A6: Differential Equations 2 - Material for the year 2020-2021

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Prof. Renaud Lambiotte
General Prerequisites: 

It is recommended to take Integral Transforms in parallel with Differential Equations 2.

Course Term: 
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.