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 by successive approximation and Gronwall’s inequality; phase plane analysis; method of characteristics for first order semilinear 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: Existence Theory for ODEs: Picard’s Theorem for first-order scalar ODEs with proof of existence via Picard iteration. Gronwall’s inequality leading to uniqueness and continuous dependence on the initial data. Examples of nonuniqueness. Existence until potential time of blow up for smooth right hand sides. Discussion of blow-up versus global existence via comparison with explicitly solvable ODEs. Extension of Picard’s theorem to systems (statement, but no proof) and applications to second order ODEs. (5 lectures)
Phase plane analysis: Phase planes, critical points, definition of stability, classification of critical points and linearisation, Bendixson-Dulac criterion. (4 lectures)
First-order semilinear PDEs in two independent variables: Solution by method of characteristics. Well-posedness and domain of definition for the Cauchy problem. Second-order semilinear PDEs in two independent variables: Classification and reduction to normal form (done in detail only for the hyperbolic case). Solution of simple hyperbolic IVPs via reduction to normal form. Maximum Principle leading to uniqueness and continuous dependence on the data for elliptic and parabolic PDEs. (7 lectures)
