An efficient algorithm for solving Hilbert type singular integral equations of the second kind

A simple and efficient method for solving Hilbert type singular integral equations of the second kind is given. In order to slide over the singularity of the equation, a transform is made. By improving the traditional reproducing kernel method, which requests that the image space of operator is W21 and that the operator is bounded, the exact solution and the approximate solution of Hilbert type singular integral equations of the second kind are presented. The advantage of the approach lies in the fact that, on one hand, the approximate solution gn(x)gn(x) is continuous. On the other hand, gn(x)gn(x) and gn′(x) converge uniformly and rapidly to the exact solution g(x)g(x) and its derivatives g′(x)g′(x) respectively. Numerical experiments show the efficiency of our method.