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.