Many nonlinear problems readily yield to asymptotic analysis and perturbation methods which, by construction, comprise complex and lengthy hierarchies of equations that are typically tractable to a combination of tedious algebra, intuition and problem-specific interpretation. As such, multi-purpose 'pseudo-exact' (in the sense that the error is well understood) solvers for ODEs and PDEs are desirable, so that automation of this process is sought.
Many problems have been conquered by achieving such automation using the algebraic manipulator 'Maple', with specific emphasis on multiple timescales. For example, in a recent project, new light has been shed on the classic problem of nonlinear surface waves in viscous coating flow on a rotating cylinder, in which nonlinear partial differential equations with thousands of terms are linearised and efficiently solved in a matter of seconds by bespoke algorithms. A current PhD project has led to the successful automation of a multiple-timescale solution procedure that is applicable to a wide variety of evolution equations arising in thin-film fluid mechanics.
There is great current interest in developing automated computer-algebra methods for solving problems involving integral equations, because many differential equations may be recast, for efficient solution, into this form. In a recent project (that reached the finals of the IMA/LMS Science Engineering and Technology awards), an automated procedure was developed for obtaining novel closed-form error formulae for a broad class of numerical boundary-integral-equation (BIE) methods employing different interpolation polynomials. A current PhD project is examining minimax optimisation of errors in BIE methods via the use of Chebyshev polynomials, and the results of this fundamental study will be used in the practical application of guaranteeing optimal approximations in the modelling of time-dependent free-surface phenomena.
Research in the Polymers and Industrial Mathematics group focuses on the mechanics of polymers and other complex fluids, free-surface flows and inverse problems. We are also concerned with the development and implementation of novel numerical and computational solution methods for both ordinary and partial differential equations, from fundamental aspects (the theoretical analysis of numerical methods) to problem-specific aspects (the design, development and practical implementation of novel algorithms). Within the polymer area, we conduct fundamental research into fluids that have a complex microstructure, such as polymer melts and solutions and colloidal dispersions.
Our research combines methods from molecular physics and continuum mechanics to develop multiscale models that link together the microscale motion of individual molecules to the flow behaviour of the bulk material. An important class of industrial flow problems are those involving free surfaces, such as in inkjet printing, film coating and bubble growth in polymeric foams. We also work on a diverse range of inverse problems in heat transfer, porous media, fluid and solid mechanics, acoustics and medicine. This is a strongly interdisciplinary subject and much of our research involves collaborations with independent research groups in science and engineering departments both at Leeds and worldwide, as well as with industry.