Advanced Partial Differential Equations

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Demonstrate knowledge and understanding partial differential equations and how they relate to different modelling situations
• Understand the concept of well-posedness.
• Understand the existence of weak solutions and shocks.
• Understand the concept of the symbol of a PDE and the resulting classification of PDEs.
• Show logical thinking in problem solving.
• Understand similarity solutions and their applications.
• Solve linear ODEs and PDEs with the use of the Green's function method.

Common linear PDEs and their boundary conditions

Cauchy data and the Cauchy-Kowalewski expansion

Weak solutions of linear PDEs

Well-posedness

Classification of PDEs and PDE systems from their principal symbol

Scalar conservation laws and the Riemann problem

Generalised functions and the delta-function

Green's functions for ODEs

Green's functions and applications for Laplace, Poisson and Helmholtz equations

Green's functions and applications for the heat equation

Green's functions and applications for the the wave equation

Teaching and learning methods

Teaching methods include:

• 12 tutorials in the form of problem classes
• Summary lecture notes, solutions to problems and mock examination paper available on the Blackboard site for the module
• Chalk and talk lectures

Learning activities include:

• Individual Study
• Working through problems set in lecture notes at the end of each chapter
• Working through mock exam paper and the coursework assignment

Resources & Reading list

General Resources

Lecture notes. The module is based on summary lecture notes which are provided. There is no recommended book, but some suggestions are contained in an information sheet handed out at the start of the module and available on blackboard.

Textbooks

STRAUSS W.. Partial differential equations.

OCKENDON J, HOWISON S, LACEY A & MOVCHAN A. Applied Partial Differential Equations.

HOWISON S.. Practical Applied Mathematics.

LEVEQUE R.. Numerical Methods for Conservations Laws.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.

Repeat

An internal repeat is where you take all of your modules again, including any you passed. An external repeat is where you only re-take the modules you failed.

Repeat Information

Repeat type: Internal & External