MA2DE2NU-Differentiable Equations II

Module Provider: Mathematics and Statistics
Number of credits: 20 [10 ECTS credits]
Level:5
Terms in which taught: Spring term module
Pre-requisites: MA0MANU Mathematical Analysis and MA1LANU Linear Algebra and MA1DE1NU Differential Equations I
Non-modular pre-requisites:
Co-requisites:
Modules excluded:
Current from: 2021/2

Module Convenor: Dr Peter Sweby
Email: p.k.sweby@reading.ac.uk

Type of module:

Summary module description:

This module is designed to teach students of mathematics a brief introduction to partial differential equations. Students learn in the class could be helpful for them in subsequent courses and research as well as in their career and in life in the long run.

The Module lead at NUIST is Dr Vahid Darvish.

Aims:

This module is intended to teach students of mathematics fundamental knowledge of partial differential equations.

Assessable learning outcomes:

By the end of the module, it is expected that students will be able to:

1. Solve linear first-order PDE with different methods.

2. Solve the heat equation and use Duhamel’s principle to treat sources in the diffusion equation.

3. Solve the wave equation by using D’ Alembert’s method and be introduced to ideas of causality, domain of dependence, and range of influence.

4. State and apply maximum principles to conclude uniqueness results.

5. Solve wave equations, heat equations, Laplace equations and Helmholtz equations on a finite interval by separation of variables and derive simple Green’s functions.

6. Know the fundamental properties of a solution to the Laplace equations.

7. Use energy methods to give apriori estimates for certain partial differential equations.

8. Classify the 2nd order partial differential equations.

9. Know the basic method of the numeric partial differential equations.

10. Know some basic information about some famous Nonlinear PDEs.

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Additional outcomes:

By the end of the module, it is expected that students should be able to apply the above skills to further study and research.Ìý

Outline content:

• Week 1Ìý ÌýOverview of PDE, Conservation law, First order linear equations, Laplace Transform, Exercise class


• Week 2Ìý ÌýMethod of the characteristic curve, inviscid Burgers’ equation, Exercise class


• Week 3Ìý ÌýFourier Transform, Heat Equation I (heat kernel, heat equations on a whole line and half-line), Exercise class


• Week 4Ìý ÌýHeat equations II (Duhamel’s principle, heat equations on a finite rod, separation of variables), Exercise class


• Week 5Ìý ÌýHeat equations III (Weak maximum principle, L^\infty estimates, Energy estimates), Exercise class


• Week 6Ìý ÌýHeat equations IV (Heat equations on higher dimensional space, Appendix), Exercise class


• Week 7Ìý ÌýQuiz No. 1, Wave equations I (Derivation of the wave equation, D’ Alembert’s formula, Causality), Exercise class


• Week 8Ìý ÌýWave equations II (Wave equations on a finite interval, separation of variables, Energy estimates), Exercise classÌý


• Week 9Ìý ÌýWave equations III (Wave equations on higher dimensional space, Huygens principle), Exercise class


• Week 10Ìý Laplace equations I (Overview, Laplace equations on rectangles, cubes, the exterior of circles, wedges, annuli, Separation ofÌý Ìý Ìývariables), Exercise class


• Week 11Ìý Laplace equations II (Laplace equations on a ball, Poisson’s formula, Green’s identities, Green’s function), Exercise class


• Week 12Ìý Laplace equations III (Mean Value inequalities, Maximum principle, Harnack inequality, Convergence theorems, Interior estimates of derivatives, Perron’s method of subharmonic functions), Exercise class, Quiz No. 2


• Week 13Ìý Classification of 2nd order partial differential equations, A brief introduction of t he numeric method I (Finite difference methods), Exercise class


• Week 14Ìý A brief introduction of numeric method II (Finite element methods), Exercise class


• Week 15Ìý A brief introduction of Nonlinear PDE (Shock waves, Solitons, Calculus of Variations), Exercise class


• Week 16Ìý A brief introduction of weak solutions (Idea of weak solutions, Sobolev spaces, Fixed point theorems), Exercise class, Review

Brief description of teaching and learning methods:

Lectures supported by problem sheets and weekly tutorials.

Contact hours:

Ìý	Autumn	Spring	Summer
Lectures		96
Guided independent study:	Ìý	Ìý	Ìý
Ìý Ìý Wider reading (independent)		68
Ìý Ìý Wider reading (directed)		20
Ìý Ìý Peer assisted learning		16
Ìý	Ìý	Ìý	Ìý
Total hours by term	0	200	0
Ìý	Ìý	Ìý	Ìý
Total hours for module	200

Summative Assessment Methods:

Method	Percentage
Written exam	70
Class test administered by School	30

Summative assessment- Examinations:

3 hours.

Summative assessment- Coursework and in-class tests:

One examination and a number of class tests.

Formative assessment methods:

Problem sheets.

Penalties for late submission:

The Support Centres will apply the following penalties for work submitted late:

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.

The University policy statement on penalties for late submission can be found at:
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Assessment requirements for a pass:

A mark of 40% overall.

Reassessment arrangements:

One examination paper of 3 hours duration in August/September - the resit module mark will be the higher of the exam mark (100% exam) and the exam mark plus previous coursework marks (70% exam, 30% coursework).

Additional Costs (specified where applicable):

Last updated: 16 April 2021