Abstract
AbstractIn this paper, a numerical procedure is proposed for the fuzzy linear Fredholm integral equations of the second kind by using splines interpolation. Besides, the convergence conditions and an upper-bound on error are derived. Finally, the advantages of the proposed method have been shown through simulation examples and comparison with Lagrange method.
Fuzzy Inf. Eng. (2012) 3: 339-347 DOI 10.1007/s12543-012-0119-8 ORIGINAL ARTICLE Numerical Solution for Fuzzy Fredholm Integral Equations with Upper-bound on Error by Splines Interpolation Yousef Jafarzadeh Received: 10 July 2011/ Revised: 20 April 2012/ Accepted: 12 July 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, a numerical procedure is proposed for the fuzzy linear Fred- holm integral equations of the second kind by using splines interpolation. Besides, the convergence conditions and an upper-bound on error are derived. Finally, the ad- vantages of the proposed method have been shown through simulation examples and comparison with Lagrange method. Keywords Ferdholm fuzzy integral equations · Fuzzy spline interpolation · Fuzzy Lagrange interpolation · Fuzzy numbers 1. Introduction The concept of integration of fuzzy functions was first introduced by Dubois and Prade [1] and investigated by Goetschel and Voxman [2], Kaleva [3], Matloka [4]. One of the main fuzzy equations, addressed by many researchers, is fuzzy Fredholm integral equation. Generally, the complexity of fuzzy integral equations hinders an- alytical solutions. Therefore, some numerical methods have been recently proposed to fuzzy Fredholm integral equation. In [5-7], iterative methods were used, based on trapezoidal rule. Solutions were presented, based on fixed point [8], Adomian de- composition [9], and Nystrom [10] methods. In [11], a method was established to approximate the solution of linear fuzzy Fredholm integral equations by using finite and divided differences. The authors of [12] considered Lagrange interpolation based on the extension principle. In this paper, a numerical method and rate of convergence of error are proposed to approximate the solution of the following linear fuzzy Fredholm integral equation (FFIE-2) of second kind by using natural and complete splines interpolation: ˜ ˜ ˜ F(t) = f (t)+λ K(x, t)F(x)dx, Yousef Jafarzadeh () Sama Technical and Vocational Training College, Islamic Azad University, Karaj Branch, Karaj, Iran email: mat2j@yahoo.com 340 Yousef Jafarzadeh (2012) whereλ> 0, K(s, t) is an arbitrary kernel function over the square a ≤ s, t ≤ b and f (t) is a fuzzy function of t. Kaleva [13] investigated some properties of Lagrange and cubic spline interpolations. In [14, 15], the properties of natural splines and complete splines of odd degree were introduced. 