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International Journal of Mathematics and Mathematical Sciences
Volume 2013 (2013), Article ID 486013, 9 pages
Fibonacci Collocation Method for Solving High-Order Linear Fredholm Integro-Differential-Difference Equations
Department of Mathematics, Celal Bayar University, Muradiye, 45140 Manisa, Turkey
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.
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  and the Direct method based on the Fourier and block-pulse method functions by Asady et al. .
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 .
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 Subsequently, by substituting the matrix form (15) into (14), we obtain the matrix relations
2.2. Matrix Relations for the Difference Part
If we put in the relation (11), we have the matrix form It is seen that the relation between the matricesandis where By using the relations (15) and (19), we can get Thus from (14) and (21), we can find By using the expressions (14) and (22), we obtain the matrix form
2.3. Matrix Relations for the Integral Part
Let us find the matrix relation for the Fredholm integral part in (9). The kernel function can be shown by the truncated Fibonacci series, and the truncated Taylor series, where The expressions (24) and (25) can be put in matrix forms as From (11), (27), and (28) we can obtain Thus By substituting the matrix forms (22) and (27) into the integral part in (9), we can have the matrix relation as follows: so that From (5) and (32), we have where If we substitute the matrix relation (5) into (31), we have the matrix form
2.4. Matrix Relations for the Conditions
3. Method of Solution
We can construct the fundamental matrix equation corresponding for (1). For this aim, we substitute the matrix relations (23) and (35) into (9). So we obtain the matrix equation By using in (37) the collocation points defined by the system of the matrix equations is obtained or shortly the fundamental matrix equation becomes where Therefore, the fundamental matrix equation (40) corresponding for (1) can be written as where Equation (42) corresponds to a system of linear algebraic equations with unknown Fibonacci coefficients. Further, we can express the matrix form (36) conditions where To obtain the solution of (1) under the conditions (2), by replacing the row matrices (44) by the last rows of the matrices (42), we have the new augmented matrix If the last m rows of the (30) are replaced, the augmented matrix of the above system is obtained as If rank , then we can write And so, the matrix (thereby the coefficients) is uniquely determined.
4. Accuracy of Solution
We can check the accuracy of the method. The truncated Fibonacci series in (3) have to be approximately satisfying (1). For each , , or If ( is any positive integer) is prescribed, then the truncation limit is increased until the difference at each of the points becomes smaller than the prescribed .
5. Numerical Examples
In this section, several examples are given to illustrate the applicability of the method and all of them are performed on the computer MATLAB. Also, the absolute errors in tables are the values of at selected points.
Example 1. Let us consider the linear Fredholm integro-differential-difference equation given by with the boundary conditions and the approximate solution by the truncated Fibonacci series where , , , , , , , and , From (38), the collocation points for are computed and from (40), the fundamental matrix equation of the problem is where The augmented matrix for this fundamental matrix equation is calculated as From (37), the matrix forms for the boundary conditions are The new augmented matrix based on conditions can be written as Solving this system, the unknown Fibonacci coefficients are obtained as Hence, by substituting the Fibonacci coefficients matrix into (11), we have the approximate solution , which is the exact solution
Example 2 (see ). Consider the following linear Fredholm integro-differential-difference equation with variable coefficients
with the conditions , , and and the exact solution .
So, , , , , , , and The fundamental matrix equation of the problem from (40) becomes
For the solutions of the problem are found as, respectively, We compare the solutions found by the present method for , and the absolute errors in Figure 1 and Table 1. It is seen that when we increase integer , the errors decrease.
Example 3 (see ). Now consider the linear Fredholm integro-differential-difference equation given by
with the initial conditions , , and and the exact solution .
Here From (40), the fundamental matrix equation of the problem becomes The solutions obtained for are compared with the exact solution is which are given in Figure 2 and we compare the absolute errors found by present method for in Figure 3. Also, the numerical solution and absolute errors are compared for in Table 2.
The Fibonacci collocation method is used to solve the linear Fredholm integro-differential-difference equations numerically. The obtained numerical results show that the accuracy improves when is increased. Tables and figures indicate that as increases, the errors decrease more rapidly. A considerable advantage of the method is that the Fibonacci polynomial coefficients of the solution are found very easily by using computer programs. This method can also be extended to the system of linear integro-differential-difference equation with variable coefficients, but some modifications are required.
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