Thesis (M. Sc.)--University of Adelaide, Dept. of Applied Mathematics, 1992. / Includes bibliographical references (leaves 79-80).

No description available.

Thesis (Ph. D.)--University of Sydney, 1999. / Title from title screen (viewed Apr. 21, 2008). Submitted in fulfilment of the requirements for the degree of Doctor of Philosophy to the School of Mathematics and Statistics, Faculty of Science. Includes bibliography. Also available in print form.

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1999. / Includes bibliographical references (p. 203-205).

Thesis (Ph. D.)--University of Washington, 2004. / Vita. Includes bibliographical references (p. 73-77).

Thesis (Ph.D.)--Memorial University of Newfoundland, 1999. / Bibliography: leaves 121-135.

No description available.

We are interested in modeling the Darwinian evolution of a population described by two levels of biological parameters: individuals characterized by an heritable phenotypic trait submitted to mutation and natural selection and cells in these individuals influencing their ability to consume resources and to reproduce. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We are looking for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses.

This thesis is divided into two independent parts. The first part concerns the 3D Navier-Stokes equations. The second part deals with regularity issues for a family of integro-differential equations. In the first part of this thesis, we consider weak solutions of the 3D Navier-Stokes equations with L² initial data. We prove that ([Nabla superscript alpha])u is locally integrable in space-time for any real [alpha] such that 1 < [alpha] < 3. Up to now, only the second derivative ([Nabla]²)u was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in local weak-L[superscript (4/([alpha]+1))]. These estimates depend only on the L² norm of the initial data and on the domain of integration. Moreover, they are valid even for [alpha] ≥ 3 as long as u is smooth. The proof uses a standard approximation of Navier-Stokes from Leray and blow-up techniques. The local study is based on De Giorgi techniques with a new pressure decomposition. To handle the non-locality of fractional Laplacians, Hardy space and Maximal functions are introduced. In the second part of this thesis, we consider non-local integro-differential equations under certain natural assumptions on the kernel, and obtain persistence of Hölder continuity for their solutions. In other words, we prove that a solution stays in C[superscript beta] for all time if its initial data lies in C[superscript beta]. Also, we prove a C[superscript beta]-regularization effect from [mathematical equation] initial data. It provides an alternative proof to the result of Caffarelli, Chan and Vasseur [10], which was based on De Giorgi techniques. This result has an application for a fully non-linear problem, which is used in the field of image processing. In addition, we show Hölder regularity for solutions of drift diffusion equations with supercritical fractional diffusion under the assumption [mathematical equation]on the divergent-free drift velocity. The proof is in the spirit of Kiselev and Nazarov [48] where they established Hölder continuity of the critical surface quasi-geostrophic (SQG) equation by observing the evolution of a dual class of test functions. / text

Ordinary and partial differential equations are often derived as a first approximation to model a real-world situation, where the state of the system depends not only on the present time, but also on the history of the system. In this situation, a higher level of realism can be achieved by incorporating distributed delays in the mathematical models described by differential equations which results in delay Volterra integro-differential equations (denoted DVIDEs). Although DVIDEs serve as indispensable tools for modelling real systems, we still lack efficient and reliable software to approximate the solution of systems of DVIDEs. This thesis is concerned with designing, analyzing and implementing an efficient method to approximate the solution of a general system of neutral Volterra integro-differential equations with time-dependent delay arguments using a continuous Runge-Kutta (CRK) method. We introduce an adaptive stepsize selection strategy resulting in an approximate solution whose associated defect (residual) satisfies certain properties that allow us to monitor the global error reliably and efficiently. We prove the classic and optimal convergence of the numerical approximation to the exact solution. In addition, a companion system of equations is introduced in order to estimate the mathematical conditioning of the problem. A side effect of introducing this companion system is that it provides an effective estimate of the global error of the approximate solution, at a modest increase in cost. We have implemented our approach as an experimental Fortran 90 code capable of handling various kinds of DVIDEs with non-vanishing, vanishing, and infinite delay arguments.