Exercise sheet 3

Exercises and solutions in LaTex

Exercise sheet 3 pdf

Exercises and solutions in PDF

Exercise sheet 3

Exercises and solutions in LaTex

Exercise sheet 3 pdf

Exercises and solutions in PDF

Exercise sheet 3 pdf

Exercises and solutions in PDF

Exercise sheet 3

Exercises and solutions in LaTex

MA311exam 2000

Exam questions and solutions in LaTex

MATH2038 Notes, Lectures and handouts

The MATH2038 (Partial Differential Equations) course, as given in semester 2 2008/9. Syllabus has changed slightly from previous years, as has coursework weighting.

Oxidation of silicon: further tests for the interfacial silicon emission model

The classical description of Si oxidation given by Deal and Grove has well-known limitations for thin oxides (below 200 Ã). Among the large number of alternative models published so far, the interfacial emission model has shown the greatest ability to fit the experimental oxidation curves. It relies on the assumption that during oxidation Si interstitials are emitted to the oxide to release strain and that the accumulation of these interstitials near the interface reduces the reaction rate there. The resulting set of differential equations makes it possible to model diverse oxidation experiments. In this paper, we have compared its predictions with two sets of experiments: (1) the pressure dependence for subatmospheric oxygen pressure and (2) the enhancement of the oxidation rate after annealing in inert atmosphere. The result is not satisfactory and raises serious doubts about the model’s correctness

The Wheal Jane wetlands model for bioremediation of acid mine drainage

Acid mine drainage (AMD) is a widespread environmental problem associated with both working and abandoned mining operations. As part of an overall strategy to determine a long-term treatment option for AMD, a pilot passive treatment plant was constructed in 1994 at Wheal Jane Mine in Cornwall, UK. The plant consists of three separate systems, each containing aerobic reed beds, anaerobic cell and rock filters, and represents the largest European experimental facility of its kind. The systems only differ by the type of pretreatment utilised to increase the pH of the influent minewater (pH <4): lime dosed (LD), anoxic limestone drain (ALD) and lime free (LF), which receives no form of pretreatment. Historical data (1994-1997) indicate median Fe reduction between 55% and 92%, sulphate removal in the range of 3-38% and removal of target metals (cadmium, copper and zinc) below detection limits, depending on pretreatment and flow rates through the system. A new model to simulate the processes and dynamics of the wetlands systems is described, as well as the application of the model to experimental data collected at the pilot plant. The model is process based, and utilises reaction kinetic approaches based on experimental microbial techniques rather than an equilibrium approach to metal precipitation. The model is dynamic and utilises numerical integration routines to solve a set of differential equations that describe the behaviour of 20 variables over the 17 pilot plant cells on a daily basis. The model outputs at each cell boundary are evaluated and compared with the measured data, and the model is demonstrated to provide a good representation of the complex behaviour of the wetland system for a wide range of variables. (C) 2004 Elsevier B.V/ All rights reserved.

Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.

A mathematical model of the in vitro keratinocyte response to chromium and nickel exposure

A mathematical model describing the main mechanistic processes involved in keratinocyte response to chromium and nickel has been developed and compared to experimental in vitro data. Accounting for the interactions between the metal ions and the keratinocytes, the law of mass action was used to generate ordinary differential equations which predict the time evolution and ion concentration dependency of keratinocyte viability, the amount of metal associated with the keratinocytes and the release of cytokines by the keratinocytes. Good agreement between model predictions and existing experimental data of these endpoints was observed, supporting the use of this model to explore physiochemical parameters that influence the toxicological response of keratinocytes to these two metals.