In this paper, we study the solution of finite section of integral equation of the second kind defined on the half line [0,) by the preconditioned conjugate gradient (PCG) method. We would like to emphasize that is not a compact operators and that the rate of convergence cannot be expected to be superlinear.We construct three different circulant integral operators to be used as preconditioner for the method to speed up its convergence rate. we prove that if the given integral operator is close to a convolution type integral operator the PCG method will converge superlinearly. Finally, numerical results are given which support the theoretical results.
