Applied Mathematics
MP494: Partial Differential Equations (5 ECTS)
(This course will run every other year.) This course introduces the theory of partial differential equations (PDEs).Topics covered include first order PDEs, linear second order PDEs in two variables, maximum principles and well-posedness of problems, separable variable and similarity solutions.
Taught in Semester(s) I. Examined in Semester(s) I.
Workload: 36 hours (24 Lecture hours, 12 Tutorial hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- solve a first order linear partial differential equation using the method of characteristics;
- solve some nonlinear first order partial differential equations using Charpit's method;
- classify second order linear partial differential equations in two variables and reduce them to canonical form;
- calculate the general solution to some second order linear partial differential equations;
- prove the maximum principle for Laplace's equation in a planar domain and be able to apply it to prove that some problems have unique solutions;
- rigorously justify the validity of some separable variable solutions to Laplace's equation;
- prove a maximum principle for the heat equation;
- construct simple similarity solutions to some parabolic equations.
Indicative Content
This course consists of an introduction to the theory of partial differential equations (PDEs), and the following topics are considered:
- PDEs as mathematical models in the applied sciences, linear equations, wellposed problems;
- first order PDEs: solution of the Cauchy problem using the method of characteristics, an existence and uniqueness theorem, solution of a general nonlinear first order equation;
- second order linear PDEs in two independent variables: classification, reduction to canonical form, general solutions;
- elliptic equations: the Dirichlet problem and the Neumann problem, the maximum principle with applications, uniqueness of solutions, separation of variables;
- parabolic equations: a maximum principle for the heat equation, construction of some similarity solutions, travelling wave solutions to the Fisher equation.
Module Resources
- An Introduction to Partial Differential Equations, 3rd edition, Y. Pinchover & J. Rubinstein, Cambridge University Press
- A First Course in Partial Differential Equations, H.F. Weinberger, Dover Publications
- Applied Partial Differential Equations, J. Ockendon, S. Howison, A. Lacey & A. Movchan, Revised Edition, Oxford University Press
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