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Rigorous Numerics for PDEs with Indefinite Tail : Existence of a Periodic Solution of the Boussinesq Equation with Time-dependent Forcing

Czechowski A., Zgliczyński P.

Schedae Informaticae

2015

Vol. 24

143--158

We 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.

Cr - Lohner algorithm

Wilczak D., Zgliczyński P.

Schedae Informaticae

2011

Vol. 20

9-42

We 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.