A new method for solving Cauchy type singular integral equations of the second kind

The exact solution and the approximate solution of Cauchy type singular integral equations of the second kind are given. In order to remove the singularity of the solution at the endpoints and the Cauchy singularity, a transform is used. By improving the traditional reproducing kernel method, which requests the image space of the operator is W¹₂ [-1,1] and the operator is bounded, the exact solution of Cauchy type singular integral equations of the second kind is given. The advantage of the approach lies in the fact that, on the one hand, the bounded approximate solution g _n(x) is continuous; on the other hand, g_n(x) and g _n′(x), g_n″(x) converge uniformly to the bounded exact solution g(x) and its derivatives g′(x), g″(x), respectively. Some numerical experiments show the efficiency of our method.