Learning Outcomes:
On completion of this module students should be able to
1. Understand conditions guaranteeing existence and uniqueness results for ordinary differential equations and recognize examples where those conditions do not hold;
2. State and prove Picard’s theorem;
3. Transform between an initial value problem and the corresponding Volterra integral equation;
4. Transform between a boundary value problem and the corresponding Fredholm integral equation;
5. State the axiomatic properties of the Green function for a second order initial value problem and boundary value problem;
6. Understand the concept of the adjoint differential operator;
7. Recognise a Sturm-Liouville eigenvalue problem and prove the basic properties of eigenvalues and eigenfunctions;
8. Understand the relationship between the Dirac delta function and the Fourier integral;
9. Understand the fundamental properties of infinite dimensional vectors spaces.
10. Prove key results such as Bessel’s inequality, Parseval’s equality and its relationship to completeness