Past Differential Equations and Applications Seminar

27 October 2011
16:00
Peter Clarkson
Abstract
In this talk I shall discuss special polynomials associated with rational solutions of the Painlevé equations and of the soliton equations which are solvable by the inverse scattering method, including the Korteweg-de Vries, Boussinesq and nonlinear Schrodinger equations. Further I shall illustrate applications of these polynomials to vortex dynamics and rogue waves. The Painlevé equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past thirty years, and have arisen in a variety of physical applications. Further the Painlevé equations may be thought of as nonlinear special functions. Rational solutions of the Painlevé equations are expressible in terms of the logarithmic derivative of certain special polynomials. For the fourth Painlevé equation these polynomials are known as the generalized Hermite polynomials and generalized Okamoto polynomials. The locations of the roots of these polynomials have a highly symmetric (and intriguing) structure in the complex plane. It is well known that soliton equations have symmetry reductions which reduce them to the Painlevé equations, e.g. scaling reductions of the Boussinesq and nonlinear Schrödinger equations are expressible in terms of the fourth Painlevé equation. Hence rational solutions of these equations can be expressed in terms of the generalized Hermite and generalized Okamoto polynomials. I will also discuss the relationship between vortex dynamics and properties of polynomials with roots at the vortex positions. Classical polynomials such as the Hermite and Laguerre polynomials have roots which describe vortex equilibria. Stationary vortex configurations with vortices of the same strength and positive or negative configurations are located at the roots of the Adler-Moser polynomials, which are associated with rational solutions of the Kortweg-de Vries equation. Further, I shall also describe some additional rational solutions of the Boussinesq equation and rational-oscillatory solutions of the focusing nonlinear Schrödinger equation which have applications to rogue waves.
  • Differential Equations and Applications Seminar
20 October 2011
16:00
Alastair Rucklidge
Abstract
Three-wave interactions form the basis of our understanding of many nonlinear pattern forming systems because they encapsulate the most basic nonlinear interactions. In problems with two comparable length scales, such as the Faraday wave experiment with multi-frequency forcing, consideration of three-wave interactions can explain the presence of the spatio-temporal chaos found in some experiments, enabling some previously unexplained results to be interpreted in a new light. The predictions are illustrated with numerical simulations of a model partial differential equation.
  • Differential Equations and Applications Seminar
13 October 2011
16:00
Robert Mackay
Abstract
Isostatic mounts are used in applications like telescopes and robotics to move and hold part of a structure in a desired pose relative to the rest, by driving some controls rather than driving the subsystem directly. To achieve this successfully requires an understanding of the coupled space of configurations and controls, and of the singularities of the mapping from the coupled space to the space of controls. It is crucial to avoid such singularities because generically they lead to large constraint forces and internal stresses which can cause distortion. In this paper we outline design principles for isostatic mount systems for dynamic structures, with particular emphasis on robots.
  • Differential Equations and Applications Seminar
23 June 2011
16:00
Qingchang Zhong
Abstract
Systems with delays frequently appear in engineering. The presence of delays makes system analysis and control design very complicated. In this talk, the standard H-infinity control problem of time-delay systems will be discussed. The emphasis will be on systems having an input or output delay. The problem is solved in the frequency domain via reduction to a one-block problem and then further to an extended Nehari problem using a simple and intuitive method. After solving the extended Nehari problem, the original problem is solved. The solvability of the extended Nehari problem (or the one-block problem) is equivalent to the nonsingularity of a delay-dependent matrix and the solvability conditions of the standard H-infinity control problem with a delay are then formulated in terms of the existence of solutions to two delay-independent algebraic Riccati equations and a delay-dependent nonsingular matrix.
  • Differential Equations and Applications Seminar
16 June 2011
10:45
to
17:30
Oxford / Cambridge Meeting 15th Biennial Event
Abstract
15th Biennial OXFORD / CAMBRIDGE MEETING PROGRAMME FOR THE ‘WOOLLY OWL TROPHY’ Invited Judges John Harper (Victoria University of Wellington, NZ) Arash Yavari (Georgia Tech, Atlanta, USA) Sharon Stephen (University of Birmingham, UK) 10:45 Morning Coffee The Maths Inst Common Room
  • Differential Equations and Applications Seminar
9 June 2011
16:00
Colin B MacDonald
Abstract
Solving partial differential equations (PDEs) on curved surfaces is important in many areas of science. The Closest Point Method is a new technique for computing numerical solutions to PDEs on curves, surfaces, and more general domains. For example, it can be used to solve a pattern-formation PDE on the surface of a rabbit. A benefit of the Closest Point Method is its simplicity: it is easy to understand and straightforward to implement on a wide variety of PDEs and surfaces. In this presentation, I will introduce the Closest Point Method and highlight some of the research in this area. Example computations (including the in-surface heat equation, reaction-diffusion on surfaces, level set equations, high-order interface motion, and Laplace--Beltrami eigenmodes) on a variety of surfaces will demonstrate the effectiveness of the method.
