Perturbation Theory and Asymptotic Expansions

Course topics

• Some introductory examples from applications
• Expansion methods in the case of algebraic equations
• Techniques for convergence improvement of truncated expansions
• Approximate solutions of linear and nonlinear ODEs
• Asymptotic expansion of integrals
• Perturbation series for ODEs and PDEs
• Boundary layer theory - matched asymptotic expansions
• WKB theory
• Multiple scale analysis

Course Motivation

Analytic Solutions: This approach alone is virtually never successful. For all but the most trivial cases, realistic governing equations simply do not admit exact solutions in terms of elementary functions. This fundamental obstacle is not due to any limitations in our ability to perform analytic manipulations - the use of symbolic algebra packages (like Mathematica) help only very little.

Numerical Solutions: This general approach is immensely powerful - large-scale computer simulations are now often considered as the third fundamental investigative technique (besides the long established ones of theory and experiment). No other approach can come even close to solving systems of hundreds or thousands of coupled nonlinear differential equations that arise in many applications. However, single-minded number-crunching suffers from notable limitations and difficulties, e.g.

• Coding can be very complex,
• We are often interested in situations where some parameter is very small (or very large). In such limits, computer costs often become prohibitive,
• Numerics is not well suited to get 'leading behaviors' in precise analytic form, suitable for further analysis.

Perturbation / Asymptotic Analysis: By replacing an analytically unsolvable problem with a sequence of analytically solvable ones, one can often avoid the fundamental barrier that is encountered when searching for exact solutions (Now, a package like Mathematica becomes extremely useful in carrying out the difficult and lengthy - but feasible - manipulations required). Asymptotic methods are usually most powerful precisely when numerical approaches encounter their most serious difficulties, such as in cases of small parameters, phenomena on vastly different scales etc. Perturbation / asymptotic analysis can then provide accurate information in analytic forms which are very well suited for both understanding and for further analysis.

The three general approaches above all complement each other. In most applications, all three are required.

Schedule

Examination and course credits

Mathematica notebooks