Abstract
Laguerre collocation method 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.
1. 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 [1–11]. 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 . 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: under the mixed conditions where is an unknown function, the known functions , , and are defined on an interval , and also , , and are appropriate constants. Our aim is to find an approximate solution expressed in the form where are unknown coefficients, , , 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 [12–15] 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.
2. 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 , we obtain the matrix relation as whereThen, by taking into account (11), we have Moreover, it is clear that the relation between the matrix and its derivative is where Hence, by means of (13) and (14) we have the matrix relation in the form
2.1. Matrix Representation for Differential-Difference Parts
The derivative of the matrix that is defined in (13), by using relation (16), can be expressed as By substituting into (16) [16, 17], we obtain where we have from (13) Then, we consider (14) and we have whereTherefore, by using relation (16), the matrix representation of differential-difference part can be given by
2.2. Matrix Representation for Fredholm Integral Part
can be expanded by Laguerre series with respect to as follows: Then, the matrix representation of the kernel function becomes where Substituting relations (9) and (24) in the Fredholm part, we obtain where We define We also have the matrix relation . Hence, the matrix representation of Fredholm integral part can be given by
2.3. 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 where
3. Method of Solution
We are ready to construct the fundamental matrix equation corresponding to (1). For this purpose, by substituting the matrix relations (22) and (30) into (1), we obtain Then, we put equally spaced collocation points in (5) and relation (35) obtains Briefly, the fundamental matrix equation is gained as follows: where The fundamental matrix equation (35) for (1) corresponds to a system of algebraic equation for the unknown coefficients [18]. Briefly we can write (37) as so that, for , Then, the matrix form of conditions (2) is where To obtain the solution of (1) under conditions (2), by replacing the rows in matrix (39) by the last m rows of matrix (41), we have the required augmented matrix If , then we can write Thus, the coefficients , , are uniquely determined by (43) [19].
4. 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 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 Briefly, the error problem is By solving problem (52) with the technique introduced in Section 3, we get the approximation to . Consequently, by means of the polynomials and , (), we obtain the corrected Laguerre polynomial solution . Also, we construct the corrected error function and the estimated error function [20–22].
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 , kernel function is , and the coefficients of the conditions are , . The problem is defined on .(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.(5)End of Algorithm 1.
Algorithm 2. Consider the following.(1)Input data: the approximate solution is , the error function , and truncated Laguerre series (3).(2)We check for that equals truncated Laguerre series. Develop the algorithm to demonstrate , the residual function of the Laguerre collocation method, via absolute error function .(3)The conditions are constructed by absolute error function. Then, obtain the estimated error function .(4)Output: the corrected error function is found.(5)End of Algorithm 2.
Algorithm 3. Consider the following.Input data: the comparison between approximate and exact solutions is demonstrated by the graphics.Output: with(plots): plot1 := plot(([y(x)], x = 0..1), style = point, color = magenta, legend = [typeset(“Exact Solution”)]); plot2 := plot (([y(N,M)(x), x = 0..1]), style = line, color = blue, legend = [typeset(“Approximate Solution”)]); display(plot(plot1,plot2)).Input: the comparison between error and residual error solutions is demonstrated by the graphics.Output: with(plots): plot3 := plot(([EN(x)], x = 0..1), style = point, color = magenta, legend = [typeset(“Error Function”)]); plot4 := plot(([E(N,M)(x), x = 0..1]), style = line, color = blue, legend = [typeset(“Residual Error Function”)]); display(plot(plot3,plot4)).End of Algorithm 3.
5. 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 1 (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: Here, Then, for , the collocation points are , , , and and the problem is defined by and matrices for conditions are and the matrices are computed as Then, we have the augmented matrix Subsequently, we have and we obtain the approximate solution of the problem for as We follow the same steps for and : and for The exact solution of this problem is .
Table 1 shows the comparison between exact and approximate solutions for , , 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 , , and , , . The results support the idea that when and values are chosen large enough, the absolute error and residual error are decreased.
Example 2 (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 , and its exact solution is . We obtained the approximate solution of the problem for , , 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 4 shows the comparison of absolute error function of Example 2 for the same values.
Figure 5 shows us the comparison between absolute error and absolute residual error graphics.
Table 4 shows the corrected absolute errors by our method for , , and , , .
Example 3 (see [25]). Let us find the Laguerre series solution of the following second order linear functional Fredholm integro-differential equation with variable coefficients: with initial conditions and and its exact solution is. We obtained the approximate solution of the problem for, , and which are tabulated and graphed.
Table 5 shows the exact solution comparison with approximate solutions for , , 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 4 (see [24]). Consider the linear functional Fredholm integro-differential equation with conditions , and exact solution .
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.
6. Conclusion
In recent years, the studies of linear functional Fredholm integro-differential equation have attracted the attention of many mathematicians and physicists. [26–28]. Laguerre collocation methods are used to solve the linear integro-differential 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 computation 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 of linear functional Fredholm integro-differential equations with variable coefficients and their residual error analysis, but some modifications are required.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.