# Mathematics

## MTH3008 - Partial Differential Equations (2019)

MODULE TITLE CREDIT VALUE Partial Differential Equations 15 MTH3008 Prof Vadim N Biktashev (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.

In this module, you will learn how PDEs can be well-posed or ill-posed, and will find out about a range of analytical techniques used to solve PDEs. The module will strengthen your ability to interpret theoretical mathematical concepts, and acquire a deeper understanding of how mathematics relates to real world problems. The module builds on material in the Differential Equation module MTH2003, in particular separation of variables and Fourier series. The module Vector Calculus and Applications MTH2004 is also a prerequisite, in particular definitions and use of gradient, divergence, and the divergence theorem.

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 MTH2003 and MTH2004. 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. Examples of PDE models. First order PDEs. Linear, quasilinear and nonlinear cases;

- Second-order linear PDEs and their classification into elliptic, hyperbolic and parabolic classes. Domains, boundary conditions and well-posedness;

- Hyperbolic equations: method of characteristics, canonical form, wave equation in one and three dimensions, conservation of energy, D'Alembert and Kirchhoff formulas;

- Parabolic equations: canonical form, uniqueness and stability theorems, diffusion equation on finite and infinite domains, solution by transform methods;

- Elliptic equations: canonical form, uniqueness theorem, Laplace equation in finite and infinite domains, solution by transform methods;

- Green's function methods for solving non-homogeneous linear equations;

- Selected examples of solutions to nonlinear PDEs.

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 All Oral at lecture sessions. Solutions posted on ELE

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Coursework – based on questions submitted for assessment 20 2 assignments, 30 hours total All Annotated script and written/verbal feedback
Written Exam – closed book 80 2 hours All Written/verbal on request, SRS

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

Type Author Title Edition Publisher Year ISBN Search
Set Ockendon, J., Howison, S. , Lacey, A. & Movchan, A. Applied Partial Differential Equations Oxford University Press 2003 978-0198527718 [Library]
Set Sneddon I.M. Elements of Partial Differential Equations McGraw-Hill 1957 [Library]
Set Smith, G.D. Numerical Solution of Partial Differential Equations: Finite Difference Methods 3rd Oxford University Press 1985 978-0198596509 [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]
Set Logan, D.J. Applied Partial Differential Equations 2nd Springer 2004 978-0387209531 [Library]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH2003, MTH2004
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 6 No Tuesday 10 July 2018 Friday 30 August 2019
KEY WORDS SEARCH Partial differential equations; parabolic equations; elliptic equations; hyperbolic equations; boundary value; initial value.