Validated enclosures of Fourier coefficients in Banach spaces of analytic functions

CDSNS Colloquium
Friday, May 10, 2024 - 3:30pm for 1 hour (actually 50 minutes)
Skiles 254
Jean-Philippe Lessard – McGill University – jp.lessard@mcgill.ca
Alex Blumenthal

Please Note: Streaming available via Zoom:

This presentation introduces a methodology for generating computer-assisted proofs (CAPs) aimed at establishing the existence of solutions for nonlinear differential equations featuring non-polynomial analytic nonlinearities. Our approach combines the Fast Fourier Transform (FFT) algorithm with interval arithmetic and a Newton-Kantorovich argument to effectively construct CAPs. A key highlight is the rigorous management of Fourier coefficients of the nonlinear term Fourier series, achieved through insights from complex analysis and the Discrete Poisson Summation Formula. We demonstrate the effectiveness of our method through two illustrative examples: firstly, proving the existence of periodic orbits in the Mackey-Glass (delay) equation, and secondly, establishing the existence of periodic localized traveling waves in the two-dimensional suspension bridge equation.

This is joint work with Jan Bouwe van den Berg (VU Amsterdam, The Netherlands), Maxime Breden (École Polytechnique, France) and Jason D. Mireles James (Florida Atlantic University, USA)