OverviewLinear PDEs. Dispersion relations. Review of d'Alembert's solutions of the wave equation.
Quasi-linear first-order PDEs. Total differential equations. Integral curves and integrability conditions. The method of characteristics.
Shock waves. Discontinuous solutions. Breaking time. Rankine-Hugoniot jump condition. Shock waves. Rarefaction waves. Applications of shock waves, including traffic flow.
General first-order nonlinear PDEs. Charpit's method, Monge Cone, the complete integral.
Nonlinear PDEs. Burgers' equation; the Cole-Hopf transformation and exact solutions. Travelling wave and scaling solutions of nonlinear PDEs. Applications of travelling wave and scaling solutions to reaction-diffusion equations. Exact solutions of nonlinear PDEs. Applications of nonlinear waves, including to ocean waves (e.g. rogue waves, tsunamis).
Level 7 Students only. Further applications of shock waves and nonlinear waves.
This module appears in:
Method of assessment
80% examination and 20% coursework.
M.J. Ablowitz, Nonlinear Dispersive Waves, Cambridge (2011)
J. Bellingham and A.C. King, Wave Motion, Cambridge (2000)
P.G. Drazin and R.S. Johnson, Solitons: an Introduction, Cambridge (1989)
R. Knobel, An Introduction to the Mathematical Theory of Waves, A.M.S. (2000)
J.D Logan, An Introduction to Partial Differential Equations, Wiley (1994)
I.N. Sneddon, Elements of Partial Differential Equations, McGraw-Hill (1957)
The intended subject specific learning outcomes. On successfully completing the level 7 module students will be able to:
1 demonstrate systematic understanding of linear and nonlinear PDEs;
2 demonstrate the capability to solve complex problems using a very good level of skill in calculation and manipulation of the material in the following areas: Fourier transforms for linear differential equations, shock waves, exact solutions of nonlinear PDEs;
3 apply a range of concepts and principles in PDEs in loosely defined contexts, showing good judgment in the selection and application of tools and techniques;
4 make effective and well-considered use of MAPLE.
The intended generic learning outcomes. On successfully completing the level 7 module students will be able to:
1 work competently and independently, be aware of their own strengths and understand when help is needed;
2 demonstrate a high level of capability in developing and evaluating logical arguments;
3 communicate arguments confidently with the effective and accurate conveyance of conclusions;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make effective use of information technology skills such as using online resources (Moodle);
7 demonstrate an increased level of skill in numeracy and computation.