Study information

# Partial Differential Equations - 2024 entry

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.

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 1– based on questions submitted for assessment 10 15 hours All Annotated script and written/verbal feedback
Coursework 2– based on questions submitted for assessment 10 15 hours All Annotated script and written/verbal feedback
Written Exam – closed book 80 2 hours (Summer) 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
Written Exam * Written Exam (2 hours) (80%) All August Ref/Def Period
Coursework 1 * Coursework 1 (10%) All August Ref/Def Period
Coursework 2 * Coursework 2 (10%) All August Ref/Def Period

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

RE-ASSESSMENT NOTES

Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 40%.

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
Set Ockendon, J., Howison, S. , Lacey, A. & Movchan, A. Applied Partial Differential Equations Oxford University Press 2003 978-0198527718
Set Sneddon I.M. Elements of Partial Differential Equations McGraw-Hill 1957
Set Smith, G.D. Numerical Solution of Partial Differential Equations: Finite Difference Methods 3rd Oxford University Press 1985 978-0198596509
Set Williams, W.E. Partial Differential Equations Clarendon Press 1980 978-0198596332
Set Tveito, A, Winther, R Introduction to partial differential equations: A computational approach 2008 Springer-Verlag 2009 978-3540887041
Set Logan, D.J. Applied Partial Differential Equations 2nd Springer 2004 978-0387209531
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH2003
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 6 No Tuesday 12th March 2024 Tuesday 12th March 2024
KEY WORDS SEARCH Partial differential equations; parabolic equations; elliptic equations; hyperbolic equations; boundary value; initial value.