## 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 Differentiable Equations I
Non-modular pre-requisites:
Co-requisites:
Modules excluded:
Current from: 2020/1

Module Convenor: Dr Peter Sweby

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 Zijin Li

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.

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

Outline content:

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

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

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

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

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

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

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

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

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

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

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

• Week 12 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 13 Classification of 2nd order partial differential equations, A brief introduction of the numeric method I (Finite difference methods), Exercise class

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

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

• Week 16 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:
 Autumn Spring Summer Lectures 96 Tutorials 32 Guided independent study: Wider reading (independent) 36 Wider reading (directed) 20 Peer assisted learning 16 Total hours by term 0 0 Total hours for module 200

Summative Assessment Methods:
 Method Percentage Written exam 70 Set exercise 30

Summative assessment- Examinations:

3 hours.

Summative assessment- Coursework and in-class tests:

One examination and a number of assignments.

Formative assessment methods:

Problem sheets.

Penalties for late submission:

The Module Convenor 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: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
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).