MMath Mathematics and Statistics

This course unit provides students with the methodology to study problems which arise in applied mathematics.  Often, the analytical solution to such a problem involves approximation in terms of a small parameter. We consider asymptotic expansions, such that the error made is controlled.

This course unit introduces students to important topics in applied mathematics, developing their understanding of asymptotic methods and calculus of variations. The syllabus is motivated by physical applications of historical importance.

• Apply asymptotic methods to obtain perturbation expansions of algebraic equations.
• Compute asymptotic expansions of integrals containing a parameter.
• Calculate asymptotic solutions to ODEs containing a parameter, matching the inner and outer solutions as appropriate.
• Interpret variational problems and solve the corresponding Euler-Lagrange equations.
• Calculate solutions to standard variational problems (e.g. the brachistochrone, Fermat’s principle, geodesics, minimal surface, the isoperimetric problem, the hanging chain).
• Formulate equations of motion in classical mechanics from a Lagrangian, via Hamilton’s Principle and describe solutions to mechanical problems using the Euler-Lagrange equations and/or perturbation theory.
• Estimate eigenvalues via the Rayleigh-Ritz method and apply to physical problems.

Individual feedback via marking comments, and cohort feedback document circulated.