Past Differential Equations and Applications Seminar

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We develop a nonlinear delay-differential equation for the human cardiovascular control system, and use it to explore blood pressure and heart rate variability under short-term baroreflex control. The model incorporates an intrinsically stable heart rate in the absence of nervous control, and features baroreflex influence on both heart rate and peripheral resistance. Analytical simplications of the model allow a general investigation of the r\^{o}les played by gain and delay, and the effects of ageing. View diagram:  Download PDF

The classical gravity-capillary water-wave problem is the

study of the irrotational flow of a three-dimensional perfect

fluid bounded below by a flat, rigid bottom and above by a

free surface subject to the forces of gravity and surface

tension. In this lecture I will present a survey of currently

available existence theories for travelling-wave solutions of

this problem, that is, waves which move in a specific

direction with constant speed and without change of shape.

The talk will focus upon wave motions which are truly

three-dimensional, so that the free surface of the water

exhibits a two-dimensional pattern, and upon solutions of the

complete hydrodynamic equations for water waves rather than

model equations. Specific examples include (a) doubly

periodic surface waves; (b) wave patterns which have a

single- or multi-pulse profile in one distinguished

horizontal direction and are periodic in another; (c)

so-called 'fully-localised solitary waves' consisting of a

localised trough-like disturbance of the free surface which

decays to zero in all horizontal directions.

I will also sketch the mathematical techniques required to

prove the existence of the above waves. The key is a

formulation of the problem as a Hamiltonian system with

infinitely many degrees of freedom together with an

associated variational principle.

When modelling a physical or biological system, it has to be decided

what framework best captures the underlying properties of the system

under investigation. Usually, either a continuous or a discrete

approach is adopted and the evolution of the system variables can then

be described by ordinary or partial differential equations or

difference equations, as appropriate. It is sometimes the case,

however, that the model variables evolve in space or time in a way

which involves both discrete and continuous elements. This is best

illustrated by a simple example. Suppose that the life span of a

species of insect is one time unit and at the end of its life span,

the insect mates, lays eggs and then dies. Suppose the eggs lie

dormant for a further 1 time unit before hatching. The `time-scale' on

which the insect population evolves is therefore best represented by a

set of continuous intervals separated by discrete gaps. This concept

of `time-scale' (or measure chain as it is referred to in a slightly

wider context) can be extended to sets consisting of almost arbitrary

combinations of intervals, discrete points and accumulation points,

and `time-scale analysis' defines a calculus, on such sets. The

standard `continuous' and `discrete' calculus then simply form special

cases of this more general time scale calculus.

In this talk, we will outline some of the basic properties of time

scales and time scale calculus before discussing some if the

technical problems that arise in deriving and analysing boundary

value problems on time scales.