About this course





MATH4403 - Partial Differential Equations

COURSE CONTENT:

This course will cover theoretical and practical aspects of PDEs, but the focus is on theoretical expositions. The course will progress through the following topics:

  • The maximum principle for differential equations,
  • Maximum principle for elliptic equations,
  • Hopf's maximum principle,
  • Sub and superharmonic functions,
  • Dirichlet problem,
  • Harnack inequality,
  • Perron method,
  • Irregular and regular boundary points,
  • Maximum principle for parabolic equations,
  • Boundary value problems for parabolic equations,
  • Cauchy problem,
  • Boundary value problems for hyperbolic equations.
All of these are either the focus of current research, or are important tools for current research in analysis.

WHERE IS IT USED:

The theory in this course is widely used by researchers looking into PDEs. Publications on PDEs form a huge part of pure and applied literature. The variational techniques that are shown are easily adapted to numerical techniques (to form, for example, the finite element method). Elliptic PDEs are used in biological pattern formation and spacial stress analysis; Parabolic PDEs are used in reaction-diffusion equations and in mathematical finance; Hyperbolic PDEs are used in vibrating systems.

WHO IS INTERESTED:

Any student who wants to know the theory behind differential equations should attend. The course will be most useful for honours students who want a career in academia or industry where PDEs are used.

WHAT DO I NEED:

You should have a good knowledge of the third-year PDE course. In particular, you should know the standard methods of attack like the method of characteristics and separation of variables.

WHEN IS IT AVAILABLE:

Semester 2 odd numbered years.

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