1. use knowledge of partial differential equations (PDEs), modelling, the general structure of solutions, and analytic and numerical methods for solutions.
2. formulate physical problems as PDEs using conservation laws.
3. understand analogies between mathematical descriptions of different (wave) phenomena in physics and engineering.
4. classify PDEs, apply analytical methods, and physically interpret the solutions.
5. solve practical PDE problems with finite difference methods, implemented in code, and analyse the consistency, stability and convergence properties of such numerical methods.
6. interpret solutions in a physical context, such as identifying travelling waves, standing waves, and shock waves.

1. Olver, P.J. (2014), Introduction to Partial Differential Equations, Springer. [Available online from the Barr Smith Library]
2. Agarwal, R. P. & O'Regan, D. (2009), Ordinary and Partial Differential Equations With Special Functions, Fourier Series, and Boundary Value Problems, Springer. [Available on line from the Barr Smith Library]
3. Iserles, A. (2009), A first course in the numerical analysis of differential equations, Cambridge University Press. [Available online from the Barr Smith Library]
4. Ockendon, J.R. et al (2003) Applied Partial Differential Equations, Oxford University Press.
5. Billingham, J. and  King, A.C. (2000) Wave motion, Cambridge University Press.
6. Kreyszig, E. (2011), Advanced engineering mathematics, 10th edn, Wiley.

Tutorial work is integrated into lecture class times.

In more detail, the course includes material from the following.

1. Conservation of mass and momentum: Car traffic has waves; Conservation of fluid; Momentum PDE for ideal gases; The wave equation.
2. Traffic flow and the method of characteristics: Non-uniqueness; Shocks.
3. Separation of variables: Linearity empowers analysis; Separation of variables generates boundary value problems.
4. Wonderful Sturm–Liouville boundary value problems: Self-adjoint operators form Sturm-Liouville problems; Eigenfunctions expand inhomogeneous solutions.
5. Discretising 1D space: Lagrange’s theorem underpins the method of lines; Find equilibria; Numerical linearisation characterises solution dynamics; PDE-free patch dynamics.
6. Modelling shallow water waves: Conservation derives the PDEs; Small amplitude waves; Compute seiches in 1D.
7. 6 PDEs with at least three independent variables: Vibration of a rectangular membrane; The self-adjoint Sturm-Liouville nature of Helmholtz-like PDEs.
8. Computational integration: 1D heat/diffusion PDE raises fundamental issues; Crank–Nicholson schemes are reasonably stable and accurate; Invoke sparse matrices for implicit schemes; Crank–Nicholson discretises wave systems; Second order PDEs in 2D.
9. General wave dynamics: Water waves in finite depth; The dispersion relation of waves; Energy travels at the group velocity; Wave propagation in multi-dimensions.
10. Shocking classification of PDEs: Change of variables transforms the PDE; Reduction to the hyperbolic canonical form; Elliptic and parabolic canonical form.