On the Numerical Solution of Singular Integral Equation with Degenerate Kernel Using Laguerre Polynomials ()
1. Introduction
Recent years, there has been a growing interest in the Fredholm and Volterra integral equations. This is an important branch of modern mathematics and arises frequently in many applied areas which include engineering, mechanics, physics, chemistry, astronomy, biology [1] [2] . There are several methods for approximating the solution of linear and non-linear integral equations [3] - [8] . We consider the singular integral equation of the second kind with smooth kernel:
(1)
where
is a continuous function for
and the kernel
is a function defined on the domain
and
is the unknown function that will be determined. In [9] , Laguerre polynomials are used to derive numerical Solutions of Volterra integral Equations. In [10] , repeated Simpson’s and Trapezoidal quadrature rule was used to solve the linear Volterra Integral equations of the second kind. Since the integral equation is called singular when one or both limits of integration become infinite or when the kernel becomes infinite at one or more points in the domain of the integration. In our study, we are interested in the case where one limits of integration become infinite. For solving singular integral equations, many methods with enough accuracy and efficiency have been used before much research, see [8] - [13] . In [14] , the author uses Toeplitz matrices method as numerical method to solve a singular integral equation, where many definite integrals cannot be computed in closed form, and must be approximated numerically. In [1] , the orthogonal polynomials are used to solve numerically Nonlinear Volterra Fredholm Integral Equations. In [13] , the Legendre and Chebyshev collocation method is presented to solve numerically the Voltterra-Fredholm Integral Equations with singular kernel. In this paper, we use numerical technique based on projection method, to reduce the singular integral Equations to a linear system of algebraic equations which will be solved using Gauss elimination or iterative methods. The paper is organized as follows. In section 2, we recall some properties related to Laguerre polynomials. In section 3, a system of algebraic equations will be presented based on Laguerre polynomials. In Section 4, we present a strategy to compute the exact solution for a singular integral equation with degenerate kernel. In section 5, we give a practical example to certify the validity of the proposed technique and then we conclude.
2. Laguerre Method
Sequences of orthogonal polynomials appear frequently used as applications in mathematics, mathematical physics, engineering and computer science, in particular during the resolution of partial differential equations (Laplace, Schrödinger) by the method of separation of variables, also these polynomials can be used to solve integral equations of first and second kind [1] [11] . One of the most common set of orthogonal polynomials is the Laguerre polynomials. Many families of orthogonal polynomials are known, which have in common a certain number of simple properties. The Laguerre differential equation is given by
The solutions of this equation are the Laguerre polynomials, expressed by the equation following differential:
Then we get an approximation of the exactly integral, let say:
(2)
is an approximation of the exact integral. This type of approximation must be chosen so that the integral (2) can be evaluated (either explicitly or by an efficient numerical technique). The functions
will be called interpolating elements. In this paper, the interpolating function
will be assumed to be the interpolating polynomial
(3)
where
are Laguerre polynomials of degree j, n is the number of Laguerre polynomials, and
are unknown parameters, to be determined.
3. System of Algebraic Equations
Consider the following systems
where
This system forms an orthonormal basis in
. In fact, we check that
The previous system is called the Laguerre polynomial system. To solve the integral Equation (1) we use the projection method. Using an approximation
of the solution of Equation (1) which is a finite linear combination of orthogonal polynomials
and also solution of the integral equation
(4)
By taking the linear combination of the Laguerre polynomials
(5)
Substituting (5) in (4) we get
(6)
Let
Then equation (6) can be written in the form
(7)
By multiplying (7) by
, we get
(8)
Using the orthogonalization condition in Equation (8) we get
(9)
The system of Equation (9) has a unique solution if
, where
this makes it possible to determine the coefficients
.
4. Exact Solution with Degenerate Kernel
Given a degenerate kernel
then Equation (1) becomes
(10)
Let c be the number defined by
(11)
Therefore
(12)
substituting (12) into (11) gives
(13)
from (13) and (12) we get:
(14)
provided that
Note that, using some degenerate kernel one can compute exactly the integral to obtain an exact solution. Sometimes
can not be evaluated exactly, for that one use quadrature rule to approximate the integral [14] .
5. Numerical Examples
To confirm the validity, the accuracy and support our theoretical presentation of the proposed method, we give some computational examples. The computations, associated with the examples are performed by MATLAB. The system of algebraic system will be solved using Gauss elimination and iterative schemes will also be applied for large system.
5.1. First Example
In this example, we consider
and
; here we have
and
according to equation (14) we have
Considering
so the exact solution of Equation (1) is
.
We used both Gauss elimination method, and SOR method iterations to solve the linear system. To plot the solution, we truncate the spacial domain to [0; 5]. The behavior of exact and numerical solution is presented in Figure 1(a). The absolute error is depicted in Figure 1(b). It is noticed that convergence to the exact solution need at least 20 Laguerre polynomials.
5.2. Second Example
Let
and
here we have
and
and by (14), the exaction solution is
the Gaussian integral gives
Figure 1. Analytical and numerical solution with n = 30. The absolute error
.
by integration by parts, we have
so the exact solution is
. For n = 6 the approximate solution
with
the coefficients
are the solution of the system
To plot the solution, we truncate the spacial domain to [0; 1]. The behavior of exact and numerical solution is depicted in Figure 2. Note that in example 2, the solution converge to the exact solution using only few Laguerre polynomail (n = 4).
Figure 2. Example 2, case of n = 5. Comparison between exact solution and numerical solution.
6. Conclusion
A very simple and efficient method based on the Laguerre polynomial basis has been developed to solve singular integral equations. The result obtained confirms the strategy proposed. The method presented is tested and confirmed by two examples. A few numbers of Laguerre polynomials are needed to get convergence to the exact solution. Further, one can apply the proposed method to more general kernels or systems of integral equation.