Fibonacci Collocation Method for Solving High-Order Linear Fredholm Integro-Differential-Difference Equations

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2013, Article ID 486013, 9 pages http://dx.doi.org/10.1155/2013/486013 Research Article Fibonacci Collocation Method for Solving High-Order Linear Fredholm Integro-Differential-Difference Equations AyGe Kurt, Salih YalçJnbaG, and Mehmet Sezer Department of Mathematics, Celal Bayar University, Muradiye, 45140 Manisa, Turkey Correspondence should be addressed to Salih Yalçınbaş; Received 8 May 2013; Accepted 27 June 2013 Academic Editor: Irena Lasiecka Copyright © 2013 Ayşe Kurt et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A new collocation method based on the Fibonacci polynomials is introduced for the approximate solution of high order-linear Fredholm integro-differential-difference equations with the mixed conditions. The proposed method is analyzed to show the convergence of the method. Some further numerical experiments are carried out to demonstrate the method. 1. Introduction The integro-differential-difference equations (IDDEs) have been developed very rapidly in recent years. This is an important branch of mathematics which has a lot of interest in many application fields such as engineering, mechanics, physics, astronomy, chemistry, biology, economics, and potential theory, electrostatics [1–14]. Since some IDDEs are hard to solve numerically, they are solved by using the approximated methods. Several numerical methods were used such as the successive approximations, Adomian decomposition, Haar Wavelet, and Tau and Walsh series methods [15–20]. Additionally the Monte Carlo method for linear Fredholm integro-differential-difference equation has been presented by Farnoosh and Ebrahimi [21] and the Direct method based on the Fourier and block-pulse method functions by Asady et al. [22]. Since the beginning of 1994, the Taylor and Chebyshev matrix methods have also been used by Sezer et al. to solve linear differential, Fredholm integral, and Fredholm integro-differential equations [23–35]. Lately, the Fibonacci collocation method has been used to find the approximate solutions of differential, integral, and integro-differential equations [36]. In