%0 Journal Article
%T Fibonacci Collocation Method for Solving High-Order Linear Fredholm Integro-Differential-Difference Equations
%A Ay£¿e Kurt
%A Salih Yal£¿£¿nba£¿
%A Mehmet Sezer
%J International Journal of Mathematics and Mathematical Sciences
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/486013
%X 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¨C14]. 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¨C20]. 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¨C35]. 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 th-order Fredholm integro-differential-difference equations, where , are the integer, , , under the mixed conditions where , , , and are functions defined on , ; , , , and are suitable constants. Our aim is to obtain an approximate solution of (1) expressed in the truncated Fibonacci series form: where , , are the unknown Fibonacci coefficients. Here is positive integer such that and , , are the Fibonacci polynomials defined by 2. Fundamental Matrix Relations Firstly, we can write the Fibonacci polynomials in the matrix form as follows: where If is even, if is odd, Let us show (1) in the following form: where 2.1. Matrix Relations for the Differential Part Firstly, we consider the solution and its th derivate in the matrix form: Then, from relations (5) and (11), we can obtain the following matrix form: Similar to (13), from relations (5), (11), and (12), we can find matrix form as To find the matrix in terms of the matrix , we can use the following relation: where
%U http://www.hindawi.com/journals/ijmms/2013/486013/