Małgorzata Moczurad, Piotr Zgliczyński, Włodzimierz Zwonek
We give an improved lower bound for the error of any quadrature computing $\int_{-1}^1 f(x) dα(x)$ of analytic functions bounded in the neighborhood of $[-1,1]$.
Daniel Wilczak, Piotr Zgliczyński
The Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions and with parameter $ν=0.1212$ is considered. We give a computer-assisted proof the existence of symbolic dynamics and countable infinity of periodic orbits with arbitrary large periods.
Małgorzata Moczurad, Piotr Zgliczyński
We consider the algorithm for verified integration of piecewise analytic functions given by Petras. The analysis of the algorithm contained in Patras' paper is limited to a narrow class of functions and gives upper bounds only. We present an estimation of the complexity (measured by a number of evaluations of an integrand) of the algorithm, both upper and lower bounds, for a wider class of functions. We show examples with complexity $Θ(|\ln\eps|/\eps^{p-1})$, for any $p >1$, where $\eps$ is the desired accuracy of the computed integral.
Jacek Cyranka, Piotr Zgliczyński
We prove the existence of globally attracting solutions of the viscous Burgers equation with periodic boundary conditions on the line for some particular choices of viscosity and non-autonomous forcing. The attract- ing solution is periodic if the forcing is periodic. The method is general and can be applied to other similar partial differential equations. The proof is computer assisted.