A practical method for singular integral equations of the second kind

A convenient and efficient numerical method is presented for the treatment of Cauchy type singular integral equations of the second kind. The solution is achieved by splitting the Cauchy singular term into two parts, allowing one of the parts to be determined in a closed-form while the other part is evaluated by standard Gauss–Jacobi mechanical quadrature. Since the Cauchy singularity is removed after this manipulation, the quadrature abscissas and weights may be readily available and the placement of the collocation points is flexible in the present method. The method is exact when the unknown function can be expressed as the product of a fundamental function and a polynomial of degree less than the number of the integration points. The proposed algorithm can also be extended to the case where the singularities are complex and is found equally effective. The proposed algorithm is easy to implement and provides a shortcut for programming the numerical solution to the singular integral equation of the second kind.