 CARMA COLLOQUIUM
 Speaker: Prof Robert Corless, University of Western Ontario
 Title: Computation and Application of Mathieu Functions: a Survey from a Historical Point of View
 Location: Room SR118, SR Building (and online via Zoom) (Callaghan Campus) The University of Newcastle
 Time and Date: 4:00 pm, Thu, 25^{th} Mar 2021

Join via Zoom, or join us in person (max room capacity is 9 people).
3:30pm for pretalk drinks + snacks, and 4pm for the talk
You can watch a video version at https://youtu.be/0rEEopdSyQ instead, or in addition!
 Abstract:
A full paper describing this talk can be found at https://arxiv.org/abs/2008.01812. Mathieu functions of period π or 2π, also called elliptic cylinder functions, were introduced in 1868 by Émile Mathieu together with socalled modified Mathieu functions, in order to help understand the vibrations of an elastic membrane set in a fixed elliptical hoop. These functions still occur frequently in applications today: our interest, for instance, was stimulated by a problem of pulsatile blood flow in a blood vessel compressed into an elliptical crosssection. This talk surveys and recapitulates some of the historical development of the theory and methods of computation for Mathieu functions and modified Mathieu functions and identifies some gaps in current software capability, particularly to do with double eigenvalues of the Mathieu equation. We demonstrate how to compute Puiseux expansions of the Mathieu eigenvalues about such double eigenvalues, and give methods to compute the generalized eigenfunctions that arise there. In examining Mathieu's original contribution, we bring out that his use of antisecularity predates that of Lindstedt. For interest, we also provide short biographies of some of the major mathematical researchers involved in the history of the Mathieu functions: Émile Mathieu, Sir Edmund Whittaker, Edward Ince, and Gertrude Blanch.
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 CARMA OANT SEMINAR
 Speaker: Prof Robert Corless, University of Western Ontario
 Title: Highorder, highaccuracy solution of a nonlinear PDE arising in a twodimensional heat transfer model
 Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
 Access Grid Venue: UniSA
 Time and Date: 3:00 pm, Mon, 9^{th} Dec 2013
 Abstract:
A classical nonlinear PDE used for modelling heat transfer between concentric cylinders by fluid convection and also for modelling porous flow can be solved by hand using a loworder perturbation method. Extending this solution to higher order using computer algebra is surprisingly hard owing to exponential growth in the size of the series terms, naively computed. In the mid1990's, socalled "Large Expression Management" tools were invented to allow construction and use of socalled "computation sequences" or "straightline programs" to extend the solution to 11th order. The cost of the method was O(N^8) in memory, high but not exponential.
Twenty years of doubling of computer power allows this method to get 15 terms. A new method, which reduces the memory cost to O(N^4), allows us to compute to N=30. At this order, singularities can reliably be detected using the quotientdifference algorithm. This allows confident investigation of the solutions, for different values of the Prandtl number.
This work is joint with Yiming Zhang (PhD Oct 2013).
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 CARMA COLLOQUIUM
 Speaker: Prof Robert Corless, University of Western Ontario
 Title: First Encounters of a Chebfun Novice
 Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
 Time and Date: 1:00 pm, Tue, 3^{rd} Apr 2012
 Abstract:
Symbolic and numeric computation have been distinguished by definition: numeric computation puts numerical values in its variables as soon as possible, symbolic computation as late as possible. Chebfun blurs this distinction, aiming for the speed of numerics with the generality and flexibility of symbolics. What happens when someone who has used both Maple and Matlab for decades, and has thereby absorbed the different fundamental assumptions into a "computational stance", tries to use Chebfun to solve a variety of computational problems? This talk reports on some of the outcomes.
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