2. Main Results Definition 2.1 A fuzzy number is a function u : R → I = [0, 1] having the properties: 1) u is normal; 2) u is fuzzy convex set; 3) u is upper semicontinuous on R; 4) The support x ∈ R : u(x) ≥ 0 is a compact set. The set of all fuzzy numbers is denoted by R . For 0 < r ≤ 1, consider the level r 0 sets [u] = {x ∈ R : u(x) ≥ r} and [u] = support{x ∈ R : u(x) ≥ 0} and we have r r r r u = in f [u] and u = sup[u] . − + Definition 2.2 Two LR fuzzy numbers m ˜ = (m,α,β) and n ˜ = (n,γ,δ) are said to be equal if and only if m = n,α = γ andβ = δ. Dubois and Prade show exact formulas for and⊕: m ˜ n ˜ = (m,α,β) (n,γ,δ) = (m− n,α+γ,β+δ), m ˜ ⊕ n ˜ = (m,α,β)⊕ (n,γ,δ) = (m+ n,α+γ,β+δ). Scalar multiplication (hm, hα, hβ), h > 0, h⊗ m ˜ = (hm,−hβ,−hα), h < 0. Definition 2.3 Let D : R × R → R . Then the Hausdorff distance between fuzzy F F r r r r r r numbers is D(u, v) = sup max{| u − v |,| u − v |} = sup d ([u] , [v] ), r∈[0,1] r∈[0,1] H − − + + r r r where [u] = [u , u ] and d is the Hausdorff distance between intervals. Let (R , D) H F − + be a metric space, see [2]. It has properties D(u⊕ w, v⊕ w) = D(u, v), ∀u, v, w ∈ R , D(k⊗ u, k⊗ v) = |k| D(u, v), ∀u, v ∈ R ,∀k ∈ R, D(u⊕ v, w⊕ e) ≤ D(u, w)+ D(v, e), ∀u, v, w, e ∈ R , D(a⊗ u, b⊗ u) ≤ |a− b| D(u, 0), ∀u ∈ R , a, b ∈ R, ab ≥ 0. And let f, g : R → R be fuzzy number valued functions. The distance between f, gis defined by D ( f, g):= sup D( f (x), g(x)). x∈R ∗ ≥0 We denote f = D ( f, 0), the function . : R → R and λ ∈ R has the usual F F F properties of the norm and D(u, v) ≤ u + v . F F ω ( f ;δ) = sup{D( f (x), f (y));|x− y| ≤ δ,∀x, y ∈ R} represents the modulus of con- tinuity of f . Fuzzy Inf. Eng. (2012) 3: 339-347 341 Definition 2.4 A function s :[x , x ] → R is called a polynomial natural spline of 0 n odd degree l = 2m− 1, m ≥ 2, provided that it possesses the following properties: l−1 a) s ∈ C [x , x ], 0 n b) s(x) is polynomial of degree l for x ∈ [x, x ); i = 0,··· , n− 1, i i+1 (v) (v) c) s (x ) = s (x ) = 0; v = m,··· , 2m− 2. 0 n (v) (v) If Condition c) in Definition 2.4 is replaced with s (x ) = s (x ) = 0; v = 0 n 1,··· , m− 1, then s is called polynomial complete spline. We denote the family of these splines by S (x , x ). If s ∈ S (x , x ), i = 0,··· , n, l 0 n i l 0 n interpolates the data (x , f ); j = 0,··· , n, where f = δ , Kronecher delta, j j j ij 1, i = j, δ = ij ⎪ 0, i j, then S (x) = s (x) ⊗ u , where S (x) ∈ S (x , x ), interpolates (x, u ), u u ···u i i u u ···u l 0 n i i 0 1 n 0 1 n i=0 i = 0,··· , n. Now we consider the fuzzy splines interpolation in the given points a = t < t < 0 1 ··· < t = b. Such that ˜ ˜ K(x, t)F(x) ≈ s (x)K(t, t)F(t ) i i i i=0 and replace the interpolating function in Fredholm integral equation, then the follow- ing iterative method is obtained y ˜ = f (t), y ˜ (t) = f (t)⊕λ A K(t, t)⊗ y ˜ (t ), k ≥ 1, k i i k−1 i i=0 where A = s (x)dx. i i The following theorem provides conditions for the existence of a unique solution for Fredholm integral equation and the rate of convergence of error. Theorem 2.1 [2,3] Let K(s, t) be continuous for a ≤ s, t ≤ b and f (t) a fuzzy contin- uous function. Ifλ< , where M = max|K(s, t)|,a ≤ s, t ≤ b, then the iterative M(b−a) procedure ˜ ˜ F (t) = f (t), ˜ ˜ F (t) = f (t)+λ K(x, t)F (x)dx, k ≥ 1. k k−1 Converges to the unique solution of fuzzy above integral equation. Specifically, ˜ ˜ ˜ ˜ sup D(F(t), F (t)) ≤ sup D(F (t), F (t)), k 1 0 1− L a≤t≤b a≤t≤b where L = λM(b− a). In the following theorem, an upper bound for distance between the exact (F(t)) and the approximate (y ˜ (t)) solutions have been obtained from the iterative method by the splines interpolation. 342 Yousef Jafarzadeh (2012) Theorem 2.2 Under the hypotheses of Theorem 2.1, we consider the following iter- ative procedure y ˜ = f (t), y ˜ (t) = f (t) ⊕ λ A k(t, t) ⊗ y ˜ (t ), k ≥ 1, where A = s (x)dx, s ∈ S (t , t ) k i i k−1 i i i i l 0 n i=0 interpolates the data (t , f );j = 0,··· , n, where f = δ , Kronecher delta, and j j j ij Δ : a = t < t <··· < t = b is a partition of the interval [a, b]. Then 0 1 n L L 2A ˜ ˜ D(F(t), Y (t)) ≤ sup D(F (t), F (t))+ (ω(˜ y , v(Δ))+ m (1+ ), (1) k 1 0 m l 1− L 1− L b− a a≤t≤b where A = max|A|, and m = max{m ,··· , m }, k ≥ 1 i l 0 k−1 and m = sup y ˜ (t) i = 0,··· , k− 1. i i a≤t≤b Proof We have ˜ ˜ ˜ D(F (t), y ˜ (t)) = D( f (t)⊕λ⊗ K(s, t)⊗ F (s)ds, k k k−1 f (t)⊕λ⊗ K(t, t)⊗ y ˜ (t )⊗ A ). i k−1 i i i=0 Using properties of metric space, ˜ ˜ D(F (t), y ˜ (t)) = λD( K(s, t)⊗ F (s)ds, K(t, t)⊗ y ˜ (t )⊗ A ) k k k−1 i k−1 i i i=0 = λD( K(s, t)⊗ F (s)ds, k−1 i−1 i=1 (K(t, t)⊗ y ˜ (t )⊗ A )⊕ K(t , t)⊗ y ˜ (t )⊗ A ) i k−1 i i 0 k−1 0 0 i=1 = λD( K(s, t)F (s)ds, k−1 i−1 i=1 ((t − t )K(t, t)˜ y (t )A )⊕ K(t , t)˜ y (t )A ). i i−1 i k−1 i i 0 k−1 0 0 t −t i i−1 i=1 Since cds = (b− a)⊗ c,wehave n n t t i i ˜ ˜ D(F (t), y ˜ (t)) ≤ λD( K(s, t)F (s)ds, K(t, t)⊗y ˜ (t )A ds) k k k−1 i k−1 i i t −t t t i i−1 i−1 i−1 i=1 i=1 +λD(K(t , t)˜ y (t )A , 0) 0 k−1 0 0 ≤ λ D(K(s, t)F (s), K(t, t)˜ y (t )A )ds k−1 i k−1 i i t t −t i−1 i i−1 i=1 +λD(K(t , t)˜ y (t )A , 0). 0 k−1 0 0 Using triangle inequality, we have ˜ ˜ D(F (t), y ˜ (t)) ≤ λ (D(K(s, t)F (s), K(s, t)˜ y (t ))ds k k k−1 k−1 i i−1 i=1 + D(K(s, t)˜ y (t ), K(t, t)˜ y (t )A ))ds k−1 i i k−1 i i t −t i i−1 +λD(K(t , t)˜ y (t )A , 0). 0 k−1 0 0 It follows that ˜ ˜ (F (t), y ˜ (t)) ≤ λ ( |K(s, t)| D(F (s), y ˜ (t )))ds k k k−1 k−1 i i−1 i=1 + (|K(s, t)| y ˜ (t ) + K(t, t)A y ˜ (t ) )ds k−1 i F i i k−1 i F t −t t i i−1 i−1 +|λ(K(t , t)A | y ˜ (t ) . 0 0 k−1 0 F Fuzzy Inf. Eng. (2012) 3: 339-347 343 Hence, we obtain ˜ ˜ D(F (t), y ˜ (t)) ≤ λ (M D(F (s), y ˜ (t ))ds k k k−1 k−1 i i−1 i=1 + m (M + K(t, t)A )ds)+|λK(t , t)A | m k−1 i i 0 0 k−1 t −t t i i−1 i−1 ≤ λ (M D(F (s), y ˜ (s))+ D(˜ y (s)), y ˜ (t ))ds k−1 k−1 k−1 k−1 i i−1 i=1 + m (M + K(t, t)A )ds+|λK(t , t)A | m . k−1 i i 0 0 k−1 t t −t i−1 i i−1 Therefore, ˜ ˜ D(F (t), y ˜ (t)) ≤ λ (M (D (F , y ˜ )+ω(˜ y , v(Δ)))ds k k k−1 k−1 k−1 i−1 i=1 ˜ ˜ +m (M + A)ds)+λMAm , k−1 k−1 t t −t i−1 i i−1 where v(Δ) = max {|t − t |}. We conclude that i=1,···,n i i−1 ˜ ˜ A A ˜ ˜ D(F (t), y ˜ (t)) ≤ LD (F , y ˜ )+ Lω(˜ y , v(Δ))+ Lm (1+ )+ Lm k k k−1 k−1 k−1 k−1 k−1 b−a b−a 2A ≤ LD (F , y ˜ )+ Lω(˜ y , v(Δ))+ Lm (1+ ). k−1 k−1 k−1 k−1 b−a Then ∗ ∗ 2A ˜ ˜ D (F , y ˜ ) ≤ LD (F , y ˜ )+ Lω(˜ y , v(Δ))+ Lm (1+ ), k−1 k−1 k−2 k−2 k−2 k−2 b−a . . . . . . 