Course Overview: The aim of this course is to introduce students reading mathematics to some of the basic theory of ordinary and partial differential equations.
Much of the study of differential equations in the first year consisted of finding explicit solutions of particular ODEs or PDEs. However, for many differential equations which arise in practice one is unable to give explicit solutions and, for the most part, this course considers what information one can discover about solutions without actually finding the solution. Does a solution exist? Is it unique? Does it depend continuously on the initial data? How does it behave asymptotically? What are appropriate data? In this course some techniques will be developed for answering such questions.
The course will furnish undergraduates with the necessary skills to pursue any of the applied options in the third year and will also form the foundation for deeper and more rigorous courses in differential equations, the part B courses on Distribution Theory and on Fourier Analysis and the Part C courses Functional Analytic Methods for PDEs and Fixed Point Methods for Nonlinear PDEs.
Learning Outcomes: Students will have learnt a range of different techniques and results used in the study of ODEs and PDEs, such as: Picard's theorem proved both by successive approximation and the contraction mapping theorem; Gronwall's inequality; phase plane analysis; method of characteristics for first order semi-linear PDEs; classification of second order semi-linear PDEs and their reduction to normal form using characteristic variables; well posedness; the maximum principle and some of its consequences. They will have gained an appreciation of the importance of existence and uniqueness of solution and will be aware that explicit analytic solutions are the exception rather than the rule.
Course Synopsis: Picard's Existence Theorem: Picard's Theorem for first-order scalar ODEs with proof. Gronwall's inequality leading to uniqueness and continuous dependence on the initial data. Examples of blow-up and nonuniqueness, discussion of continuation and global existence. Proof of Picard's Theorem via Contraction Mapping (Theorem CMT to be covered in Metric Spaces course) and relationship between the two proofs; extension to systems. Application to scalar second order ODEs, with particular reference to linear equations. (5 lectures)
Phase plane analysis: Phase planes, critical points, definition of stability, classification of critical points and linearisation, Bendixson-Dulac criterion. (4 lectures)
PDEs in two independent variables: First order semi-linear PDEs (using parameterisation). Classification of second order, semilinear PDEs; Normal form; Ideas of uniqueness and wellposedness. Illustration of suitable boundary conditions by example. Poisson's Equation and the Heat Equation: Maximum Principle leading to uniqueness and continuous dependence on the initial data.
(7 lectures)