  • Differential Equations and Applications Seminar
2 June 2011
16:00
Abstract
Voltammetry is a powerful method for interrogating electrochemical systems. A voltage is applied to an electrode and the resulting current response analysed to determine features of the system under investigation, such as concentrations, diffusion coefficients, rate constants and thermodynamic potentials. Here we will focus on ac voltammetry, where the voltage signal consists of a high frequency sine-wave superimposed on a linear ramp. Using multiple scales analysis, we find analytical solutions for the harmonics of the current response and show how they can be used to determine the system parameters. We also include the effects of capacitance due to the double-layer at the electrode surface and show that even in the presence of large capacitance, the harmonics of the current response can still be isolated using the FFT and the Hanning window.
  • Differential Equations and Applications Seminar
26 May 2011
16:00
Demetrios Papageorgiou
Abstract
Flows involving immiscible liquids are encountered in a variety of industrial and natural processes. Recent applications in micro- and nano-fluidics have led to a significant scientific effort whose aim (among other aspects) is to enable theoretical predictions of the spatiotemporal dynamics of the interface(s) separating different flowing liquids. In such applications the scale of the system is small, and forces such as surface tension or externally imposed electrostatic forces compete and can, in many cases, surpass those of gravity and inertia. This talk will begin with a brief survey of applications where electrohydrodynamics have been used experimentally in micro-lithography, and experiments will be presented that demonstrate the use of electric fields in producing controlled encapsulated droplet formation in microchannels. The main thrust of the talk will be theoretical and will mostly focus on the paradigm problem of the dynamics of electrified falling liquid films over topographically structured substrates. Evolution equations will be developed asymptotically and their solutions will be compared to direct simulations in order to identify their practicality. The equations are rich mathematically and yield novel examples of dissipative evolutionary systems with additional effects (typically these are pseudo-differential operators) due to dispersion and external fields. The models will be analysed (we have rigorous results concerning global existence of solutions, the existence of dissipative dynamics and an absorbing set, and analyticity), and accurate numerical solutions will be presented to describe the large time dynamics. It is found that electric fields and topography can be used to control the flow.Time permitting, I will present some recent results on transitions between convective to absolute instabilities for film flows over periodic topography.
  • Differential Equations and Applications Seminar
19 May 2011
16:00
Abstract
The notion of critical mass in research is one that has been around for a long time without proper definition. It has been described as some kind of threshold group size above which research standards significantly improve. However no evidence for such a threshold has been found and critical mass has never been measured -- until now. We present a new, simple, sociophysical model which explains how research quality depends on research-group structure and in particular on size. Our model predicts that there are, in fact, two critical masses in research, the values of which are discipline dependent. Research quality tends to be linearly dependent on group size, but only up to a limit termed the 'upper critical mass'. The upper critical mass is interpreted as the average maximum number of colleagues with whom a given individual in a research group can meaningfully interact. Once the group exceeds this size, it tends to fragment into sub-groups and research quality no longer improves significantly with increasing size. There is also a lower critical mass, which small research groups should strive to achieve for stability. Our theory is tested using empirical data from RAE 2008 on the quantity and quality of research groups, for which critical masses are determined. For pure and applied mathematics, the lower critical mass is about 2 and 6, respectively, while for statistics and physics it is 9 and 13. The upper critical mass, beyond which research quality does not significantly improve with increasing group size, is about twice the lower value.
  • Differential Equations and Applications Seminar
12 May 2011
16:00
Nikolai Brilliantov
Abstract
We develop a theory of impact of viscoelastic spheres with adhesive interactions. We assume that the collision velocities are not large to avoid the fracture and plastic deformation of particles material and microscopic relaxation time is much smaller than the collision duration. The adhesive interactions are described with the use of Johnson, Kendall and Roberts (JKR) theory, while dissipation is attributed to the viscoelastic behavior of the material. For small impact velocities we apply the condition of a quasi-static collision and obtain the inter-particle force. We show that this force is a sum of four components, having in addition to common elastic, viscous and adhesive force, the visco-adhesive cross term. Using the derived force we compute the coefficient of normal restitution and consider the application of our theory to the collisions of macro and nano-particles.
  • Differential Equations and Applications Seminar

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