Fibonacci Collocation Method for Solving Linear Differential -difference Equations

This study presents a new method for the solution of mth-order linear differential-difference equations with variable coefficients under the mixed conditions. We introduce a Fibonacci collocation method based on the Fibonacci polynomials for the approximate solution. Numerical examples are included to demonstrate the applicability of the technique. The obtained results are compared by the known results.


INTRODUCTION
The study of the differential-difference equations developed very rapidly in recent years [1][2][3].These equations play an important role in various branches of science such as engineering, mechanics, physics, biology, control theory etc. Differentialdifference equations occur also frequently as a mathematical model for problems [3][4].
Since some equations are hard to solve analytically, they are solved by using the approximate methods by many authors [5][6][7].Approximate solutions of linear differential, difference, differential-difference, integral and integro-differentialdifference, pantograph equations have been found using the Taylor collocation method and Chebyshev polynomial method by Sezer et.al. [8][9][10][11][12][13][14].Also, the Fibonacci matrix method has been used to find the approximate solutions of differential and integrodifferential equations [15].
In this paper, we consider the approximate solution of the mth-order linear differential-difference equation with variable coefficients, Our aim is to find an approximate solution of (1) expressed in the truncated Fibonacci series form 1 ( ) ( )

FUNDAMENTAL MATRIX RELATIONS
We can write the Fibonacci polynomials () Fx in the matrix form as follows Let us show Eq(1) in the form where  (x) x T y = X C A (8) Similar to Eq. ( 8), from relations (4), ( 6) and ( 7), we can obtain () To find the matrix (x) X in terms of the matrix (x) X , we can use the following relation (10) where 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 .
Consequently, by substituting the matrix form (10) into Eq.(9), we obtain the matrix relations

Matrix relations for the difference part Q(x)
If we put rr xx   in the relation ( 6), we have the matrix form Also, it is seen that the relation between the matrices () x X and () By using the relations ( 10) and ( 13), we can get Thus from ( 9) and ( 14), we can find ) By using the expressions ( 9) and ( 15), we obtain the matrix forms 0 ( ) ( )( )

Matrix relations for the conditions
By means of ( 11), the corresponding matrix forms for the conditions (2) can be shown as

METHOD OF SOLUTION
In this part, we construct the fundamental matrix equation corresponding to Eq. (1).For this purpose, we substitute the matrix relations ( 16) and (17) into Eq.( 5).So we obtain the matrix equation By using in Eq. ( 19) the collocation points i x defined by, ( 1), the system of the matrix equations is obtained as () Therefore, the fundamental matrix equation ( 21) corresponding to Eq. ( 1) can be written in the augmented form where 00 ( ) ( , )( ) .

WA = G
If the last m rows of the (22) are replaced, the augmented matrix of the above system is obtained as follows , then we can write .

   -1 A (W) G
(25) Hence, the matrix A (thereby the coefficients 12 , ,..., N a a a ) is uniquely determined.Further the Eq. ( 1) with conditions (2) has a unique solution.This solution is given by the truncated Fibonacci series (3).
k is any positive integer) is prescribed, then the truncation limit N is increased until the difference () i Ex at each of the points i x becomes smaller than the prescribed 10 k  .

NUMERICAL EXAMPLES
In this section, several numerical examples are given to show the accuracy and the efficiency of this method.( ) 1, ( ) 1, ( ) , ( ) --2 5 From Eq. ( 20), the collocation points (20) for 4 N  is computed , , 0 33 and from Eq. ( 21), the fundamental matrix equation of the problem becomes

P X(T ) C + Q Xβ( )(T ) C + Q Xβ( )(T ) C A G
where The augmented matrix for this fundamental matrix equation is calculated  

WG
From Eq. ( 23), the matrix forms for the conditions are .
The new augmented matrix based on conditions can be calculated as Solving this system, Fibonacci coefficients are obtained as Hence, by substituting the Fibonacci coefficients matrix into Eq.( 6), We obtain the solution 2 ( ) 1 y x x  , which is the exact solution.

CONCLUSIONS
Differential-difference equations with variable coefficients are usually difficult to solve analytically; therefore, approximate solutions are required.To have the best approximate solution for the equation, we take more terms from the Fibonacci expansion of functions, that is, the accuracy improves when N is increased.A considerable advantage of this method is that Fibonacci coefficients of the solution are obtained very easily by using the computer programs.We use the MATLAB program to obtain the solution of equations.
In this study, examples, tables and figures indicate that the present method is convenient, reliable and effective.So, we can say that the Fibonacci collocation method can be a suitable method for solving analytic solutions to linear differential-difference equations.
22) corresponds to a system of N linear algebraic equations with unknown Fibonacci coefficients 12 , ,..., To obtain the solution of Eq. (1) under the conditions (2), by replacing the row matrices (23) by the last m rows of the matrices (22), we have the new augmented matrix . 

Example 5 . 1 . [ 6 ]
Let us first consider the linear differential-difference equation given by

Table 1 .
Comparison of the absolute errors of Example 5.2

Table 2 .
Comparison of the absolute errors of Example 5.3