2A ∗ ∗ ˜ ˜ D (F , y ˜ ) ≤ LD (F , y ˜ )+ Lω(˜ y , v(Δ))+ Lm (1+ ). 1 1 0 0 0 0 b−a So we have k ∗ k ˜ ˜ D(F (t), y ˜ (t)) ≤ L D (F , y ˜ )+ Lω(˜ y , v(Δ))+···+ L ω(˜ y , v(Δ)) k k 0 0 k−1 0 2A k +(1+ )(Lm +··· L m ). k−1 0 b−a Since D (F , y ˜ ) = 0) and if ω(˜ y , v(Δ)) = max{ω(˜ y , v(Δ)),··· ,ω(˜ y , v(Δ))},0 ≤ 0 0 m 0 k m ≤ k, we obtain 2 k D(F (t), y ˜ (t)) ≤ (L+ L +···+ L )ω(˜ y , v(Δ)) k k m 2A 2 k +(1+ )(L+ L ···+ L )m b−a L(1−L ) 2A ≤ (ω(˜ y , v(Δ))+ (1+ )m ). m l 1−L b−a 1−L 1 Since ≤ , 0 < L < 1, we have 1−L 1−L L 2A D(F (t), y ˜ (t)) ≤ (ω(˜ y , v(Δ))+ (1+ )m ). k k m l 1− L b− a Therefore, ∗ ∗ ∗ ˜ ˜ ˜ ˜ D (F(t), y ˜ (t)) ≤ D (F(t), F (t))+ D (F (t), y ˜ (t)) k k k k L L 2A ˜ ˜ ≤ D (F (t), F (t))+ (ω(˜ y , v(Δ))+ (1+ )m ) . 1 0 m l 1− L 1− L b− a 344 Yousef Jafarzadeh (2012) According to the above relation in the Theorem 2.2, the first term of the above relation tends to zero as k tends to ∞; if the second term of the above relation is a small value, the convergence of the iterative procedure has been obtained. 3. Algorithm and Examples 3.1. Algorithm (This algorithm is used for cubic spline with triangular fuzzy num- bers.) 1) Read a, b, n, m,λ, h,ε, i = 0,··· , n. 2) Put t = a+ ih for i = 0,··· , n. i i ⎪ s (x), x ∈ [x , x ], 0 0 1 ⎪ . 3) Compute S (x) = ⎪ s (x), x ∈ [x, x ], i i i i+1 ⎪ . s (x), x ∈ [x , x ], n−1 n−1 n 3 3 (x − x) (x− x ) i+1 i s (x) = m + m +λ (x− x )+μ, for x ∈ [x, x ], i i i+1 i i i i i+1 6h 6h i+1 i+1 where S (t ) = y = 1 and i i i S (t )) = y = 0j i, j = 0,··· , n, i j j m h i+1 μ = y − , i i y − y h (m − m ) i+1 i i+1 i+1 i λ = − , h 6 i+1 h = x − x , m = m = 0, i i i−1 0 n ⎡ ⎤ ⎡ ⎤ −1 ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ m b c d ⎢ 1 ⎥ ⎢ 1 1 ⎥ ⎢ 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ m a b c d ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 2 2 2 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ , ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ . . . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ m a b d n n−1 n−1 n−1 h 6 y − y y − y i i+1 i i i−1 where a = , b = 2, c = 1− a and d = ( − ). i i i i i h + h h + h h h i i+1 i i+1 i+1 i 4) Compute A = S (t)dt. i i 5) Set y ˜ = f (t) and for k = 0, 1,··· , compute y ˜ = y ˜ ⊕ λ K(t, t )A ⊗ y ˜ until 0 k+1 k i i k i=0 D(F(t), y ˜ (t))<ε. k+1 3.2. Examples Example 1 The following fuzzy Fredholm integral equation ˜ ˜ ˜ F(t) = 1⊕ tsF(s)ds, (2) 0 Fuzzy Inf. Eng. (2012) 3: 339-347 345 where 1 = (1, 0.5, 1.5) is a triangular fuzzy number with spread 0.5. The exact solu- tion of (2) is given by ˜ ˜ F(t) = 1(1+ t). Now, we earn an approximate solution by the splines interpolation method with n = 4, h = 0.25, t = 0, t = 0.25, t = 0.5, t = 0.75, t = 1. The approximate 0 1 2 3 4 112 56 168 ˜ ˜ solution converges to y ˜ = 1⊕ ( , , )⊗ t F(t). 