International Journal of Mathematics and Mathematical Sciences

Volume 2013 (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

Ayşe Kurt, Salih Yalçınbaş, and Mehmet Sezer

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.

Abstract.

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 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

The corresponding matrix form for the conditions (2) can be shown, by means of (17), as

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 [26]). 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.

Table 1: Comparison of the absolute errors of Example 2.

Exact solution
Present method

Absolute errors

Absolute errors

−1
−0.8414711
−1.114125
0.2726539
−3.078521
2.23705

−0.8
−0.7173561
−0.869866
0.1525099
−1.875847
1.158491

−0.6
−0.5646425
−0.633677
0.06903453
−1.054038
0.4893957

−0.4
−0.3894183
−0.4110374
0.0216191
−0.5333391
0.1439208

−0.2
−0.1986693
−0.2014897
0.00282041
−0.2163871
0.01771782

0
0

0.2
0.1986693
0.2016525
0.00298318
0.2155338
0.01686445

0.4
0.3894183
0.4137037
0.02428533
0.5200945
0.1306762

0.6
0.5646425
0.6476974
0.08305494
0.9903835
0.4257410

0.8
0.7173561
0.9164897
0.1991336
1.689322
0.9719655

1
0.841471
1.235210
0.3937393
2.667579
1.826108

Figure 1: Numerical and exact solutions of Example 2 for .

Example 3 (see [30]). 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.

Table 2: Comparison of the absolute errors of Example 3.

Exact solution
Present method

Absolute errors

Absolute errors

−1
0.3678794
0.368349

0.3678795

−0.8
0.449329
0.4496545

0.4493289

−0.6
0.5488116
0.5490096

0.5488116

−0.4
0.67032
0.6704069

0.67032

−0.2
0.8187308
0.8187468
0.1605797
0.8187308

0
1.0
1.0
0
1.0
0

0.2
1.221403
1.221369

1.221403

0.4
1.491825
1.491453

1.491825

0.6
1.822119
1.820404

1.822119

0.8
2.225541
2.220096

2.225537

1
2.718282
2.704284

2.71826

Figure 2: Numerical and exact solutions of Example 3 for .

Figure 3: Comparison of the absolute errors of Example 3 for .

6. Conclusion

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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