On mixed collocation methods for Volterra integral equations with periodic solution (Q1360536)

The authors consider linear Volterra integral equations of the form \[ y(x)= f(x)+ \int_{-\infty}^X K(x-s)y(s)ds, \quad 0\leq x\leq X, \qquad\text{with }y(x)= \psi(x) \text{ in }-\infty<x\leq 0, \] where \(K\in L^1(0,X)\), \(f\) is a continuous, periodic function and \(\psi(x)\) is a given bounded and continuous function. After the existence and uniqueness proof for periodic solutions, a mixed (trigonometric + polynomial term) collocation method is described as an extension of results for Volterra integro-differential equations. A convergence analysis for the exact (not discretized) method, with results for the order of convergence follows. Finally, one finds numerical results for a test example with comparison of polynomial and mixed collocation methods based on Lobatto and Gauss points as collocation parameters.