Overview

Differential equations have played a central role in Applied Mathematics for more than a century. With the advent of the computer their importance has increased rather than diminished. Throughout science, engineering and far beyond, scientific computation is taking place in efforts to understand and control our natural environment and to develop new technological processes. At the heart of this modelling lie differential equations. The investigation of differential equations arising in applications has led to many deep mathematical problems. These are often pursued without any applications in mind but sometimes turn out to be of great importance to new applications.

Within the school, the following people have a particular interest in differential equations: John Appleby, Jurgen Burzlaff, John Carroll, Turlough Downes, Brien Nolan, Eugene O'Riordan, David Reynolds.

Analytical, asymptotic and/or numerical techniques are used to determine both the qualitative and the quantitative properties of the solutions to various classes of differential equations.  Individual members of this group tend to concentrate on either analytical, asymptotic or numerical approaches to examine differential equations. Below are listed the current main areas of interest within the broad area of differential equations.

Functional Differential Equations

It is well-established how to use ordinary differential equations to model the evolution of physical, biological and economic systems, in which the response of the system depends purely on the current state of the system. However, in many applications the response of the system can be delayed, or depend on the past history of the system in a more complicated way. Dynamical systems which respond in this way are called Functional Differential Equations (FDEs). Furthermore, in applications it is typical for the system to be perturbed by random noise or for the mathematical model to contain some unknown parameters. In these cases, it is more appropriate to model the dynamics of the system using Stochastic Functional Differential Equations (SFDEs). Areas of the sciences in which functional differential equations are applied include: materials with memory (viscoelastic materials); demography and population dynamics, artificial neural networks with transmission delays and inefficient markets in mathematical finance.

Singularly Perturbed Differential Equations

Differential equations with a small perturbation parameter multiplying the highest order derivative terms are said to be singularly perturbed. Singularly perturbed differential equations are ubiquitous in mathematical problems in the sciences and engineering. For example the Navier-Stokes equations of fluid flow at high Reynolds number, the equations governing flow in porous media, the drift-diffusion equations of semiconductor device physics and mathematical models of liquid crystal materials all involve singular perturbation parameters. In general, the solutions to singularly perturbed differential equations are not smooth. Thin layers (regions where the solution changes rapidly) appear in the solutions of such problems. Although the layer regions are small, their influence on the overall solution is very significant. If one wishes to identify the essential features of the solutions to such problems, it is crucial to determine the location and width of all the layers present. Classical numerical methods usually give unsatisfactory numerical results when layers are present. In particular, the pointwise errors in numerical approximations generated from standard numerical algorithms depend inversely on a power of the singular perturbation parameter. It is of both theoretical and practical interest to construct numerical methods of guaranteed accuracy for such singularly perturbed differential equations irrespective of the size of the perturbation parameter.

Nonlinear Partial Differential Equations

The nonlinear terms in a differential equation describe the interactions present in the phenomenon modelled by the equation. Since some interactions play a part in almost any realistic situation, the corresponding differential equations should be nonlinear. Only when the interactions are so small that they can be neglected, will a linear differential equation be a good approximation. To a mathematician, nonlinear partial differential equations present new challenges, and have already yielded some surprises. Some of their solutions, for example, exhibit an unexpected stability only associated with particle-like objects. They are therefore sometimes called solitons. To prove the existence of solutions to nonlinear partial differential equations, to find soliton-like solutions and to study their properties requires a range of new mathematical techniques. The gravitational interaction - as described by Einstein's general relativity - gives rise to non-linear partial differential equations. This non-linearity has many interesting consequences: the occurrence of black holes, space-time singularities and cosmological models with a big bang singularity. A combination of analytic and numerical methods for ordinary and partial differential equations are required to properly understand these phenomena.

Stochastic Differential Equations

In modelling physical, biological or economic process, it is often important to recognise that the evolution of the phenomenon of interest is subject to random forces (i.e., forces which are unpredictable in advance). In this case, instead of modelling the change in the process using a deterministic differential equation, a stochastic differential equation (SDE) is used instead. An especially interesting class of SDE is the Ito stochastic differential equation, which is widely used in modelling the changes in stock prices, interest rates, or volatility in models of financial markets. In this case, the presence of randomness in the equation mimics the fact that the evolution of stock prices is unpredictable. The presence of randomness complicates mathematical analysis, but leads to a very rich class of dynamical behaviour. For example, the presence of noise in a system can stabilise or destabilise a system, cause oscillation, or change the rate at which the system grows or stabilises. It can also cause the model to fluctuate persistently, as might happen during an epidemic of an infectious disease or a period of high stock market volatility. The manner in which solutions fluctuate, grow, decay or oscillate is the subject of stochastic dynamical systems. It is of particular interest to understand, for a given model, at which point the presence of noise becomes critical; such questions are important in engineering, where vibration may cause damage or instability to a structure or device. Since many stochastic models are difficult to analyse using pen-and-paper methods alone, it is also necessary to have reliable methods for simulating solutions of stochastic differential equations. This gives rise to the disciplines of stochastic numerical analysis and Monte Carlo simulation.

Contact

If you are interested in post-graduate study in differential equations, please send an email to Prof. Eugene O'Riordan.