MATH8702 Methods of Applied Mathematics GE

This topic includes: First and second order differential equations in the phase plane. Linear approximations at equilibrium points. Index of a point; limit cycles; averaging, regular and singular perturbation methods. Stability and Liapunov's method. Bifurcation. Basic ideas of calculus of variations. The Euler-Lagrange equations; eigenvalue problems. Applications to second and higher order differential and partial differential equations. Rayleigh-Ritz and Galerkin methods and discrete models.

This topic aims to:

• Provide theoretical and practical understanding of applied mathematics
• Improve ability to analyse and solve applied problems using mathematics
• Improve ability to work individually and as part of a group to solve problems
• Improve capacity to undertake lifelong learning

1. Understand phase plane analysis
2. Understand equilibrium points and linearization
3. Calculate stability of equilibrium points and application of the Lyapunov method
4. Calculate averaging, regular and singular perturbation methods to solve differential equations
5. Understand bifurcation of solutions to differential equations
6. Formulate solutions to ordinary and partial differential equations as problems in calculus of variations
7. Find approximate solutions to differential equations through the Rayleigh-Ritz and Galerkin methods
8. Understand the role and theoretical basis of discrete models and be familiar with basic practical methods