Fast preconditioned iterative methods for convolution-type integral equations (Q1577723)

The authors study the numerical solution of the Fredholm-type integral equation \[ x(t)+ \sum^m_{j=1} \mu_jx(t- t_j)+ \int^\tau_0 k(t- s)x(s) ds= 0,\quad t\in [0,\tau].\tag{1} \] Here, \(k\) and \(g\) are piecewise smooth; the points of discontinuity of \(k\) are supposed to be at \(t_j\in [-\tau,\tau]\), and \(x(t)= 0\) for \(t\not\in[0,\tau]\). The proposed numerical scheme has the property that the discretization matrix characterizing the approximating operator equation has Toeplitz structure. The resulting equation is then solved by using the preconditioned conjugate gradient method with Toeplitz-like preconditioners. Three numerical examples illustrate the accuracy and the performance of this method.