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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 so-called 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 cross-section. 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 anti-secularity 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.

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 low-order 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 mid-1990's, so-called "Large Expression Management" tools were invented to allow construction and use of so-called "computation sequences" or "straight-line 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 quotient-difference 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).

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.