Smoothing the solutions of history-dependent dynamical systems

Abstract

We introduce and analyze a fully discrete numerical method for solving second order integrodifferential equations modelling dynamical systems whose evolution depends also on the past states. We show it is fourth order convergent. Then from the numerical approximate values for the unknown function, we reconstruct by means of deficient quartic splines a smooth solution of the problem. The reconstruction scheme is shown to be cubically convergent. Both methods when assuming analytic evaluation of the integrals are convergent with superquartic and supercubical rates respectively. For ordinary differential equations the scheme for the calculation of the solution at the grid knots gains one order of convergence. Numerical experiments confirm our theoretical findings.