Synopsis for B5b: Applied Partial Differential Equations

Level: H-level Method of Assessment: Written examination.
Weight: Half-unit (OSS paper code 2B44)

Recommended Prerequisites:

Calculus of Variations and Fluid Mechanics from Part A are desirable but not essential. The introductory Michaelmas Term course B568a is a prerequisite for both parts of the course, and the material in that course will be assumed to be known.

Overview

This course continues the Part A Differential Equations course, and extends some of the techniques of B5a to partial differential equations. In particular, first-order conservation laws are solved and the idea of a shock is introduced; general nonlinear first-order partial differential equations are solved, the classification of second-order partial differential equations is extended to systems, with hyperbolic systems being solved by characteristic variables. Then Green's function, maximum principle and similarity variable methods are demonstrated for partial differential equations.

Learning Outcomes

Students will know a range of techniques to solve PDEs including non-linear first-order and second-order and their classification. They will be able to demonstrate various principles for solving PDEs including Green's function, maximum principle and similarity solutions.

Synopsis

First-order equations: conservation laws and shocks. Charpit's equations; eikonal equation. [4 lectures] Systems of partial differential equations, characteristics. Shocks; viscosity solutions; weak solutions. [4 lectures] Maximum principles, well-posed problems for the heat equation and for Laplace's equation. [3 lectures] Similarity solutions. [2 lectures] Fundamental solution for the heat equation and for Laplace's equation via delta functions and similarity solutions. [3 lectures]

1. Dr Norbury's web notes.
2. Institute lecture notes are now available (JN).
3. M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations (Springer-Verlag, New York, 2004).
4. J. P. Keener, Principles of Applied Mathematics: Transformation and Approximation (revised edition, Perseus Books, Cambridge, Mass., 2000).
5. J. R. Ockendon, S. D. Howison, A. A. Lacey and A. B. Movchan, Applied Partial Differential Equations (revised edition, Oxford University Press, Oxford, 2003).