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Applied and Computational Math Seminar

Computer-assisted proofs for differential equations with non-polynomial nonlinearities via the FFT.

Jean-Philippe Lessard (McGill University)

Location:  Hill 425
Date & time: Friday, 19 April 2024 at 1:00PM - 2:00PM

This presentation introduces a methodology for generating computer-assisted proofs (CAPs) to establish the existence of solutions for nonlinear differential equations with non-polynomial analytic nonlinearities. Our approach integrates the Fast Fourier Transform (FFT) algorithm with interval arithmetic and a Newton-Kantorovich argument to construct CAPs effectively. Notably, to rigorously manage the Fourier coefficients of the nonlinear term Fourier series, we leverage insights from complex analysis and the Discrete Poisson Summation Formula. We showcase the applicability of our method through two examples: firstly, verifying the existence of periodic orbits in the Mackey-Glass (delay) equation, and secondly, proving the existence of periodic localized traveling waves in the two-dimensional suspension bridge equation.