a 3 2 2 1 p → u − (3u u + 3u u )K dt i2 j0 j2 j1 j0 ij j=1 2 2 3 2 − (6u u + 4u u )K dt = 0, j2 j0 j1 j0 ij j=1 n
k−1 k−l−1 k 1 p → u − K (u u u )dt ik jl jm j,k−l−m−1 ij j=1 l=0 m=0 n
k−1 k−l−1 k−l−m−1 − K (u u u u )dt = 0. jl jm jv j,k−l−m−v−1 ij j=1 l=0 m=0 v=0 Setting p = 1 in Equation (15) results the solution of Equation (10). 4. Applying the Method Let us consider the nonlinear Volterra-Fredholm integral equation as x b 1 r 2 s u(x) = f (x)+ K (x, t)[u(t)] dt+ K (x, t)[u(t)] dt, (16) a a 1 2 a ≤ x, t ≤ b, r, s ∈ N, where K (x, t), K (x, t) are continuous kernel functions over the square a ≤ x, t ≤ b, and f (x) is a function of x ∈ [a, b]. If f (x) is a fuzzy function this equation may only poses a fuzzy solution. Some propositions for the existence of a unique solution to the fuzzy Volterra-Fredholm integral equation of the second kind, i.e., to Equation (16), where f (t) is a fuzzy function, can be found in [4,10,27]. Now consider parametric form of a fuzzy nonlinear Volterra-Fredholm integral equation with respect to Deﬁnition 1. Let ( f (x, r), f (x, r)) and (u(x, r), u(x, r)), 0 ≤ r ≤ 1 and x ∈ [a, b] are parametric form of f (x) and u(x), respectively. Then para- metric form of Equation (16) is as follows: u(x, r)= f (x, r)+ v (x, t, u(r, t), u(r, t))dt + v (x, t, u(r, t), u(r, t))dt, (17) u(x, r)= f (x, r)+ v (x, t, u(r, t), u(r, t))dt + v (x, t, u(r, t), u(r, t))dt, a Fuzzy Inf. Eng. (2011) 2: 147-156 153 where ⎪ 1 1 K (x, t)u(r, t), K (x, t) ≥ 0, v (x, t, u(r, t), u(r, t)) = 1 ⎪ 1 1 K (x, t)u(r, t), K (x, t) < 0, ⎪ 2 2 K (x, t)u(r, t), K (x, t) ≥ 0, v (x, t, u(r, t), u(r, t)) = 1 ⎪ 2 2 K (x, t)u(r, t), K (x, t) < 0, (18) ⎪ 1 1 K (x, t)u(r, t), K (x, t) ≥ 0, v (x, t, u(r, t), u(r, t)) = 2 ⎪ 1 1 K (x, t)u(r, t), K (x, t) < 0, 2 2 K (x, t)u(r, t), K (x, t) ≥ 0 v (x, t, u(r, t), u(r, t)) = 2 ⎪ 2 2 K (x, t)u(r, t), K (x, t) < 0 for each 0 ≤ r ≤ 1 and a ≤ x ≤ b, we can see that Equation (17) is a system of nonlinear Volterra-Fredholm integral equations in crisp case for each 0 ≤ r ≤ 1 and a ≤ x ≤ b. Now we are able to use Homotopy perturbation Method for solving the obtained nonlinear Volterra-Fredholm integral equations system as mentioned in previous sec- tion. 5. Numerical Examples Example 1 Let us solve the following fuzzy nonlinear Volterra-Fredholm integral 1 2 2 equation for K (x, t) = (t + x )/25, r = 2, K (x, t) = 0, (problem reduced to the fuzzy nonlinear Volterra integral equation): 2 4 1 2 u(x) = xr− r x (3+ 4x)+ K (x, t)(u(t)) dt, 2 4 1 2 u(x) = x(2− r)− (r− 2) x (3+ 4x)+ K (x, t)(u(t)) dt. The exact solution is U(x, r) = (xr, x(2− r)). Let 2 4 u (x) = xr− r x (3+ 4x), 2 4 u (x) = x(2− r)− (r− 2) x (3+ 4x), and U ≈ U + U +···+ U , numerical results are shown in following table (Table 1) 0 1 5 by using metric of Deﬁnition 2, so are depicted in following ﬁgure for x = 1. Example 2 Let 1 2 2 K (x, t) = sinπt/2, r = 1, K (x, t) = cos πx/2, s = 1, 0 ≤ x, t ≤ 1. 154 Hossein Attari · Allahbakhsh Yazdani (2011) 0.8 0.6 0.4 0.2 exact approximate 0 0.5 1 1.5 2 Fig.1 Exact and approximate solution for x= 1 Table 1: Occurred error by choosing u = u . i=0 xx = 0 x = 0.25 x = 0.50 x = 0.75 x = 1.0 −17 −16 −16 −08 Error 0 5.5511× 10 1.1102× 10 2.2204× 10 1.8000× 10 −17 Please note that the zero element in Table 1 means that the error is less than 10 . Therefore, r cos πx r(−2πx+ sin 2πx) u(x) =− + r sinπx+ π 8π x 1 1 2 + K (x, t)u(t)dt+ K (x, t)u(t)dt, 0 0 (−2+ r) cos πx (−2+ r)(−2πx+ sin 2πx) u(x) = + (−2+ r) sinπx+ π 8π x 1 1 2 + K (x, t)u(t)dt+ K (x, t)u(t)dt. 0 0 The exact solution is U(x, r) = (r sinπx, (2− r) sinπx). Let r cos πx r(−2πx+ sin 2πx) u (x) = − + r sinπx+ , π 8π (−2+ r) cos πx (−2+ r)(−2πx+ sin 2πx) u (x) = + (−2+ r) sinπx+ , π 8π and U ≈ U + U +···+ U , numerical results are shown in following table (Table 0 1 8 2): r Fuzzy Inf. Eng. (2011) 2: 147-156 155 Table 2: Occurred error by choosing u = u . i=0 xx = 0 x = 0.25 x = 0.50 x = 0.75 x = 1.0 −04 −04 −05 −04 −04 Error 2.2343× 10 1.3455× 10 4.5896× 10 1.8699× 10 3.3159× 10 6. Conclusion In this paper, the homotopy perturbation method is used to solve nonlinear fuzzy Volterra-Fredholm integral equation computationally. Therefore, as mentioned above, one can interpret the fuzzy nonlinear integral equation as system of nonlinear inte- gral equations, and solves it by appropriate methods such a homotopy perturbation method. Acknowledgments The authors would like to acknowledge Dr. S.H. Nasseri (Department of Mathemat- ics, Mazandaran University) for his review and valuable comments. The authors also thank the anonymous referee for his suggestions and comments to improve an earlier version of this work. References 1. Abbasbandy S, Babolian E, Alavi M (2007) Numerical method for solving linear Fredholm fuzzy integral equations of the second kind. 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Journal
Fuzzy Information and Engineering
– Taylor & Francis
Published: Jun 1, 2011
Keywords: Fuzzy arithmetic; Fuzzy nonlinear; Volterra-Fredholm integral equation; Homotopy perturbation method