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DRPS : Course Catalogue : School of Mathematics : Mathematics

Undergraduate Course: Multi-scale Methods in Mathematical Modelling (MATH11141)

Course Outline
SchoolSchool of Mathematics CollegeCollege of Science and Engineering
Credit level (Normal year taken)SCQF Level 11 (Year 5 Undergraduate) AvailabilityAvailable to all students
SCQF Credits10 ECTS Credits5
SummaryThe aim of this course is to introduce a unified framework for the systematic simplification of a variety of problems that all share the common feature of possessing multiple scales in their description. Multiscale systems are ubiquitous across various scientific areas, including chemical and biological processes or material science, and are characterised by nontrivial interactions between a wide range of spatial and temporal scales. The high complexity of multi-scale systems implies that accurate description of the underlying problem is either impossible or practically intractable and, instead, a coarse-grained approach must be used. The set of techniques discussed in this course - commonly referred to as averaging and homogenisation - is applicable to problems characterised by separation of scales and described by either ODEs, PDEs or SDEs.

The driving principle behind this approach is to derive systematic approximations of the original highly heterogeneous system so that the simplified description, which effectively 'averages out' the
microscopic features, provides an accurate description of the system properties at the 'macro' scales of interest. The main advantage of this approach is that the resulting equations are much more amenable to rigorous analysis and numerical implementation. We will also discuss conditions which are necessary for the solution to the full equations to converge to the averaged/homogenised description in the limit of the scale of the small-scale inhomogeneities tending to zero.
Course description Motivating examples
Basics of ODEs and probability
Multiple-scale perturbation expansions; singular perturbations
Slow and fast dynamics in ODEs; Dimension reduction in ODEs; The Fredholm Alternative Invariant manifolds and 'slow' manifolds in ODEs; chaos & shadowing lemmas
Averaging and Homogenisation for ODEs (Hamiltonian & dissipative systems)
Averaging and Homogenisation for PDEs
Convergence Theorems
Optional/additional: Averaging and Homogenisation for SDEs
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: Honours Differential Equations (MATH10066)
Prohibited Combinations Other requirements None
Information for Visiting Students
High Demand Course? Yes
Course Delivery Information
Not being delivered
Learning Outcomes
- Motivate and explain the concepts of multi-scale expansions and the singular perturbations.
- Describe with the concepts of averaging and homogenisation in ODEs and PDEs, as well as the limitations of the approach.
- Ability to identify and apply suitable transformations to reduce various problems encountered in practice to the general framework considered in the course.
- Explain and apply the key aspects of the solvability conditions via the Fredholm alternative and the need for considering them in the context of averaging and homogenisation.
- The ability to explain the concept of and to determine invariant manifolds and 'slow' manifolds in systems of ODEs.
- Ability to apply homogenisation and averaging techniques to simple, low-dimensional ODEs and PDEs.
- Optional/additional: Understanding the key aspects of convergence theorems underlying the developed framework.
Reading List

G. A. Pavliotis and A. M. Stuart. Multiscale Methods: Averaging and Homogenization, Springer, 2008. (Main course text)

D. Cioranescu and P. Donato. An Introduction to Homogenization. Oxford University Press, New York, 1999.

M. H. Holmes, Introduction to Perturbation Methods, Springer, 2012.
Additional Information
Graduate Attributes and Skills Not entered
Course organiserDr Michal Branicki
Tel: (0131 6)50 4878
Course secretaryMr Thomas Robinson
Tel: (0131 6)50 4885
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