TAM 549: Asymptotic Methods

Class Description:

Mathematical equations arising from engineering and physical problems are, more often than not, intractable analytically. Engineers and Scientists are therefore forced to make approximations. Perturbation and Asymptotic methods are among the most important tools available to Engineers and Scientists, for obtaining rational and reliable approximations. Taking advantage of the relative magnitude of the different controlling parameters and/or the disparate scales, a complicated problem is replaced by a set of simpler problems that can be either solved analytically or, if a numerical solution is required, can be solved by a relatively simple strategy. This procedure permits the construction of reasonably accurate solutions with deep physical insight. An awareness of the structure of the solution obtained by perturbation methods is often helpful even when a direct numerical simulation of the full problem is adopted. Perturbation and numerical methods, therefore, complement one another.

The goal in this course is to convey the main ideas of perturbation theory by illustrating the approach on examples taken from the various engineering disciplines and the physical sciences.  The discussion will cover regular perturbation problems, singular perturbation problems of secular- type and layer-type, linear and nonlinear stability and bifurcation theory. The methods of multi-scale, Krylov-Bogoliubov averaging, strained coordinates, matched asymptotic expansions, homogenization, WKB, Laplace's method for integrals and the method of steepest descent will be introduced, along with the notions of ordering, asymptotic expansions, limit process expansions, uniform expansions and distinguished limits.

Books on Reserve -

Holmes, M. H. - "Introduction to Perturbation Methods", Springer-Verlag, 1995

Van Dyke, M. "Perturbation Methods in Fluid Mechanics", Parabolic Press

Kevorkian J. and Cole J. D. "Multiple Scale and Singular Perturbation Methods" Springer-Verlag

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