# Perturbation Theory and Asymptotic Expansions

A graduate student course at the Dept. of Information Technology, Division of Scientific Computing, given by visiting professor Bengt Fornberg, May-June 2004.

## 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

The governing equations for most phenomena in nature and in the sciences can be formulated in terms of PDEs, ODEs, integral equations, or in combinations of these. The main approaches to obtain insights from such equations are

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

All room numbers refer to building 2, MIC, Polacksbacken.
 Mon 2004-05-17 10:15-12:00 Room 2145 Tue 2004-05-18 10:15-12:00 Room 2145 Wed 2004-05-19 08:15-10:00 Room 2144 Wed 2004-05-19 10:15-12:00 Room 2145 Thu 2004-05-27 10:15-12:00 Room 2145 Fri 2004-05-28 10:15-12:00 Room 2145 Wed 2004-06-02 10:15-12:00 Room 2145 Thu 2004-06-03 10:15-12:00 Room 2145 Fri 2004-06-04 10:15-12:00 Room 2145

## Examination and course credits

The form of the examination is not yet determined. The course will give 3 graduate student credit points. For more information, contact Elisabeth Larsson, at the address below.

## Mathematica notebooks

Regular perturbation; polynomial roots Reg_Pert_Pol_Roots.nb
ODE expansion around regular point ODE_Exp_Reg_Pt.nb
ODE expansion around irregular singular point LINDIF01.nb
Padé convergence acceleration for a Stieltjes function Pade_01.nb
Conversion from Taylor- to continued fraction expansion Taylor_to_cf.nb
Fourier-Laplace method to get ODE solution in integral form FourierLaplace.nb
Laplace integral - integration by parts Laplace_int_part.nb
Laplace's method - higher order terms Laplace_high_order.nb
Laplace integrals - Watson's lemma Watson_02.nb
Gamma function expansion using steepest descent Gamma_As.nb
Perturbation expansions for the projectile problem:
--- By plugging in an assumed series and equating coefficients Proj_Pert_1.nb
--- By parametric differentiation and by iteration Hw09_P3.nb
First example on boundary layers B_Layer_01.nb
Second example on boundary layers B_Layer_02.nb
Numerical solution of boundary value problems Num_ODE.nb