Numerical solution for the Fredholm integral equation of the second kind with Toeplitz kernels by using preconditioners

In this paper, first we consider the Fredholm integral equations of the second kind with Toeplitz kernel for examplesk(s,t)=12|s-t|,cos(s-t)1+|s-t|,sin|s-t|1+|s-t|,11+|s-t|,1(1+|s-t|)2,…Then by using quadrature method the integral equations reduce to a Block Toeplitz system. The condition number of the coefficient matrix of the system is large. We use preconditioners to decrease the condition number of the new system. In the case that the coefficient matrix of above system is positive definite, we can use CG method for solving the system.