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Explosions and unbounded growth in nonlinear delay differential equations: Numerical and asymptotic analysis

McCarthy, Michael J. (2011) Explosions and unbounded growth in nonlinear delay differential equations: Numerical and asymptotic analysis. PhD thesis, Dublin City University.

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Abstract

This thesis investigates the asymptotic behaviour of a scalar, nonlinear dierential equation with a fixed delay, and examines whether the properties of this equation can be replicated by an appropriate discretisation. We begin by considering equations for which the solution explodes in finite time. Existing work on such explosive equations has dealt with devising numerical schemes for equations with polynomially growing instantaneous feedback, and methods to deal with delayed feedback have not been fully explored. We therefore set out a discretised scheme which replicates all the qualitative features of the continuous-time solution for a more general class of equations. Next, for non-explosive equations which exhibit extremely rapid growth, the rate of growth of the solution depends on the comparative asymptotic nonlinearities of the coefficients of the equation and the magnitude of the delay. Thus we set out conditions on these parameters which characterise the growth rate of the solution, and investigate numerical methods for recovering this rate. Using constructive comparison principles and nonlinear asymptotic analysis, we extend the numerical methods devised for explosive equations for this purpose.

Item Type:Thesis (PhD)
Date of Award:November 2011
Refereed:No
Supervisor(s):Appleby, John
Uncontrolled Keywords:asymptotic behaviour; explosive equations; non-explosive equations
Subjects:Mathematics > Differential equations
Mathematics > Numerical analysis
DCU Faculties and Centres:DCU Faculties and Schools > Faculty of Science and Health > School of Mathematical Sciences
Use License:This item is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 3.0 License. View License
Funders:Irish Research Council for Science Engineering and Technology
ID Code:16640
Deposited On:01 Dec 2011 11:31 by John Appleby. Last Modified 01 Dec 2011 11:31

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