# MAM3040W - Applied Mathematics 3040

72 credits at NQF level 7

## Entry Requirements:

MAM2000W and either MAM2046W or both MAM2047H and MAM2048H

## Course Outline:

This course introduces students to advanced topics in Applied Mathematics. 3MP: METHODS OF MATHEMATICAL PHYSICS (MAM3043F in EBE) Fourier-transform solution of linear PDEs on the line. Long-term asymptotic behaviour: methods of Laplace, stationary phase, steepest descents. Nonlinear waves: Riemann invariants. Effect of dissipation; Cole-Hopf transform for the Burgers equation; travelling fronts for the KPP equation. Effect of dispersion: KdV, sine-Gordon equation. Elliptic integrals, elliptic functions; cnoidal waves, solitons. Multisoliton solutions: Hirota method, Baecklund transformations. 3CV: METHODS OF FUNCTIONS OF COMPLEX VARIABLES Complex calculus, calculus of residues, special functions, applications to physics. 3AN: ADVANCED NUMERICAL METHODS (MAM3050S in EBE) Advanced methods for ODEs, boundary value problems, differential eigenvalue problems. Numerical solution of PDEs by methods of finite differences, finite elements and spectral methods. 3GR: INTRODUCTION TO GENERAL RELATIVITY (MAM3049S in EBE) This course introduces special relativity, taught in a blended learning fashion (online lectures and tutorials) and general relativity including tensors, the metric tensor, symmetries, curvature, Einstein's field equations and solutions of Minkowski space and Black Holes. Application, description of fluids, equations of fluid flow for simple fluids, analytical techniques.