Exponential asymptotics for integro-differential equations

Lead Research Organisation: Imperial College London
Department Name: Dept of Mathematics


A large number of physical problems involve the influence of ``small'' effects. An example is the effect of surface tension at the interface between two viscous fluids of differing viscosities. While apparently ``small'', such effects can often be surprisingly influential in determining global dynamical properties of the system. Since the effect is small, mathematically it is natural to attempt an asymptotic expansion in the parameter governing its influence on the problem.However, in many cases, standard asymptotic analysis fails and one has to examine exponentially small terms in order to ascertain the global effect of the small perturbation. This subject of ``exponential asymptotics'' has become a very active field of applied mathematics.Recent work, however, has focussed on rendering many of the mathematical ideas more rigorous. This proposal seeks to continue with the task of putting the subject of exponential asymptotics, especially as it pertains to the analysis of integro-differential equations, on a more rigorous footing. An initial case study, consisting of the selection problem for the rising Taylor bubble, will be analysed.


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Description The most significant output of this collaborative grant, with Professor Saleh Tanveer of Ohio State University, is a novel numerical scheme for the challenging problem of computing fields (e.g. electrostatic) near close-to-touching objects (e.g. conductors). The idea of the numerical scheme, which we call the hybrid basis scheme is simple: it relies on the strategic use of a mixture of basis sets specially tailored to significantly overcome the numerical challenges, often due to exponentially/logarithmically small singular terms, that arise when the objects are very close together.

The seeds for these developments were laid during Prof. Tanveer's sabbatical visit to Imperial partially funded by this Mathematics Small Grant; the ideas were recently brought to fruition, and submitted as an article, at the end of Prof. Tanveer's more recent second sabbatical to Imperial (funded by other sources).
Exploitation Route The idea of the hybrid basis scheme we have proposed is very general, and powerful. There is broad scope to adapt the basic idea to many different scenarios.
Sectors Aerospace, Defence and Marine,Other

URL http://arxiv.org/abs/1409.2466
Description The findings have only just been published (as an arxiv article) and should appear in a published journal soon. It is too early to register impact
First Year Of Impact 2014
Sector Aerospace, Defence and Marine,Digital/Communication/Information Technologies (including Software),Manufacturing, including Industrial Biotechology,Other
Impact Types Economic