Introductory description

The theory of partial differential equations (PDE) is important both in pure and applied mathematics. On the one hand they are used to mathematically formulate many phenomena from the natural sciences (electromagnetism, Maxwell's equations) or social sciences (financial markets, Black-Scholes model). On the other hand since the pioneering work on surfaces and manifolds by Gauss and Riemann partial differential equations have been at the centre of many important developments on other areas of mathematics (geometry, Poincare-conjecture).

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

Subject of the module are four significant partial differential equations (PDEs) which feature as basic components in many applications:
The transport equation, the wave equation, the heat equation, and the Laplace equation. We will discuss the qualitative behaviour of solutions and, thus, be able to classify the most important partial differential equations into elliptic, parabolic, and hyperbolic type. Possible initial and boundary conditions and their impact on the solutions will be investigated. Solution techniques comprise the method of characteristics, Green's functions, and Fourier series.

Learning outcomes

By the end of the module, students should be able to:

• At the end, you will be familiar with the notion of well-posed PDE problems and have an idea what kind of initial or boundary conditions may be imposed for this purpose.
• You will have studied some techniques which enable you to solve some simple PDE problems.
• You will also understand that properties of solutions to PDEs sensitively depend on the type.

Indicative reading list

A script based on the lecturer's notes will be provided. For further reading you may find the following books useful (sections of relevance will be pointed out in the script or in the lectures):
S Salsa: Partial differential equations in action, from modelling to theory. Springer (2008).
A Tveito and R Winther: Introduction to partial differential equations, a computational approach. Springer TAM 29 (2005).
W Strauss: Partial differential equations, an introduction. John Wiley (1992).
JD Logan: Applied partial differential equations. 2nd edt. Springer (2004).
MP Coleman: An introduction to partial differential equations with MATLAB. Chapman and Hall (2005).
M Renardy and RC Rogers: An introduction to partial differential equations, Springer TAM 13 (2004).
LC Evans: Partial differential equations. 2nd edt. American Mathematical Society GMS 19 (2010).

Subject specific skills

At the end, students will be familiar with the notion of well-posed PDE problems and have an idea what kind of initial or boundary conditions may be imposed for this purpose. Students will have studied some techniques which enable you to solve some simple PDE problems. They will also understand that properties of solutions to PDEs sensitively depend on the type.

Transferable skills

The module provides technical competence in solving basic partial differential equations that feature at least as building blocks in applications. There are aspects of critical thinking and creativity related to analysing and solving PDE problems.