3,045 research outputs found
Characterising small solutions in delay differential equations through numerical approximations
This paper discusses how the existence of small solutions for delay differential equations can be predicted from the behaviour of the spectrum of the finite dimensional approximations.Manchester Centre for Computational Mathematic
An algorithm to detect small solutions in linear delay differential equations
This is a PDF version of a preprint submitted to Elsevier. The definitive version was published in the Journal of computational and applied mathematics and is available at www.elsevier.comThis preprint discusses an algorithm that provides a simple reliable mechanism for the detection of small solutions in linear delay differential equations.This article was submitted to the RAE2008 for the University of Chester - Applied Mathematics
Hydro-dynamical models for the chaotic dripping faucet
We give a hydrodynamical explanation for the chaotic behaviour of a dripping
faucet using the results of the stability analysis of a static pendant drop and
a proper orthogonal decomposition (POD) of the complete dynamics. We find that
the only relevant modes are the two classical normal forms associated with a
Saddle-Node-Andronov bifurcation and a Shilnikov homoclinic bifurcation. This
allows us to construct a hierarchy of reduced order models including maps and
ordinary differential equations which are able to qualitatively explain prior
experiments and numerical simulations of the governing partial differential
equations and provide an explanation for the complexity in dripping. We also
provide a new mechanical analogue for the dripping faucet and a simple
rationale for the transition from dripping to jetting modes in the flow from a
faucet.Comment: 16 pages, 14 figures. Under review for Journal of Fluid Mechanic
Numerical treatment of oscillary functional differential equations
NOTICE: this is the authorās version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 234(2010), doi: 10.1016/j.cam.2010.01.035This preprint is concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear v-methods will also be oscillatory. We conclude with some general theor
Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advances
NOTICE: this is the authorās version of a work that was accepted for publication in Journal of computational and applied mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of computational and applied mathematics, 235(2011), doi: 10.1016/j.cam.2011.04.041This article discusses the oscillatory behaviour of the differential equation of mixed type
Performance Modelling and Optimisation of Multi-hop Networks
A major challenge in the design of large-scale networks is to predict and optimise the
total time and energy consumption required to deliver a packet from a source node to a
destination node. Examples of such complex networks include wireless ad hoc and sensor
networks which need to deal with the effects of node mobility, routing inaccuracies, higher
packet loss rates, limited or time-varying effective bandwidth, energy constraints, and the
computational limitations of the nodes. They also include more reliable communication
environments, such as wired networks, that are susceptible to random failures, security
threats and malicious behaviours which compromise their quality of service (QoS) guarantees.
In such networks, packets traverse a number of hops that cannot be determined
in advance and encounter non-homogeneous network conditions that have been largely
ignored in the literature. This thesis examines analytical properties of packet travel in
large networks and investigates the implications of some packet coding techniques on both
QoS and resource utilisation.
Specifically, we use a mixed jump and diffusion model to represent packet traversal
through large networks. The model accounts for network non-homogeneity regarding
routing and the loss rate that a packet experiences as it passes successive segments of a
source to destination route. A mixed analytical-numerical method is developed to compute
the average packet travel time and the energy it consumes. The model is able to capture
the effects of increased loss rate in areas remote from the source and destination, variable
rate of advancement towards destination over the route, as well as of defending against
malicious packets within a certain distance from the destination. We then consider sending
multiple coded packets that follow independent paths to the destination node so as to
mitigate the effects of losses and routing inaccuracies. We study a homogeneous medium
and obtain the time-dependent properties of the packetās travel process, allowing us to
compare the merits and limitations of coding, both in terms of delivery times and energy
efficiency. Finally, we propose models that can assist in the analysis and optimisation
of the performance of inter-flow network coding (NC). We analyse two queueing models
for a router that carries out NC, in addition to its standard packet routing function. The
approach is extended to the study of multiple hops, which leads to an optimisation problem
that characterises the optimal time that packets should be held back in a router, waiting
for coding opportunities to arise, so that the total packet end-to-end delay is minimised
Theory and numerics for multi-term periodic delay differential equations, small solutions and their detection
We summarise a theoretical treatment that analyses whether the equation has small solutions. We consider discrete equations that arise when a numerical method with fixed step size is applied to approximate the solution to (ā ) and we develop a corresponding theory. Our results show that small solutions can be detected reliably by the numerical scheme. We conclude with some numerical examples
Two-dimensional turbulence in magnetised plasmas
In an inhomogeneous magnetised plasma the transport of energy and particles
perpendicular to the magnetic field is in general mainly caused by quasi
two-dimensional turbulent fluid mixing. The physics of turbulence and structure
formation is of ubiquitous importance to every magnetically confined laboratory
plasma for experimental or industrial application. Specifically, high
temperature plasmas for fusion energy research are also dominated by the
properties of this turbulent transport. Self-organisation of turbulent vortices
to mesoscopic structures like zonal flows is related to the formation of
transport barriers that can significantly enhance the confinement of a fusion
plasma. This subject of great importance in research is rarely touched on in
introductory plasma physics or continuum dynamics courses. Here a brief
tutorial on 2D fluid and plasma turbulence is presented as an introduction to
the field, appropriate for inclusion in undergraduate and graduate courses.Comment: This is an author-created, un-copyedited version of an article
published in European Journal of Physics. IOP Publishing Ltd is not
responsible for any errors or omissions in this version of the manuscript or
any version derived from it. The definitive publisher authenticated version
is available online at doi: 10.1088/0143-0807/29/5/00
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