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

The integro-differential-difference equations (IDDEs) have been developed very rapidly in recent years.This is an important branch ofmathematicswhich has a lot of interest inmany 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,


Fundamental Matrix Relations
Firstly, we can write the Fibonacci polynomials   () in the matrix form as follows: where If  is even, Let us show (1) in the following form: where International Journal of Mathematics and Mathematical Sciences 3 2.1.Matrix Relations for the Differential Part ().Firstly, we consider the solution y() and its th derivate y () () 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 y () () matrix form as To find the matrix X () () in terms of the matrix X(), we can use the following relation: () = X (1) () where Subsequently, by substituting the matrix form ( 15) into ( 14), we obtain the matrix relations

Matrix Relations for the Difference Part 𝑄(𝑥).
If we put  →    +   in the relation (11), we have the matrix form It is seen that the relation between the matrices X() and 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

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 ,  = 0, 1, . . ., , ,  = 0, 1, . . ., .
International Journal of Mathematics and Mathematical Sciences 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

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  1 ,  2 , . . .,   .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 W = rank [ W; G] = , then we can write And so, the matrix A (thereby the coefficients  1 ,  2 , . . .,   ) is uniquely determined.

Accuracy of Solution
We can check the accuracy of the method.The truncated Fibonacci series in If max(10 −  ) = 10 − ( 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 10 − .

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 integrodifferential-difference equation given by with the boundary conditions and the approximate solution () by the truncated Fibonacci series where From ( 38), the collocation points for  = 3, are computed and from (40), the fundamental matrix equation of the problem is where International Journal of Mathematics and Mathematical Sciences 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 () =  2 −  + 1, which is the exact solution Example 2 (see [26]).Consider the following linear Fredholm integro-differential-difference equation with variable coefficients with the conditions (0) = 0,   (0) = 1, and   (0) = 0 and the exact solution The fundamental matrix equation of the problem from (40) becomes  (66) We compare the solutions found by the present method for  = 8,  = 9 and the absolute errors in Figure 1 and Table 1.
It is seen that when we increase integer , the errors decrease.

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-differentialdifference equation with variable coefficients, but some modifications are required.
Exact solutionPresent method for N = 8 Present method for N = 9
Exact solutionPresent method for N = 6 Present method for N = 9
Present method for N = 6 Present method for N = 9

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

Table 2 :
Comparison of the absolute errors of Example 5.3.