This course consists of the 6 lectures Introduction to Applied Mathematics and a further 21 lectures on Mathematical Methods, details of which are given below.

Examination questions will be set on the whole course.

Synopsis

Linear Ordinary Differential Equations

Initial value problems, fundamental matrices, Volterra integral equations, stability, Floquet theory.

Boundary value problems for second order equations, Fredholm integral equations, Greens functions, Sturm Liouville theory, eigenfunction expansions, comparison theorems, Rayleigh Ritz method.

Equations with delay.

Transforms and Distributions

Fourier Transform from Fourier Series, Laplace Transforms, generalisations.

Distributions as limits and functionals, Greens functions.

Nonlinear Ordinary Differential Equations

Gradient systems and Lyapunov functions, Hamiltonian Systems.

Reading List

1. Coddington and Levinson, Theory of Ordinary Differential Equations. Chapters 3,7,8.
2. Mattheij and Molenaar, Ordinary Differential Equations in Theory and Practice. Chapters 4,5.
3. Hildebrand, Methods of Applied Mathematics. Chapter 4.
4. Stakgold, Green's Functions and Boundary Value Problems. Chapters 1,2,3.
5. Jordan and Smith, Nonlinear Ordinary Differential Equations. Chapters 8,9,10.
6. Collins, Differential and Integral Equations. Chapters 1-4,8,9,11,12,14,15.