A general approach for rigorously integrating PDEs using semigroup theory

Akitoshi Takayasu
University of Tsukuba

In this talk we introduce a general rigorous PDE integrator that proves the existence of a solution to the Cauchy problem of time-dependent PDEs. We derive a fixed-point formulation to prove the existence of a solution locally in time, which is based on the solution map of a linearized problem called evolution operator. Using rigorous numerics we validate the contraction of the fixed-point form on a neighborhood of a numerically computed approximate solution. Then we extend the time interval to exist the solution via time stepping. The main advantage of our approach is that the rigorous integrator can be applied to a general class of PDEs, even performed for higher spatial dimensional PDEs.
This is joint work with Jean-Philippe Lessard and Gabriel Duchesne.