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Simulating maths of the real world
No full textAPPLIED PARTIAL DIFFERENTIAL EQUATIONS. first edition. By John Ockendon. Sam Howison, Andrew Lacey and Alexander Movchan. Oxford University Press. 427pp. Pounds 25.00. ISBN 0 19 853243 1.<nip> SECTION:Features NO PHYSICAL FILE As one scours the league tables of mathematics departments rated subjectively as providing an "excellent" mathematics education, I wonder just how many of those institutions produce graduates who can provide the solution to anything but the simplest of first-order linear partial differential equations (PDEs)? In one such department (from which I escaped), it is entirely possible to obtain a mathematics degree without ever having seen the equations that are essential to the successful modelling of electromagnetism, elasticity, fluid dynamics, mathematical biology, quantum mechanics and stock options. To such graduates, Maxwell might just be a dead publisher and a Green function could be a social event for environmentalists. Nevertheless, these graduates' degrees have been officially "approved" by the Quality Assurance Agency. PDEs are generally perceived by undergraduates as being difficult, dry and unprofitable. In the United Kingdom, A level dilution and the "point-and-click" mentality have all contributed to this. However, hope for the future of undergraduate PDE courses may lie in the exponential increase of interest in the modelling of biological phenomena or continuous-time finance. Commercially, multimedia interest in simulation has led to at least one UK MSc course covering PDE modelling for computer games. For departments still offering serious PDE courses, Applied Partial Differential Equations provides a comprehensive and challenging text. It is amply illustrated with traditional and less familiar examples rooted in the work of the late Alan Tayler, an Oxford mathematician dedicated to dragging mathematics away from standard classroom examples and into the more complex industrial world.Able mathematics undergraduates at the more prestigious UK institutions, those at the MSc/PhD level or established researchers should all find something they can take from the text. All the usual suspects are lined up for undergraduate courses, from first-order linear techniques through second-order classification and characteristics, similarity and transform solutions and Green function representations. Numerical techniques and perturbation techniques are not covered. This is perhaps disappointing for more advanced readers, since the practising PDE user will most often have need of these. However, as they note, the expanse of these topics necessitates their exclusion. To compensate, the authors include welcome chapters on free-boundary problems and non quasi-linear equations. The book concludes with a collection of miscellaneous topics containing insightful "tricks of the trade".The novel problems that motivate the text include fibre drawing, granular flow, option pricing, sandpiles and ray scattering. Each application is explained in terms amenable to the mathematically literate newcomer and several are drawn from recent research work by the authors' groups. Some unfortunate comments do appear, for example in the discussion of geometric optics, that the "large eigenvalues of the circle have an unsurprising distribution". In reality, the locally normalised nearest-neighbour-spacing of the eigenvalues is random, even though the corresponding classical behaviour is integrable. It is also incorrectly implied that the analogous problem in a rectangular domain is ergodic. However, these do not alter the overall enjoyment of the book.There are numerous footnotes providing helpful physical explanations, including one on the possible origin of spontaneous human combustion. The end of each chapter is well furnished with interesting exercises. The very first problem invites the reader to solve a PDE model for the number of proof-reading errors and to test it against their experience of the book so far. More advanced sections are starred for the avoidance of first-time readers, or those departments not rated as "excellent" by QAA-inspired journalists.Christopher Howls is lecturer in applied mathematics, University of Southampton
Weyl series for Aharonov-Bohm billiards
No full textFollowing a conjecture of Berry and Howls (1994) concerning the geometric information contained within the high orders of Weyl series, we examine such series for the average spectral properties of two- and three-dimensional quantum ball billiards threaded by a single flux line at the centre. We adapt a Mellin-based scheme of Bordag et al (1996) to generate the Weyl series. It is shown that for a circular billiard, only a single Weyl series term is changed and thus the flux line only induces a simple constant shift in the average properties of the spectrum, although the fluctuations about this average will still be flux dependent. This implies that the late terms in the expansion are dominated by the diametrical periodic orbit of the unfluxed circle, rather than the shorter diffractive orbits encountering both the billiard boundary and the flux line. For a spherical billiard with flux the late terms suffer modifications which can be linked to diffractive orbits. The origins of the differences between the structure of the series are traced to the interaction of the geometry and symmetry breaking
Hyperasymptotics for multidimensional integrals, exact remainders and the global connection problem
No full textThe method of steepest descents for single dimensional Laplace-type integrals involving an asymptotic parameter k was extended by Berry & Howls in 1991 to provide exact remainder terms for truncated asymptotic expansions in terms of contributions from certain non-local saddlepoints. This led to an improved asymptotic expansion (hyperasymptotics) which gave exponentially accurate numerical and analytic results, based on the topography of the saddle distribution in the single complex plane of the integrand. In this paper we generalize these results to similar well-behaved multidimensional integrands with quadratic critical points, integrated over infinite complex domains. As previously pointed out the extra complex dimensions give rise to interesting problems and phenomena. First, the conventionally defined surfaces of steepest descent are no longer unique. Second, the Stokes's phenomenon (whereby contributions from subdominant saddles enter the asymptotic representation) is of codimension one. Third, we can collapse the representation of the integral onto a single complex plane with branch cuts at the images of critical points. The new results here demonstrate that dimensionality only trivially affects the form of the exact multidimensional remainder. Thus the growth of the late terms in the expansion can be identified, and a hyperasymptotic scheme implemented. We show by a purely algebraic method how to determine which critical points contribute to the remainder and hence resolve the global connection problem, Riemann sheet structure and homology associated with the multidimensional topography of the integrand
Book review. Number crunching: taming unruly computational problems from mathematical physics to science fiction
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