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UID:e892d0476c13db855fb3a7656c4eb2af
CATEGORIES:Applied and Computational Math Seminar
CREATED:20240416T221749
SUMMARY:Computer-assisted proofs for differential equations with non-polynomial nonlinearities via the FFT.
LOCATION:Hill 425
DESCRIPTION:This presentation introduces a methodology for generating computer-assisted
  proofs (CAPs) to establish the existence of solutions for nonlinear differ
 ential equations with non-polynomial analytic nonlinearities. Our approach 
 integrates the Fast Fourier Transform (FFT) algorithm with interval arithme
 tic and a Newton-Kantorovich argument to construct CAPs effectively. Notabl
 y, to rigorously manage the Fourier coefficients of the nonlinear term Four
 ier series, we leverage insights from complex analysis and the Discrete Poi
 sson Summation Formula. We showcase the applicability of our method through
  two examples: firstly, verifying the existence of periodic orbits in the M
 ackey-Glass (delay) equation, and secondly, proving the existence of period
 ic localized traveling waves in the two-dimensional suspension bridge equat
 ion. 
X-ALT-DESC;FMTTYPE=text/html:<div class="x_elementToProof" style="border: 0px; font-style: inherit; font
 -variant: inherit; font-weight: inherit; font-size: 15px; line-height: inhe
 rit; font-family: Calibri, Helvetica, sans-serif; margin: 0px; padding: 0px
 ; vertical-align: baseline; color: #000000; text-align: left; text-indent: 
 0px;">This presentation introduces a methodology for generating computer-as
 sisted proofs (CAPs) to establish the existence of solutions for nonlinear 
 differential equations with non-polynomial analytic nonlinearities. Our app
 roach integrates the Fast Fourier Transform (FFT) algorithm with interval a
 rithmetic and a Newton-Kantorovich argument to construct CAPs effectively. 
 Notably, to rigorously manage the Fourier coefficients of the nonlinear ter
 m Fourier series, we leverage insights from complex analysis and the Discre
 te Poisson Summation Formula. We showcase the applicability of our method t
 hrough 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.</div><div class="x_elementToProof" style="border: 0px; font-styl
 e: inherit; font-variant: inherit; font-weight: inherit; font-size: 15px; l
 ine-height: inherit; font-family: Calibri, Helvetica, sans-serif; margin: 0
 px; padding: 0px; vertical-align: baseline; color: #000000; text-align: lef
 t; text-indent: 0px;">&nbsp;</div>
CONTACT:Jean-Philippe Lessard (McGill University)
DTSTAMP:20260831T164448
DTSTART;TZID=America/New_York:20240419T130000
DTEND;TZID=America/New_York:20240419T140000
SEQUENCE:0
TRANSP:OPAQUE
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