Abstract
We study Legendre polynomials and develop new operational matrix of integration. Based on the operational matrix, we develop a new method to solve a coupled system of Fredholm integral equations of the form , , where , , , and are real constants and . The method reduces the coupled system to a system of easily solvable algebraic equations without discretizing the original system. As an application, we provide examples and numerical simulations demonstrating that the results obtained using the new technique match very well with the exact solutions of the problems. To show the efficiency of the method, we compare our results with some of the results already studied with other available methods in the literature.
1. Introduction
Fredholm integral equations are frequently encountered in many physical processes such as dynamic stiffness of rigid rectangular foundations [1], soil mechanics and rock mechanics [2], diffraction of waves by randomly rough surface in two dimensions [3], thermoelasticity [4], and scattering problem [5], to name a few. For systems of such equations, various techniques such as extrapolation method, Galerkin discretization, collocation methods, and quadrature, iterative, spline, orthogonal polynomial, and multiple grid methods have been proposed to determine desired solutions (see, e.g., [6–9] and the references quoted there). These methods include approximate analytical and numerical approaches.
Recently, approximate solutions to system of integral equations have attracted the attention of many authors and they obtained solutions using various available techniques in the literature. For example, system of integral equations has been studied with wavelets techniques in [10, 11], with Adomian decomposition method in [12, 13], with Tau method in [14], with chebesheve polynomial and block pulse function in [15, 16], and with Taylor expansion and some modified methods based on taylor series expansion in [17–25].
In this paper, we use shifted Legendre polynomials and develop a new operational matrix of integration. Based on the operational matrix of integration, we develop a simple method to find solutions of the coupled system of Fredholm integral equations. The method reduces the coupled system to a system of easily solvable algebraic equations without discretizing the original system of equations. Besides simplicity, the method yields accurate results even for small value of resulting in the reduction of the system to small system of algebraic equations. It is verified by examples and their numerical simulations demonstrating that the results obtained using the new technique match very well with the exact solutions of the problems. To show the efficiency of the method over some of the well-known techniques, we compare our results with some of the results already studied with other available methods such as Taylor series approximation method [19] and block pulse method [16]. We find that the new techniques provide highly accurate solutions as compared to Taylor series approximation method and block pulse method.
2. Main Results: New Operational Matrix of Integrations
The Legendre polynomials defined on are given by the following recurrence relation: The transformation transforms the interval to and the polynomials transformed to the so called shifted Legendre polynomials given as [26] follows: where , . The orthogonality condition is
Consequently, any can be approximated by shifted Legendre polynomial as follows: In vector notation, we write where , is the coefficient vector, and is terms vector function. In case of function of two variables, that is, , we write The orthogonality condition of is found to be In vector notation, (6) can be written as where and are column vectors containing Legendre polynomial and is the coefficient matrix whose entries are obtained by using (6).
2.1. Error Analysis
For sufficiently smooth function on , the error of the approximation is given by where We refer the reader to [27] for the proof of the above result.
Lemma 1. Let and ; then where is the Legendre coefficient vector of and the matrix , where .
Proof. In view of (5) and (6), we have Using (12), we obtain which implies that Using the orthogonality relation, we get where . In matrix form, we have
3. System of Fredholm Integral Equations
Consider the following coupled system of Fredholm integral equations: where , , , and are real constants, , , and , are unknown functions to be determined. Approximating and in terms of Legendre polynomials, we obtain Using Lemma 1, we have the following approximations: Using (18) and (19) in the coupled system (17), we obtain the following system of algebraic equations which can be written as The transpose of the above system is given by which can further be written as where Hence it follows that which is a generalized Sylvester type equation and can easily be solved for the unknown and by any computational software.
4. Illustrative Examples
Example 1. Consider the following system of Fredholm integral equation: The exact solutions of the system are and . The solutions (, ) obtained via our technique for (small enough) are compared with the exact solutions of the problem in Figure 1, where dots represent the exact solutions and the curves are for the solutions obtained via the new method. From Figure 1, it follows that our solutions matchs very well with the exact solution of the problem even for small value , which shows the effectiveness of our technique.
Example 2. For comparison purposes, consider the following coupled system of Fredholm integral equations: The exact solutions of the system are and . We obtain the approximate solutions of the system for different values of and compare the results with the exact solutions of the system. For and , the comparison is shown in Figure 2, where dots represent the exact solutions of the system and doted curves (red and yellow) represent the approximate solution ( and ) obtained via our technique for while Blue and orange dots represent the approximate solutions ( and ) obtained via our technique for . It is clear that the approximate solutions approach rapidly the exact solutions as the values of increase. It also shows that the approximate solutions are very close to the exact ones for . For example, error of approximation in both (red doted curve) and (blue doted curve) is less than for as shown in Figure 3, which is much more acceptable number and demonstrates high accuracy of the new technique. Further, we compare our results with some other available results in the literature. We compare the absolute errors (red line) with the absolute error obtained in [19] using Taylor series approximation and also with absolute error obtained in [16] using numerical solution with block pulses. The results are shown in Figures 4 and 5. From these analyses, it is clear that the absolute error in our method even for small value of is much smaller than those obtained in [16, 19] even for much larger values of such as . It is a clear indication that the new techniques provide highly accurate solutions as compared to Taylor series approximation method and block pulse method.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.