Current Projects for PhD, Masters and Honours Students
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Motivation: Why do research?
Undertaking a research project can be: challenging; innovative; signficant; engaging; interesting; and fun. At the heart of the learning process is the art of discovery and creation. Research students will learn many new things; but perhaps some of the most amazing aspects are that students will discover and create new facts and ideas; and advance the current knowledge in their field of study. This is what makes a research project unique and a lot of fun. |
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Current and Past Research Students
Tomasia Kulik, PhD Program, 2007. Tomasia is currently researching quantitative and qualitative properties of solutions to nonlinear, integral equations on time scales.
Atiya Zaidi, PhD Program, 2006. Atiya is currently researching quantitative and qualitative properties of solutions to nonlinear, dynamic equations on time scales.
Sebastian Aland, Professional Practicum Program, 2007. Sebastian researched dyanmic equations that feature a delay. Such investigations are especially important for understanding certain population dynamics. He is now completing his studies in TU-Dresden, Germany.
Petr Stehlik, Professional Practicum Program, 2004. Petr researched the qualitative properties to difference inclusions. He is now teaching at the Department of Mathematics, University of West Bohemia, Czech Republic.
Publications:
Stehlik, P.; Tisdell, C. C. On boundary value problems for second-order discrete inclusions. Boundary Value Problems 2005(2005), no. 2, 153--164.
Lit Hau Tan, Honours in Applied Mathematics, 2005. Lit Hau researched the basic properties of dynamic equations on time scales. In addition, he investigated existence theory to ordinary differential equations. He is now studying in Europe.
Publications:
Tisdell, C. C.; Tan L. H. On vector boundary value problems without growth restrictions. JIPAM. J. Inequal. Pure Appl. Math. 6 (2005), no. 5, Article 137, 10 pp. (electronic).
Tan, L. H. Investigations into dynamic equations on time scales. UNSW Honours Thesis, Department of Applied Mathematics, 2006.
Erin Rae Peterson, Master of Science and Technology in Mathematics, 2005. Erin did a project on the qualitative properties of difference equations and discrete systems. She is now working for US company Corning Data Services, Inc. -an IBM and Oracle business partner.
Any of the following projects would be ideal for students who have an interest in differential equations, difference equations and nonlinear analysis.
It is important to note that projects can be adjusted to suit the individual needs, experience and interests of the student - all enquiries are welcome!
Supervision is not limited to the following ideas, so students are very welcome to make enquiries about other avenues of research.
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Dynamic Equations on Time Scales
Dynamic equations on time scales (measure chains) is a new and exciting theory that models phenomena whose behaviour can change smoothly at one level, but irregularly at another. The field was first created by Stefan Hilger in 1990 in an attempt to unify the ideas of differential calculus and difference calculus. In fact, the field of dynamic equations on time scales contains the theory of differential equations and the theory of difference equations both as special cases. Due to the short history of time scales, there are many open questions and significant problems to work on. Moreover, because of the versatility of time scales, it has potential to model a wide range of continuous and discrete phenomena under the one framework. The theory is expected to apply to such "stop-start" behaviour as: insect population models, stock markets, crop harvests and combustion. In addition, numerical techinques involving time scales are becoming increasingly more popular and versatile. This project will lead to advancements in basic theory and applications of dynamic equations on time scales. Students who undertake this project will be well-equipped to contribute to this research area. This project is supported by funding from the Australian Research Council's Discovery Projects. Further info: Time Scales. MATH5215 Dynamic Equations on Time Scales @ UNSW. |
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Difference Equations: Theory, Method and Application
The field of difference equations provides a natural framework for modelling discrete processes and is also used in the numerical approximation of solutions to ordinary differential equations. Difference equations are particularly useful in economics and ecology, where many situations are (sometimes crudely) described by simple difference equations. A striking point of interest is that even simple difference equations can possess a very rich spectrum of dynamical behaviour. This project will lead to advancements in theory and applications of difference equations. Students who undertake this project will be well-equipped to contribute to this research area. Further info: Simple mathematical models with very complicated dynamics by R M May. |
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Theory and Application of Impulsive Systems
"Impulsive" systems are differential equations that can accurately model processes in which abrupt, instantaneous changes occur. Impulsive behaviour is seen in such phenomena as: bursting rhythm models in medicine; stimulated neural networks; and motions of missiles or aircraft. Therefore the need to better understand impulsive systems is naturally motivated. The outcomes of this project will provide a deeper understanding of qualitative and quantitative aspects of the impulsive differential equations. This project is supported by funding from the Australian Academy of Science. Further info: Impulsive Systems. Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. |
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In the last century most of the sciences, engineering and technology have
triggered a multitude of nonlinear complex phenomena. These problems involve
singular (with respect to dependent variable), second order ordinary
differential equations together with some boundary conditions over finite or
infinite intervals. Singular BVPs arise in diverse fields such as gas diffusion through porous media, thermal self-ignition of a chemically active mixture of gases in a vessel, catalytic theory, chemically reacting systems and adiabatic tubular reactors, diffusion of heat generated by positive temperature-dependent sources, fluid dynamics, electrical potential theory, steady-state of oxygen diffusion in a cell with Michaelis-Menten kinetics, cell membrane and heat conduction in the human brain. This project aims to advance current knowledge on the subject by through modern and advanced methods from nonlinear analysis. This project is supported by funding from the John Yu Fellowship to Europe Program. Further info: Link 1 Link 2. |
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Functional differential equations (also called delay equations) arise in
situations where the certain characteristics of solutions may not only depend
on the current time, but also depends on previous (or future) times. In
particular, these types of equations arise in control theory, where the
variational problems are complicated by the effect of time delays in signal
transmission. A particularly important potential application of FDEs are in the field of dynamic economic modelling. This project aims to advance and refine current ideas concerning qualitative and quantitative mathematics results and their application to dynamic economic models. Further info: FDEs and on BVPs for FDEs |
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The theory of differential inclusions is a very general and wide ranging way
of treating differential equations that may have discontinuities in the right-hand side, or even when the right-hand side is inaccurately known.
Traditional methods are rather
ill-equipped to treat these equations. This project will progress the current state of affairs of differential inclusions with respect to existence of solutions. Important applications of differential inclusions lie in the trajectory of an r-stage rocket and also in pricing models in economics. This project is supported by funding from the Australian Research Council's Linkage International Projects. Further info: Differential Inclusions. Differential inclusion solver. |
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Further information can be obtained from:
Dr Chris Tisdell
School of Mathematics
UNSW Sydney 2052
AUSTRALIA
cct@unsw.edu.au
Return to Chris Tisdell's
personal webpage.
Return to University of New South Wales
School of Mathematics.