DSpace Collection:
https://dspace.lboro.ac.uk/2134/4642
2016-02-08T00:03:08ZSome properties of a class of stochastic heat and wave equations with multiplicative Gaussian noises
https://dspace.lboro.ac.uk/2134/19611
Title: Some properties of a class of stochastic heat and wave equations with multiplicative Gaussian noises
Authors: Tian, Kuanhou
Abstract: We study stochastic partial differential equations of the form Lu = ∆u+σ(u)N˙ with particular initial conditions. Here L denotes a first or second order partial differential operator, N˙ a Gaussian noise, ∆ the Laplacian operator, and σ : R → R a Lipschitz continuous function. In the first case we choose Lu := ∂tu, we study the stochastic fractional heat equation of the form ∂tu(t, x) = ∆ α/2u(t, x) +λ σ(u(t, x))N˙ for t > 0, x ∈ B(0, R) ⊂ R d . Here ∆ α/2 is the infinitesimal generator of a symmetric α-stable process killed at the boundary of the ball B(0, R), λ is a positive parameter called the level noise and N˙ denotes space-time white noise when d = 1 or white-colored noise in the case of Riesz Kernel of order β when d ≥ 1. We start to show the existence and uniqueness of solutions, the main task is to study how the second moment of the solution u(t, x) and excitation index of the solution grows as λ tends to infinity for a fixed t > 0. This study was initiated by [KK13] and [KK15]. Our results are significant extensions of those results and that of [FJ14]. In the second case we choose Lu := ∂ttu+2η∂tu, and study the stochastic damped wave equation of the form ∂ttu(t, x)+2η∂tu(t, x) = ∆u(t, x)+σ(u(t, x))W˙ for t > 0, x ∈ R, where η represents the positive damping parameter and W˙ space-time white noise. We study second moment and p-th moment of solution to show the intermittency properties, and existence and uniqueness of solutions will be proved. The study of the intermittency properties in stochastic partial differential equations was initiated by [FK09], our results are significant extension of results in [CJKS13]. In the end, we also show the result of excitation index for the stochastic damped wave equation.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2015-01-01T00:00:00ZUse of microcomputers in mathematics in Hong Kong higher education
https://dspace.lboro.ac.uk/2134/19483
Title: Use of microcomputers in mathematics in Hong Kong higher education
Authors: Pong, Tak-Yun G.
Abstract: Since the innovation of computers some 40 years ago and the
introduction of microcomputers in 1975, computers are
playing an active role in education processes and altering
the pattern of interaction between teacher and student in
the classroom. Computer assisted learning has been seen as
a revolution in education. In this research, the author has
studied the impact of using microcomputers on mathematical
education, particularly at the Hong Kong tertiary level, in
different perspectives.
Two computer software packages have been developed on the
microcomputer. The consideration of the topic to be used in
the computer assisted learning was arrived at in earlier
surveys with students who thought that computers could give
very accurate solutions to calculations. The two software
packages, demonstrating on the spot the error that would be
incurred by the computer, have been used by the students.
They are both interactive and make use of the advantages of
the microcomputer's functions over other teaching media,
such as graphics facility and random number generator, to
draw to the students' attention awareness of errors that may
be obtained using computers in numerical solutions.
Much emphasis is put on the significance and effectiveness
of using computer packages in learning and teaching.
Measurements are based on questionnaires, conversations with
students, and tests on content material after the packages
have been used. Feedback and subjective opinion of using
computers in mathematical education have also been obtained
from both students and other teachers.
