# MAM3040W

72 HEQF credits at level 7

**Course co-ordinator(s):** Dr. A. Pototsky

**Entrance requirements:** MAM2046W or MAM2047H and MAM2048H; and MAM2000W

**Course outline:** A total of five 30-lecture modules is offered, four of which make up MAM3040W. The module 3MP constitutes core (compulsory) material while the other three modules can be chosen from 3CV, 3FD, 3AN and 3GR. Modules 3AN and 3CV will be offered in the first semester, and modules 3FD, 3GR and 3MP in the second semester. A written project will be a compulsory component of the course.

**Syllabuses:**

3MP METHODS OF MATHEMATICAL PHYSICS (coded as MAM3043S for Engineering students)

The Fourier-transform and Laplace-transform solution of linear PDEs on the line; the influence function; the Parseval identity. The long-term asymptotic behaviour of solutions: the methods of Laplace, stationary phase and steepest descents. Nonlinear waves: the method of characteristics; Riemann invariants. The effect of dissipation; the Cole-Hopf transform for the Burgers equation; travelling fronts for the KPP equation. The effect of dispersion: KdV, nonlinear Schroedinger and sine-Gordon equation. Elliptic integrals and elliptic functions; cnoidal waves and solitons; kinks and breathers for the sine-Gordon equation. Multisoliton solutions: the Hirota method and Baecklund transformations.

3CV METHODS OF FUNCTIONS OF COMPLEX VARIABLES

Complex calculus, calculus of residues, special functions, applications to physics.

3AN ADVANCED NUMERICAL METHODS (coded as MAM3050F for Engineering Students). Advanced methods for ODEs boundary value problems, differential eigenvalue problems. Numerical solution of PDEs by the methods of finite differences, finite elements and spectral methods.

3GR INTRODUCTION TO GENERAL RELATIVITY (coded as MAM3049S for Engineering students) Note: This module is also available to interested parties on the Internet. Further details may be obtained from the website http://www.mth.uct.ac.za/omei/gr. Christoffel relations, geodesics, curvature, the Riemann tensor. The energy-momentum tensor in electrodynamics and fluid dynamics. Principle of equivalence, Einstein's field equations. Black holes, gravitational waves.

3FD FLUID DYNAMICS (coded as MAM3054S for Engineering students)

Description of fluids, equations of fluid flow for simple fluids, analytical techniques. Applications.

**Lecture periods:** Every day of the week in 3rd period

**Tutorials:** One tutorial per week, Thursday, 6th and 7th periods.

**DP requirements:** A class record of 30% or more is required in each module of the course.

**Assessment:** For modules 3GR and 3FD the year mark counts 25% and the examination counts 75%. For modules 3MP, 3AN and 3CV, the year mark counts 35% and the examination counts 65%. A subminimum of 40% is required for the written project. The papers for module 3AN and 3CV are written in June and modules 3FD, 3GR and 3MP are written in October/November. All papers are no longer than 2 hours, except 3GR which is no longer than 3 hours.