# Mathematics

## ECM3708 - Partial Differential Equations (2015)

MODULE TITLE CREDIT VALUE Partial Differential Equations 15 ECM3708 Prof Andrew Gilbert (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 0
 Number of Students Taking Module (anticipated) 88
DESCRIPTION - summary of the module content

A PDE is a differential equation in which the unknown function is a function of multiple independent variables and the equation involves its partial derivatives.  The order is defined similarly to the case of ordinary differential equations, but further classification into elliptic, hyperbolic, and parabolic equations, especially for second order linear equations, is of utmost importance. Some partial differential equations do not fall into any of these categories over the whole domain of the independent variables and they are said to be of mixed type.

On this module, you will learn which types of PDEs can be solved exactly, and which require a numerical approach. Furthermore, you will discover how PDEs can be well-posed or ill-posed, and will find out about a range of analytical and numerical techniques used to solve PDEs.

The module describes how computers can be utilised to model equations from physics. In addition, you will strengthen your ability to interpret theoretical mathematical concepts, and acquire a deeper understanding of how mathematics relates to real world problems.

AIMS - intentions of the module

Partial differential equations (PDEs) form a central part of mathematics. The laws of physics are formulated in terms of PDEs, so the subject is of great practical importance. However, the range of application of PDEs goes beyond the physical world into the modelling of subjects as diverse as ecology and economics. This leads to interesting connections between subjects that at first seem unrelated. The purpose of this module is to develop some of the main analytical and numerical techniques used to solve PDEs, building on the work done in ECM2702 and ECM2706. We will illustrate the topic using a range of real world examples.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
1 demonstrate understanding of the classification of linear partial differential equations (PDEs) of first and second order;
2 apply a range of analytical techniques and a wider knowledge and appreciation of applications of PDEs in mathematics;
3 exhibit detailed knowledge of specific parabolic, elliptic and hyperbolic second order PDEs.
Discipline Specific Skills and Knowledge:
4 complete extended multi-step calculations using a variety of mathematical techniques;
5 translate unfamiliar problems into ones that can be tackled by familiar techniques;
6 show a knowledge of the relevance of PDEs in applications.
Personal and Key Transferable/ Employment Skills and  Knowledge:
7 illustrate self management and time-management skills;
8 express complex abstract arguments in a logical and coherent manner;
9 use learning resources, including e-learning resources to extend their knowledge.

SYLLABUS PLAN - summary of the structure and academic content of the module

- introduction and derivation of PDE models, revision of first PDEs methods, linear, quasilinear and nonlinear cases;
- domains, boundary conditions and well-posedness;
- Green’s function to solve boundary value PDEs;
- second-order linear PDEs and their classification into elliptic, hyperbolic and parabolic problems;
- elliptic problems: uniqueness theorem for Dirichlet and Neumann problems;
- Laplace and Poisson equations;
- Green's function methods for solution of Laplace and Poisson equations in two and three dimensions;
- hyperbolic problems: method of characteristics, wave and Helmholtz equations, inhomogeneous Helmholtz problems;
- parabolic problems: diffusion equation on finite, infinite and semi-infinite domains, solution by transform methods;
- examples from selected applications;
- examples of phenomena from nonlinear PDEs such as Burger’s (shock solutions) and Korteweg-deVries (solutons);
- introduction to basic numerical methods for PDEs, finite difference and finite element methods.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures/example classes Guided independent study 30 Assessment preparation Guided independent study 57 Study of notes and formative examples Guided independent study 30 Wider reading

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Examples sheets 5 during semester Oral Oral at lecture sessions. Solutions posted on ELE
Class test 1 hour All Solutions posted on ELE

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours All Solutions posted on VLE, oral feedback from lecturer
Coursework – assessed questions from problem sheets 20 Selected questions All Solutions posted on VLE, oral feedback from lecturer

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Logan, D.J. Applied Partial Differential Equations 2nd Springer 2004 978-0387209531 [Library]
Set Ockendon J., Howison S. , Lacey A. & Movchan A. Applied Partial Differential Equations Oxford University Press 2003 978-0198527718 [Library]
Set Smith G.D. Numerical Solution of Partial Differential Equations: Finite Difference Methods 3rd Oxford University Press 1985 978-0198596509 [Library]
Set Sneddon I.M. Elements of Partial Differential Equations McGraw-Hill 1957 [Library]
Set Williams W.E. Partial Differential Equations Clarendon Press 1980 978-0198596332 [Library]
Set Tveito, A, Winther, R Introduction to partial differential equations: A computational approach 2008 Springer-verlag 2009 978-3540887041 [Library]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES None None
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 6 No Friday 09 January 2015 Monday 12 January 2015
KEY WORDS SEARCH Partial differential equations; parabolic equations; elliptic equations; hyperbolic equations; boundary value; initial value.