- Lecturer: Ian Hewitt
General Prerequisites:
It is recommended to take Integral Transforms in parallel with Differential Equations 2.
Course Term: Hilary
Course Lecture Information: 16 lectures
Course Overview:
This course continues the Differential equations 1 course, with the focus on boundary value problems. The course aims to develop a number of techniques for solving boundary value problems and for understanding solution behaviour. The course concludes with an introduction to asymptotic theory and how the presence of a small parameter can affect solution construction and form.
Learning Outcomes:
Students will acquire a range of techniques for solving second order ODE's and boundary value problems. They will gain a familiarity with ideas that are applicable beyond the direct content of the course, such as the Fredholm alternative, Bessel functions, and asymptotic expansions.
Course Synopsis:
Models leading to two point boundary value problems for second order ODEs
Inhomogeneous two point boundary value problems (\(Ly=f\)); Wronskian and variation of parameters. Green's functions.
Adjoints. Self-adjoint operators. Eigenfunction expansions (issues of convergence and completeness noted but full treatment deferred to later courses). Sturm-Liouville theory. Fredholm alternative.
Series solutions. Method of Frobenius. Special functions.
Asymptotic sequences. Approximate roots of algebraic equations. Regular perturbations in ODE's. Introduction to boundary layer theory.
Inhomogeneous two point boundary value problems (\(Ly=f\)); Wronskian and variation of parameters. Green's functions.
Adjoints. Self-adjoint operators. Eigenfunction expansions (issues of convergence and completeness noted but full treatment deferred to later courses). Sturm-Liouville theory. Fredholm alternative.
Series solutions. Method of Frobenius. Special functions.
Asymptotic sequences. Approximate roots of algebraic equations. Regular perturbations in ODE's. Introduction to boundary layer theory.