149 149 149 At the point t = 0.5, exact solution is (1.375, 0.6875, 2.0625) and the approximate solution by splines interpolation method is (1.3784, 0.6879, 2.0637). In this example, ≥0 all of A are positive so for t ∈ R , sign of kernel is positive. In Table 1, we show the advantage of numerical solution for fuzzy Fredholm in- tegral equations by the splines interpolation method (just by n = 4, h = 0.25) to the Lagrange interpolation method [12]. Table 1: Comparison between the approximate solution by splines interpolation and Lagrange interpolation. r F − y F − y Spline method 0.0 0.00041 0.0012 Lagrange method 0.0 0.0051 0.0053 Spline method 0.5 0.00014 0.00031 Lagrange method 0.5 0.0021 0.0045 Spline method 1.0 0.00083 0.00083 Lagrange method 1.0 0.0027 0.0028 Example 2 Consider the following fuzzy Fredholm integral equation t˜ −1 ˜ ˜ F(t) = e ⊕ 2e t⊗ F(s)ds. (3) The exact solution of (3) is given by t 0 F(t) = e ⊕ 2t⊗ e . Now, we earn the approximate solution by the splines interpolation method with n = 4, h = 0.25, t = 0, t = 0.25, t = 0.5, t = 0.75, t = 1. The approximate solution 0 1 2 3 4 t˜ 0 converges to y ˜ = e ⊕ 2.0015t ⊗ e . Table 2 Shows the results for the approximate solution and exact solution (for t =0.5). 346 Yousef Jafarzadeh (2012) Table 2: Comparison between the approximate solution and exact solution by splines interpolation. r F − y F − y 0.0 0.00071 0.00021 0.5 0.00084 0.00041 1.0 0.00093 0.0005 4. Conclusion A numerical procedure, based on splines interpolation, is proposed to solve the fuzzy linear Fredholm integral equations of the second kind. The convergence properties, as well as an upper-bound on the error, are established for the considered method. To demonstrate the efficiency of the suggested method, some numerical examples are presented. The results are compared with the Lagrange method. Results show the superiority of the proposed approach in achieving a smaller error of approximation over the Lagrange method. Acknowledgements The author would like to thank to anonymous referees and the Editor-in-Chief for their comments which improved the paper. References 1. Dubois D, Prade H (1982) Towards fuzzy differential calculus. Fuzzy Sets and Systems 8: 1-7 2. Goetschel R, Vaxman W (1986) Elementary calculus. Fuzzy Sets and Systems 18: 31-43 3. Kaleva O (1987) Fuzzy differential equations. Fuzzy Sets and Systems 24: 301-317 4. Matloka M (1987) On fuzzy integrals. 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Fuzzy Information and Engineering – Taylor & Francis
Published: Sep 1, 2012
Keywords: Ferdholm fuzzy integral equations; Fuzzy spline interpolation; Fuzzy Lagrange interpolation; Fuzzy numbers
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