Piotr Zgliczyński
Schedae Informaticae, Volume 20, 2011, pp. 9-42
https://doi.org/10.4467/20838476SI.11.001.0287We present a Lohner type algorithm for the computation of rigorous bounds for the solutions of ordinary differential equations and its derivatives with respect to the initial conditions up to an arbitrary order.
Piotr Zgliczyński
Schedae Informaticae, Volume 24, 2015, pp. 143-158
https://doi.org/10.4467/20838476SI.15.014.3486We consider the Boussinesq PDE perturbed by a time-dependent forcing. Even though there is no smoothing effect for arbitrary smooth initial data, we are able to apply the method of self-consistent bounds to deduce the existence of smooth classical periodic solutions in the vicinity of 0. The proof is non-perturbative and relies on construction of periodic isolating segments in the Galerkin projections.