# DEGREE REGULATIONS & PROGRAMMES OF STUDY 2018/2019

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

 School School of Mathematics College College of Science and Engineering Credit level (Normal year taken) SCQF Level 11 (Year 5 Undergraduate) Availability Available to all students SCQF Credits 10 ECTS Credits 5 Summary The 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) Convergence Theorems
 Pre-requisites Students MUST have passed: Honours Differential Equations (MATH10066) Co-requisites Prohibited Combinations Other requirements None
 Pre-requisites Visiting students are advised to check that they have studied the material covered in the syllabus of any pre-requisite course listed above before enrolling High Demand Course? Yes
 Academic year 2018/19, Available to all students (SV1) Quota:  None Course Start Semester 2 Timetable Timetable Learning and Teaching activities (Further Info) Total Hours: 100 ( Lecture Hours 22, Seminar/Tutorial Hours 5, Summative Assessment Hours 2, Programme Level Learning and Teaching Hours 2, Directed Learning and Independent Learning Hours 69 ) Assessment (Further Info) Written Exam 95 %, Coursework 5 %, Practical Exam 0 % Additional Information (Assessment) Coursework 5%, Examination 95% Feedback Not entered Exam Information Exam Diet Paper Name Hours & Minutes Main Exam Diet S2 (April/May) Multi-scale Methods in Mathematical Modelling (MATH11141) 2:00
 On completion of this course, the student will be able to: Apply the method of multiple scales to ODEs.Identify and apply suitable transformations to 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.Explain the concept of invariant manifolds and 'slow' manifolds in the systems of ODEs, and determine these.Apply homogenisation and averaging techniques to simple, low-dimensional ODEs.
 Recommended: 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.
 Graduate Attributes and Skills Not entered Keywords MSM
 Course organiser Dr Benjamin Goddard Tel: (0131 6)50 5127 Email: B.Goddard@ed.ac.uk Course secretary Mr Martin Delaney Tel: (0131 6)50 6427 Email: Martin.Delaney@ed.ac.uk
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