this study, we consider the approximate solution of the 𝑚th-order Fredholm integro-differential-difference equations, 𝑚 𝑠 𝑘=0 𝑟=0 ∑ 𝑃𝑘 (𝑥) 𝑦(𝑘) (𝑥) + ∑𝑄𝑟 (𝑥) 𝑦(𝑟) (𝜇𝑟 𝑥 + 𝜏𝑟 ) (1) 𝑏 = g (𝑥) + 𝜆 ∫ 𝐾 (𝑥, 𝑡) 𝑦 (𝑡) 𝑑𝑡, 𝑎 where 𝑠 ≤ 𝑚, 𝜏𝑟 are the integer, 0 ≤ 𝑎 ≤ 𝑥, 𝑡 ≤ 𝑏, under the mixed conditions 𝑚−1 ∑ [𝑎𝑗𝑘 𝑦(𝑘) (𝑎) + 𝑏𝑗𝑘 𝑦(𝑘) (𝑏)] = 𝜆 𝑗 , 𝑗 = 1, 2, 3, . . . , 𝑚, (2) 𝑘=0 where 𝑃𝑘 (𝑥), 𝑄𝑟 (𝑥), 𝑔(𝑥), and 𝐾(𝑥, 𝑡) are functions defined on 𝑎 ≤ 𝑥, 𝑡 ≤ 𝑏; 𝑎𝑗𝑘 , 𝑏𝑗𝑘 , 𝜆, and 𝜆 𝑗 are suitable constants. Our aim is to obtain an approximate solution of (1) expressed in the truncated Fibonacci series form: 𝑁 𝑦 (𝑥) = ∑ 𝑎𝑛 𝐹𝑛 (𝑥) , 𝑛=1 (3) 2 International Journal of Mathematics and Mathematical Sciences where 𝑎𝑛 , 𝑛 = 1, 2, 3, . . . , 𝑁, are the unknown Fibonacci coefficients. Here 𝑁 is positive integer such that 𝑁 ≥ 𝑚 and 𝐹𝑛 (𝑥), 𝑛 = 1, 2, 3, . . . , 𝑁, are the Fibonacci polynomials defined by 2. Fundamental Matrix Relations Firstly, we can write the Fibonacci polynomials 𝐹𝑛 (𝑥) in the matrix form as follows: F𝑇 (𝑥) = CX𝑇 (𝑥) ⇐⇒ F (𝑥) = X (𝑥) C𝑇 , 𝐹𝑛 (𝑥) = [(𝑛−1)/2] ∑ 𝑗=0 𝑛 − 𝑗 − 1 𝑛−2𝑗−1 ( , )𝑥 𝑗 (𝑛 − 2) { , { (𝑛 − 1) [ ] = { (𝑛 2− 1) { 2 , { 2 where F (𝑥) = [𝐹1 (𝑥) 𝐹2 (𝑥) ⋅ ⋅ ⋅ 𝐹𝑁 (𝑥)] , (4) 𝑛 even, X (𝑥) = [1𝑥 ⋅ ⋅ ⋅ 𝑥𝑁−1 ] . 𝑛 odd. (6) If 𝑁 is even, 0 0 0 0 ⋅⋅⋅ 0 ] (0 ) [ ] [ 1 [ 0 0 ⋅⋅⋅ 0 ] 0 (0) ] [ ] [ 2 1 [ 0 ⋅⋅⋅ 0 ] 0 (0) (1 ) ] [ 3 2 ] [ ⋅⋅⋅ 0 ] 0 (0) 0 (1) [ ] [ .. .. .. .. .. .. ] [ ] [ . . . . . . C=[ ] 𝑛 ] [ (𝑛 − 2) ] [ 2 ) ] [( 2 ) 0 ⋅ ⋅ ⋅ 0 0 ( (𝑛 − 4) ] [ (𝑛 − 2) ] [ ] [ 2 2 𝑛 ] [ (𝑛 + 2) ] [ 𝑛−1 ] [ 2 2 0 ( (𝑛 − 2) ) 0 ( (𝑛 − 4) ) ⋅ ⋅ ⋅ ( 0 )] [ 2 [ (5) ; (7) . (8) ]𝑁×𝑁 2 if 𝑁 is odd, 0 0 0 0 ⋅⋅⋅ 0 (0 ) ] [ ] [ 1 [ 0 0 ⋅⋅⋅ 0 ] 0 (0) ] [ ] [ 2 1 0 ⋅⋅⋅ 0 ] 0 (0) (1 ) [ ] [ 3 2 ] [ ⋅⋅⋅ 0 ] 0 (0) 0 (1) [ ] [ .. .. .. .. .. .. ] [ ] [ . . . . . . C=[ ] ] [ (𝑛 + 1) (𝑛 − 1) ] [ [ 2 2 0 ( (𝑛 − 5) ) ⋅ ⋅ ⋅ 0 ] 0 ( (𝑛 − 3) ) ] [ ] [ ] [ 2 2 ] [ ] [ (𝑛 − 1) (𝑛 + 1) ] [ 𝑛−1 ] [( 2 ) 2 0 ( (𝑛 − 3) ) ⋅⋅⋅ ⋅ ⋅ ⋅ ( 0 )] [ (𝑛 − 1) [ 2 𝑠 Let us show (1) in the following form: 𝑃 (𝑥) + 𝑄 (𝑥) = g (𝑥) + 𝜆𝐼 (𝑥) , where 𝑚 𝑃 (𝑥) = ∑ 𝑃𝑘 (𝑥) 𝑦 𝑘=0 ]𝑁×𝑁 2 (9) 𝑄 (𝑥) = ∑𝑄𝑟 (𝑥) 𝑦(𝑟) (𝜇𝑟 𝑥 + 𝜏𝑟 ) , 𝑟=0 𝑏 (𝑘) (𝑥) , 𝐼 (𝑥) = 𝜆 ∫ 𝐾 (𝑥, 𝑡) 𝑦 (𝑡) 𝑑𝑡. 