The research then attempts to examine the suitability of
applying computer assisted learning in Hong Kong education
sectors. Some studies on the comments made by students who
participated in the learning process are undertaken. The
successes and failures in terms of student accomplishment
and interest in the subject area as a result of using a
software package is described. Suggestions and
recommendations are given in the concluding chapter.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.1988-01-01T00:00:00ZMathematical modelling of nonlinear ring waves in a stratified fluid
https://dspace.lboro.ac.uk/2134/18587
Title: Mathematical modelling of nonlinear ring waves in a stratified fluid
Authors: Zhang, Xizheng
Abstract: Oceanic waves registered by satellite observations often have curvilinear fronts and propagate over various currents. In this thesis, we study long linear and weakly-nonlinear ring waves in a stratified fluid in the presence of a depth-dependent horizontal shear flow. It is shown that despite the clashing geometries of the waves and the shear flow, there exists a linear modal decomposition, which can be used to describe distortion of the wavefronts of surface and internal waves, and systematically derive a 2+1-dimensional cylindrical Korteweg-de Vries (cKdV)-type equation for the amplitudes of the waves. The general theory is applied to the case of the waves in a two-layer fluid with a piecewise-constant shear flow, with an emphasis on the effect of the shear flow on the geometry of the wavefronts. The distortion of the wavefronts is described by the singular solution (envelope of the general solution) of the nonlinear first order differential equation, constituting generalisation of the dispersion relation in this curvilinear geometry. There exists a striking difference in the shape of the wavefronts: the wavefront of the surface wave is elongated in the shear flow direction while the wavefront of the interfacial wave is squeezed in this direction. We solve the derived 2+1-dimensional cKdV-type equation numerically using a finite-difference scheme. The effects of nonlinearity and dispersion are studied by considering numerical results for surface and interfacial ring waves generated from a localised source with and without shear flow and the 2D dam break problem. In these examples, the linear and nonlinear surface waves are faster than interfacial waves, the wave height decreases faster at the surface, the shear flow leads to the wave height decreasing slower downstream and faster upstream, and the effect becomes more prominent as the shear flow strengthens.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2015-01-01T00:00:00ZRadiation damage and inert gas bubbles in metals
https://dspace.lboro.ac.uk/2134/17927
Title: Radiation damage and inert gas bubbles in metals
Authors: Gai, Xiao
Abstract: Inert gases in metals can occur due to ion implantation, from a plasma in a magnetron device or as a result of being by-products of nuclear reactions. Mainly because of the nuclear applications, the properties of the inert gases, helium, argon and xenon in the body centred cubic (bcc) iron crystal are examined theoretically using a combination of molecular dynamics, static energy minimisation and long time scale techniques using empirical potential functions. The same techniques are also used to investigate argon and xenon in aluminium.
The primary interest of the work occurred because of He produced in nuclear fission and its effect on the structural materials of a fission reactor. This structure is modelled with perfectly crystalline bcc Fe. In bcc iron, helium is shown to diffuse rapidly forming small bubbles over picosecond time scales, which reach a certain optimum size. In the initial phase of He accumulation, Fe interstitials are ejected. This occurs instantaneously for bubbles containing 5 He atoms and as the more He accumulates, more Fe interstitials are ejected. The most energetically favourable He to vacancy ratios at 0 K, vary from 1 : 1 for 5 vacancies up to about 4 : 1 for larger numbers of vacancies. An existing He bubble can be enlarged by a nearby collision cascade through the ejection of Fe interstitials, allowing more He to be trapped.
Ar and Xe in bcc Fe prefer to be substitutional rather than interstitial and there are large barriers to be overcome for the inert gas atoms to diffuse from a substitutional site. Bubbles that form can again be enlarged by the presence of a nearby collision cascade or at very high temperatures. In this case the most energetically favourable vacancy ratios in the bubbles is 1: 1 for Ar and from 0.6: 1 to 0.8: 1 for Xe. For Ar and Xe, bubble formation is more likely as a direct result of radiation or radiation enhanced diffusion rather than diffusion from a substitutional site.
Ar in aluminium is also studied. Ar atoms in fcc Al prefer to be substitutional rather than interstitial and evolution into substitutional occurs over picosecond time scales at room temperature. Bubble formation can occur more easily than in bcc iron, mainly because the barriers for vacancy diffusion are much lower but the time scales for bubble accumulation are much longer than those for He. A vacancy assisted mechanism is found which allows Ar to diffuse through the lattice. Finally some preliminary results on the energetics of different geometrical structures of larger Xe bubbles in Al are investigated since experiment has indicated that these can become facetted.
Description: A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.2015-01-01T00:00:00Z