Department of Mathematical Sciences, Unit Catalogue 2006/07

MA50175 Topics in differential equations

Credits: 6

Level: Masters

Semester: 2

Assessment: CW 25%, EX 75%

Requisites:

Before taking this unit you must take MA50190

Aims & Learning Objectives:
Aims: To investigate and model phenomena associated with multiple scales; to describe mathematical methods for modelling, analysis and numerical approximation of multiscale problems; to apply these techniques to problems in materials science; and to introduce students to the literature on the subject. On completing the course, students will have an advance knowledge of a modern area of the study of differential equatios and their applications.
Objectives: At the end of the course, students should be familiar with various multiscale problems. They should be able to analyse selected problems, such as phase transitions and the formation of microstructures; they should be able to model multiscale problems; and they should be able to investigate the behaviour of multiscale materials analytically and numerically.
Content:
Review of continuum mechanics. Example of multiscale problems in materials science. Crystallographic description. Twinning and compatibility. Microstructures and their variational treatment. Averaging and relaxation. Recoverable strains. Gamma-convergence. Atomic to continuum passage. Diffusion in periodic media. Homogenisation. Two-scale methods.
References:
* K. Bhattacharya, Microstructure of Martensite. Why it forms and how it gives rise to the shape-memory effect, Oxford Series of Materials Modelling 2, Oxford University Press, 2003.
* A. Braides, Gamma-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22, Oxford University Press, 2002.
* D. Cioranescu & P. Donato, An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications 17, Oxford University Press, 1999.
* U. Hornung, Homogenization and porous media, Springer Series on Interdisciplinary Applied Mathematics 6, Springer, 1997.
Supplementary texts, in addition to the above and selected articles:
* J. Jost & Y. Li-Jost, Xianqing, Calculus of variations, Cambridge Studies in Advanced Mathematics 64, Cambridge University Press, 1998.
* S. Müller, Variational models for microstructure and phase transitions, in: Calculus of variations and geometric evolution problems (Cetraro, 1996), Springer Lecture Notes in Math. 1713, 85-210, Springer, 1999.

University | Catalogues for 2006/07