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Perturbation theory and asymptotic expansions

This course is given at the doctoral level and corresponds to 5 hp (credit points). The contents can be described as advanced mathematical methods for scientists and engineers. The mathematical tools provided in this course are such that by applying them we can reformulate and reduce difficult problems to a shape where they can more easily be dealt with numerically or even analytically.

The course will be given in the form of nine lectures (2x45 min) and two homework assignments.


The course is planned for late spring 2014 and will be given fairly intensely. It was previously given in 2004. The lectures will be given the following dates and times:

  • Thursday May 15, 10-12, room P2344.
  • Friday May 16, 10-12, room P2344.

  • Monday May 19, 10-12, room P2345.
  • Tuesday May 20, 10-12, room P2344.
  • Wednesday May 21, 10-12, room P2344.

  • Monday May 26, 10-12, room P2344.
  • Tuesday May 27, 10-12, room P2344.

  • Monday June 2, 10-12, room P2345.
  • Tuesday June 3, 10-12, room P2345.


Bengt Fornberg, Dept. of Applied Mathematics, University of Colorado, Boulder, CO, USA and

Natasha Flyer, National Center for Atmospheric Research, Boulder, CO, USA.

Course contents

  • Regular perturbation of polynomial roots
  • Singular perturbation of polynomial roots
  • Convergent versus asymptotic expansions
  • Acceleration (Padé for example): Getting the best out of your five terms
  • Approximate solution of ODEs: Local expansions around regular, regular singular, and irregular singular points
  • Asymptotic methods for integrals: Integration by parts, Watson's lemma, stationary phase, steepest descent, Airy function
  • Dimensional analysis and scaling
  • Boundary layers
  • WKB for linear problems and method of multiple scales

Course material

Some course material can be found here.


The lectures will be based on material from the book Bender and Orzag, Mathematical methods for scientists and engineers, Prentice Hall 1978, reprinted by Springer Verlag.


The examination consists of completing two "take home exams" (homework assignments).

Updated  2015-10-02 00:21:23 by Kurt Otto.