Half-bounded Numerical Solution of Singular Integral Equations with Cauchy Kernel

Problem statement: In this study, a numerical solution for singular integral equations of the first kind with Cauchy kernel over the finite segment [-1,1] is presented. The numerical solution is bounded at x =1 and unbounded at x = -1. Approach: The numerical solution is derived by approximating the unknown density function using the weighted Chebyshev polynomials of the fourth kind. Results: The force function is approximated by using the Chebyshev polynomials of the third kind. Conclusion: The exactness of the numerical solution is shown for characteristic equation when the force function is a cubic.


INTRODUCTION
Consider the Cauchy type singular integral equations of the first kind: where K(x, t) and f (x) are assumed to be real-valued functions belong to the class of Hölder on the sets is unknown function to be determined. The theories and applications of singular integral equations are described in many references as well as in Gakhov (1990), Ladopoulos (2000), Lifanov (1996) and Nandhakumar et al. (2009).
The characteristic singular integral equation is of the form (Lifanov (1996): It is well known that the analytical solution of characteristic equation (2) which is bounded at the endpoint x = 1 is given by the following formula: By solving equation (1) with respect to its characteristic part, we will find that it is equivalent to the Fredholm equation of the second kind: in the sense of obtaining the solution, which one can apply the Fredholm's theorems. Mahiub et al. (2009) presented a numerical solution of equation (1) which is unbounded at both the endpoints x = ± 1. They used Chebyshev polynomials of the first kind with the corresponding weight function to approximate the unknown density function. They proved that the numerical solution of characteristic equation is identical to the exact solution when the force function is a cubic. Abdulkawi et al. (2009a) discussed a numerical solution of equation (1) which is bounded at the both endpoints x = ± 1. They used Chebyshev polynomials of the second kind with the corresponding weight function to approximate the density function. They showed that the numerical solution of characteristic equation is identical with the exact solution when the force function is a cubic. Abdulkawi et al. (2009b) presented an approximate solution of equation (1) which is bounded at the endpoint x = -1, but unbounded at the endpoint x = 1. They used Chebyshev polynomials of the third kind with the corresponding weight function to approximate the unknown density function. They proved that the approximate solution of characteristic equation is identical to the exact solution when the force function is a cubic.
In this study, we present a numerical solution of equation (1) which is bounded at the endpoint x = 1 and unbounded at the endpoint x = -1.

MATERIALS AND METHODS
It is known that (Mason and Handscomb, 2003): Interpolating the known force function f (x) as follows: Where: Approximating the unknown density function φ by n φ , in which: where the unknown coefficients { } n j 0 b are to be determined.

Exactness of the numerical solution:
Substituting (17) into (2), we have: Substitution (17) into (8) The Chebyshev recurrence relations of the third and fourth kinds, respectively, are: and: With help of (20) and (21) It is known that the Chebyshev polynomials of the third and fourth kinds are orthogonals with respect to the weight functions 1 ω and 2 ω , respectively, i.e.: From the system (16) where K(x, t) = 0 yields: Substituting (26) into (9) for n = 3 , we get: Thus the approximate solution of equation (18) In order to obtain the exact solution of equation (18), we substitute (17) It is easy to see that: From (29)-(30) we obtain the exact solution of equation (18) which is identical to the approximate solution given by (28). The proof is complete.

RESULTS AND DISCUSSION
In this section, we discuss a particular example to show the accuracy of our numerical solution.
Let us consider the integral equation (1) and we seek the solution of this equation, which is bounded at the endpoint x = 1 and unbounded at the endpoint x = -1. Due to (8), we have: It is easy to see that: 10(W (t) V (t)) .

CONCLUSION
The Chebyshev orthogonal polynomials of the third and fourth kinds are used to approximate the known force function and unknown density function, respectively. Proposition 1 shows the exactness of the numerical solution for characteristic equation when the force function is a cubic. Particular results show that our numerical solution does not only give the exact solution for characteristic equation but also for other Cauchy singular integral equations.