𝑎 (10) International Journal of Mathematics and Mathematical Sciences 2.1. Matrix Relations for the Differential Part 𝑃(𝑥). Firstly, we consider the solution y(𝑥) and its 𝑘th derivate y(𝑘) (𝑥) in the matrix form: y (𝑥) = F (𝑥) A, 𝑇 A = [𝑎1 𝑎2 ⋅ ⋅ ⋅ 𝑎𝑁] , (11) y(𝑘) (𝑥) = F(𝑘) (𝑥) A. (12) Then, from relations (5) and (11), we can obtain the following matrix form: y (𝑥) = X (𝑥) C𝑇 A. (13) Similar to (13), from relations (5), (11), and (12), we can find y(𝑘) (𝑥) matrix form as y(𝑘) (𝑥) = X(𝑘) (𝑥) C𝑇 A. (14) 3 where 𝛽 (𝜇𝑟 , 𝜏𝑟 ) 1 𝑁−1 0 0 0 0 1 0 𝑁−1 [(0) 𝜇𝑟 𝜏𝑟 (0) 𝜇𝑟 𝜏𝑟 ⋅ ⋅ ⋅ ( 0 ) 𝜇𝑟 𝜏𝑟 ] [ ] [ ] [ ] 1 𝑁 − 1 1 0 1 𝑁−2 [ ] 0 ( 𝜏 ⋅ ⋅ ⋅ ( 𝜏 ) 𝜇 ) 𝜇 𝑟 𝑟 𝑟 𝑟 [ ]. 1 1 =[ ] .. .. .. [ ] [ ] . . d . [ ] [ 𝑁−1 𝑁−1 0 ] 𝜏𝑟 0 0 ⋅⋅⋅ ( )𝜇 𝑁−1 𝑟 [ ] (20) By using the relations (15) and (19), we can get 𝑘 X(𝑘) (𝜇𝑟 𝑥 + 𝜏𝑟 ) = X (𝑥) 𝛽 (𝜇𝑟 , 𝜏𝑟 ) (T𝑇 ) . Thus from (14) and (21), we can find 𝑘 y(𝑘) (𝜇𝑟 + 𝜏𝑟 ) = X (𝑥) 𝛽 (𝜇𝑟 , 𝜏𝑟 ) (T𝑇 ) C𝑇 A. To find the matrix X(𝑘) (𝑥) in terms of the matrix X(𝑥), we can use the following relation: X (1) (𝑥) = X (𝑥) T , 𝑚 2 X (𝑥) = X 𝑘 𝑃 (𝑥) = ∑ 𝑝𝑘 (𝑥) X (𝑥) (T𝑇 ) C𝑇 A, 𝑘=0 (15) .. . (𝑘−1) (23) 𝑠 2.3. Matrix Relations for the Integral Part. Let us find the matrix relation for the Fredholm integral part 𝐼(𝑥) in (9). The kernel function 𝐾(𝑥, 𝑡) can be shown by the truncated Fibonacci series, where 𝑁 𝑘 (17) (18) It is seen that the relation between the matrices X(𝑥) and X(𝜇𝑟 𝑥 + 𝜏𝑟 ) is X (𝜇𝑟 𝑥 + 𝜏𝑟 ) = X (𝑥) 𝛽 (𝜇𝑟 , 𝜏𝑟 ) , (24) 𝑚=0 𝑛=0 and the truncated Taylor series, 2.2. Matrix Relations for the Difference Part 𝑄(𝑥). If we put 𝑥 → 𝜇𝑟 𝑥 + 𝜏𝑟 in the relation (11), we have the matrix form y (𝜇𝑟 + 𝜏𝑟 ) = F (𝜇𝑟 𝑥 + 𝜏𝑟 ) A. 𝑁 𝑓 𝐹𝑚 (𝑥) 𝐹𝑛 (𝑡) , 𝐾 (𝑥, 𝑡) = ∑ ∑ 𝑘𝑚𝑛 (16) 𝑁 (19) 𝑁 𝑡 𝐾 (𝑥, 𝑡) = ∑ ∑ 𝑘𝑚𝑛 𝑥𝑚 𝑡𝑛 , Subsequently, by substituting the matrix form (15) into (14), we obtain the matrix relations y(𝑘) (𝑥) = X (𝑥) (T𝑇) C𝑇 A. 𝑇 𝑟=0 (𝑥) T = X (𝑥) (T ) , 0 1 0 ⋅⋅⋅ 0 0 0 [0 0 2 ⋅ ⋅ ⋅ 0 0 0 ] [ ] [0 0 0 ⋅ ⋅ ⋅ 0 0 0 ] [ ] 𝑇 T = [ .. .. .. .. .. ] . [. . . ] . . [ ] [0 0 0 ⋅ ⋅ ⋅ 0 0 𝑁 − 1] 0 ] [0 0 0 ⋅ ⋅ ⋅ 0 0 𝑇 𝑟 𝑄 (𝑥) = ∑𝑄𝑟 (𝑥) X (𝑥) 𝛽 (𝜇𝑟 , 𝜏𝑟 ) (T ) C A. 𝑇 𝑘 𝑇 (22) By using the expressions (14) and (22), we obtain the matrix form 𝑇 X(2) (𝑥) = X(1) (𝑥) T𝑇 = (X (𝑥) T𝑇) T𝑇 = X (𝑥) (T𝑇 ) (𝑘) (21) (25) 𝑚=0 𝑛=0 where 𝑡 𝑘𝑚𝑛 = 1 𝜕𝑚+𝑛 𝐾 (0, 0) ; 𝑚!𝑛! 𝜕𝑥𝑚 𝜕𝑥𝑛 𝑚, 𝑛 = 0, 1, . . . , 𝑁. (26) The expressions (24) and (25) can be (...truncated)