Fast preconditioned iterative methods for convolution-type integral equations

We consider solving the Fredholm integral equation of the second kind with the piecewise smooth displacement kernel x(t) + ∑^m_j=1μ_jx(t - t_j) + ∫^τ₀ k(t - s)x(s) ds = g(t), 0 ≤ t ≤ τ, where t_j ∈ (-τ, τ), for 1 ≤ j ≤ m. The direct application of the quadrature rule to the above integral equation leads to a non-Toeplitz and an underdetermined matrix system. The aim of this paper is to propose a numerical scheme to approximate the integral equation such that the discretization matrix system is the sum of a Toeplitz matrix and a matrix of rank two. We apply the preconditioned conjugate gradient method with Toeplitz-like matrices as preconditioners to solve the resulting discretization system. Numerical examples are given to illustrate the fast convergence of the PCG method and the accuracy of the computed solutions.