Laguerre Collocation Method for Solving Fredholm Integro-Differential Equations with Functional Arguments

Laguerre collocationmethod is applied for solving a class of the Fredholm integro-differential equations with functional arguments. This method transforms the considered problem to a matrix equation which corresponds to a system of linear algebraic equations. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. Also, the approximate solutions are corrected by using the residual correction method.


Introduction
Orthogonal polynomials occur often as solutions of mathematical and physical problems.They play an important role in the study of wave mechanics, heat conduction, electromagnetic theory, quantum mechanics, medicine and mathematical statistics, and so forth [1][2][3][4][5][6][7][8][9][10][11].They provide a natural way to solve, expand, and interpret solutions to many types of important equations.Representation of a smooth function in terms of a series expansion using orthogonal polynomials is a fundamental concept in approximation theory and forms the basis of spectral methods of solution of Fredholm integro-differential equations with functional arguments.Laguerre polynomials   () constitute complete orthogonal sets of functions on the semi-infinite interval [0, ∞).The past couple of decades have seen a dramatic increase in the application of these models to problems in biology, physics, and engineering.In the field of Fredholm integro-differential equation with functional arguments the computation of its solution has been a great challenge and has been of great importance due to the versatility of such equations in the mathematical modeling of processes in various application fields.Mainly, we deal with the following equation: where () is an unknown function, the known functions  ℎ (), (), and   (, ) are defined on an interval [0, ], and also   ,   , and   are appropriate constants.Our aim is to find an approximate solution expressed in the form where   are unknown coefficients,   (),  = 0, 1, . .., are Laguerre polynomials, and  is chosen positive integer.The relation between the powers of  and the Laguerre polynomials   () is To obtain a solution in the form (3) of the problem (1) under conditions (2), we can use the collocation points defined by [12][13][14][15]   =   ,  = 0, 1, 2, . . ., .
The remainder of the paper is organized as follows.Higher order linear mixed functional integro-differential equation with variable coefficients and fundamental relations is presented in Section 2. The method of finding approximate solution and the algorithm for the calculation are described in Section 3. Residual error analysis is described in Section 4.
To support our finding, we present result of numerical experiments using Maple 12 in Section 5. Section 6 concludes this paper with a brief summary.

Fundamental Matrix Relations
Let us write (1) in the form where the functional differential part is Fredholm integral part is We convert these parts and the mixed conditions equation (2) into the matrix form.Let us consider (1) and find the matrix forms of each term of the equation.We first consider the solution () and its derivative  () () defined by a truncated Laguerre series.Then, we can put series in the matrix form where By using (4) and taking  = 0, 1, . . ., , we obtain the matrix relation as where Then, by taking into account (11), we have Moreover, it is clear that the relation between the matrix X() and its derivative X () () is where Hence, by means of ( 13) and ( 14) we have the matrix relation in the form

Matrix Representation for Differential-Difference Parts.
The derivative of the matrix L() that is defined in (13), by using relation (16), can be expressed as By substituting  =  ℎ  +  ℎ into (16) [16,17], we obtain where we have from ( 13) Then, we consider ( 14) and we have where Therefore, by using relation (16), the matrix representation of differential-difference part can be given by Then, the matrix representation of the kernel function   (, ) becomes where K  (, ) = [  ()] ,  = 0, 1, . . ., .
Substituting relations (9) and (24) in the Fredholm part, we obtain where We define [ℎ We also have the matrix relation Q  = HM  H  .Hence, the matrix representation of Fredholm integral part can be given by

Matrix Representation of the Conditions.
Let us define the matrix form of the conditions given by (2) that can be written as Then, we have Briefly, we demonstrate the conditions in matrix form as ,  = 0, 1, 2, . . .,  − 1, where

Residual Correction and Error Estimation
In this section, we give an error estimation for the Laguerre polynomial solution (3) with the residual error function, and it supports the idea of corrected Laguerre polynomial solution with respect to the residual error function.Let us define (1) with  operator as follows: Then, we get the residual function of the Laguerre collocation method as where   () is the approximate solution which is solved via Laguerre polynomials given by (3) of problem (1) with conditions (2).Thus,   () satisfies the problem −1 Also, the error function   () can be defined as where () is the exact solution of problem ( 1) with ( 2) conditions.Then, by using ( 1), ( 2), (45), and (47), we obtain the error differential equation with the homogeneous conditions ()  (0) +    ()  () = 0,  = 0, 1, 2, . . .,  − 1. (51) Briefly, the error problem is By solving problem (52) with the technique introduced in Section 3, we get the approximation 4.1.Algorithm.In this section, we will consider each step of the method which is solved by Maple 12 and we distinguish the algorithms in three steps.Algorithm 1 demonstrates the calculation of approximate solution.Algorithm 2 shows each step of residual error function and its support to corrected error and corrected Laguerre polynomial solution.We plot the graphics of all functions in Algorithm 3.
Algorithm 1.Consider the following.
(1) Input initial data: the coefficients are  ℎ , kernel function is   (, ), and the coefficients of the conditions are   ,   .The problem is defined on [0, 1].
(2) Develop the algorithm by matrix inputs which are constructed by collocation points in (4).
(3) Use the conditions to construct augmented matrix and the system is done.
(4) Output: the system is solved and  unknown matrix is found.Then, approximate solution is found with respect to coefficients of unknown matrix in truncated Laguerre series.
Algorithm 2. Consider the following.

