Chebyshev Polynomials for Solving a Class of Singular Integral Equations

This paper is devoted to studying the approximate solution of singular integral equations by means of Chebyshev polynomials. Some examples are presented to illustrate the method.


Introduction
During the last three decades, the singular integral equation methods with applications to several basic fields of engineering mechanics, like elasticity, plasticity, aerodynamics and fracture mechanics have been studied and improved by several scientists (see [1]- [6]).Hence, it is interesting to solve numerically this type of integral equations (see [7] [8]).Chebyshev polynomials are of great importance in many areas of mathematics particularly approximation theory (see [9] [10]).
In this paper we analyze the numerical solution of singular integral equations by using Chebyshev polynomials of first, second, third and fourth kind to obtain systems of linear algebraic equations, these systems are solved numerically.The methodology of the present work expected to be useful for solving singular integral equations of the first kind, involving partly singular and partly regular kernels.The singularity is assumed to be of the Cauchy type.The method is illustrated by considering some examples.
Singular integral equation of first kind, with a Cauchy type singular kernel, over a finite interval can be represented by ( ) ( ) ( ) ( ) ( ) where ( ) ( ) , , , k t x L t x and ( ) f x are given real-valued continuous functions belonging to the class Holder of continues functions and ( ) , 0 k t t ≠ .In Equation (1.1) the singular kernel is interpreted as Cauchy principle value.Integral equation of form (1.1) and other different forms have many applications (see [1] [2] [6] [11] [12]).The theory of this equation is well known and it is presented in [13] [14].An approximate method for solving (1.1) using a polynomial approximation of degree n has been proposed in [7].
It is well known that the analytical solutions of the simple singular integral equation ( ) ( ) k t x = and ( ) , 0 L t x = , for the following four cases: 1) The solution is unbounded at both end-points 2) The solution is bounded at both end-points 3) The solution is bounded at end 1, The solution is unbounded at end 1, x = − but bounded at end 1 x = + , are given by [15].In this paper the used approximate method for solving Equation (1.1) stems from recent work [10] wherein an approximate method has been developed to solve the simple Equation (1.2).The approximate method developed below appears to be quite appropriate for solving the most general type Equation (1.1).Some examples are presented to illustrate the method.

The Approximate Solution
In this section we present the method of the approximate solution of Equation (1.1) in four cases.
 are the corresponding weight functions.Substituting the approximate solution (2.1) for the unknown function into (1.1)yields In above Equation (2.6), we next use the following Chebyshev approximation to the kernels ( ) ( ) , L t x , given by (for fixed x , cf. [7]) with known expressions for ( ) p K x and ( ) q L x .Then (2.6) gives where , ,

Numerical Examples
In this section, we consider some problems to illustrate the above method.All results were computed using FORTRAN code.
Example 1 Consider the following singular integral equation where ( ) ( ) ( ) Hence we find that relation (2.8) produces Firstly, let us consider in detail the case (I), By applying the following relations It is easy to estimate the values ( ) ( ) ( ) , , By choosing the collocation points we obtain the following system of linear equations: ∑ By solving this system for the unknown coefficients ( ) is given by Table 1.Secondly, let us consider in detail the case (II), 2 j = , for 3 n = .This results in By applying the following relations ( ) ( ) It is easy to estimate the values ( ) ( ) ( ) , , From the relations (2.3) and (3.1.9)-(3.1.12)we get , n = we obtain the following system of linear equations : ∑ By solving this system for the unknown coefficients is given by Table 1.Thirdly, let us consider in detail the case (III), 3 j = , for 3 n = .This results in By applying the following relations ( ) ( ) It is easy to estimate the values ( ) ( ) ( ) , , By choosing the collocation points ( ) ( ) ( ) we obtain the following system of linear equations : ∑ By solving this system for the unknown coefficients ( ) By applying the relations ( ) ( ) It is easy to estimate the values ( ) ( ) ( ) we obtain the following system of linear equations :   ( ) ( ) which corresponds with ( ) Hence we find that relation (2.8) produces From the relations (3.1.2)-(3.1.5)and (3.2.2) we obtain By choosing the collocation points we obtain the following system of linear equations : ∑ By solving this system for the unknown coefficients ( )  2.
Secondly, let us consider in detail the case (II), 2 j = , for 3 n = .This results in ( ) ( ) By applying the relations (3.1.9)-(3.1.12)and (3.2.6) we get , n = we obtain the following system of linear equations: we obtain the following system of linear equations: ∑ By solving this system for the unknown coefficients where (2.9) gives   is given by Table 3.
into (2.1)we obtain the approximate solutions of Equation (1.1).
1.7) we obtain the approximate solution of Equation (3.1.1)in the form Which coincides with the exact solution.The error of approximate solution (3.1.8) of Equation (3.1.1)at 20 n =

From ( 3 . 1 . 15 )
14) we obtain the approximate solution of Equation (3.1.1)in the form Which coincides with the exact solution.The error of approximate solution (3.1.15) of Equation (3.1.1)at 20 n =

From ( 3 . 1 .
28) we obtain the approximate solution of Equation (3.1.1)in the form of

From ( 3 . 2 . 4 )
we obtain the approximate solution of Equation (3.2.1) in the form of the exact solution.The error of approximate solution (3.2.5) of equation (3.2.1) at 20 n = is given by Table

From ( 3 . 2 . 8 )
we obtain the approximate solution of Equation (3.2.1) in the form of ( )

4 )
So the approximate solution of Equation (3.3.1) is given by the exact solution, the error of the approximate solution (3.3.5) of Equation(3.3.

7 )
By solving the system (3.3.2), at the collocation points ( ) the exact solution, the error of the approximate solution (3.3.9) of Equation (3.3.1) at 20 n =

Table 2 .
Illustrates errors of approximate solutions of Equation (3.2.1) in Case (I), Case (II) and Case (IV) respectively at n = 20.

Table 2 .
Similarly, doing the same operations as we did for Case (I), Case (II) and Case (IV), one can solve for Case (III) .

Table 3 .
Illustrates errors of approximate solutions of Equation (3.3.1) in Case (II) and Case (III) at n = 20.