Illustrative Examples
In this section, several numerical examples are given to illustrate the accuracy and the effectiveness of the method.All of them are performed on the computer by using Maple 12.
Example 4 (see [23]).Let us first consider the functional Fredholm integro-differential equations as follows: under the mixed conditions and we seek the approximate solution   () as a truncated Laguerre series: Then, for  = 3, the collocation points are  0 = 0,  1 = 1/2,  2 = 2/3, and  3 = 1 and the problem is defined by and matrices for conditions are ] , and the matrices are computed as Then, we have the augmented matrix  (67) The exact solution of this problem is () =  2 + 1.
Table 1 shows the comparison between exact and approximate solutions for  = 3, 10, 15 values.The results of residual error function support the evaluation of the corrected new Laguerre polynomial solutions for different  and  values.Figure 1 shows us the comparison between exact and approximate solutions for various  values.Figure 2 shows the absolute error and absolute residual error functions.Mainly, this comparison supports the idea of corrected Laguerre polynomial solution for different  and  values.
Table 2 shows the corrected absolute errors by our method for  = 3, 10, 15 and  = 7, 12, 20.The results support the idea that when  and  values are chosen large enough, the absolute error and residual error are decreased.
Example 5 (see [24]).Let us find the Laguerre series solution of the following second order linear functional Fredholm integro-differential equation with variable coefficients: with initial conditions (0) = 4,   (0) = −4 and its exact solution is () = ( − 2) 2 .We obtained the approximate solution of the problem for  = 8, 10, 18 and the matrices are computed.Hence linear algebraic system is gained.This system is approximately solved using Maple 12.
Table 3 shows us the comparison between exact solution, approximate solutions for different  values, and corrected Laguerre polynomial solutions with respect to the residual error analysis.Figure 3 shows us the comparison between exact and approximate solutions for different  values.Figure 5 shows us the comparison between absolute error and absolute residual error graphics.
Table 5 shows the exact solution comparison with approximate solutions for  = 8, 10, 20 with present method.Absolute error comparisons are also given in the same table.Figure 6 shows the comparison between approximate solutions for different  values.We observe the absolute error differences for the same  values in Figure 7.  Example 7 (see [24]).Consider the linear functional Fredholm integro-differential equation with conditions (0) = 1,   (0) = 0 and exact solution () = cos().
Table 6 shows the comparison between exact and approximate solutions and corrected Laguerre polynomial solutions for the different  and  values.Figure 8 shows the comparison of absolute error functions for the same  values.
Table 7 shows the comparison between exact and approximate solutions and corrected Laguerre polynomial solutions for the different  and  values.

Conclusion
In recent years, the studies of linear functional Fredholm integro-differential equation have attracted the attention of many mathematicians and physicists [26][27][28].Laguerre collocation methods are used to solve the linear integrodifferential equation with functional arguments numerically.It is observed that the method has the best advantage when the known functions in equation can be expanded to Laguerre series.Another advantage of the method is that the Laguerre polynomial coefficients of the solution are found very easily by using computer programs.Shorter  of linear functional Fredholm integro-differential equations with variable coefficients and their residual error analysis, but some modifications are required.

Figure 2 :
Figure 2: Comparison of error and residual error functions for  3,7 values for Example 4.

Figure 4 :
Figure 4: Error function of Example 5 for various  values.

Figure 4
Figure 4 shows the comparison of absolute error function of Example 5 for the same  values.Figure5shows us the comparison between absolute error and absolute residual error graphics.Table4shows the corrected absolute errors by our method for  = 8, 10, 18 and  = 10, 12, 22.

Figure 5 :
Figure 5: Comparison of error and residual error functions for  8,10 values.

Table 1 :
Numerical solution of Example 4 for different  values.

Table 3 :
Numerical solution of Example 5 for different  values.

Table 5 :
Numerical solution and absolute error of Example 6 for different  values.

Table 6 :
Numerical solution of Example 7 for different  values.

Table 7 :
Comparison of the actual and estimated absolute errors for  = 12, 22, 25 and  = 15, 28, 28 of Example 7. time and lower operation count result in reduction of cumulative truncation errors and improvement of overall accuracy.To get the best approximating solution of the equation, we have improved the Laguerre collocation method with the aid of the residual error function for solving the linear Fredholm integro-differential equation with piecewise intervals.For this purpose, the power of the employed method is confirmed.As a result, considerable solutions are shown in tables and figures.We assured the correctness of the obtained solutions by putting them back into the original equation with the aid of Maple 12.As a future work, the method can also be extended to the system computation