- 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

**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.

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 |

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

Elisabeth Larsson bette@it.uu.se Last modified: Sat May 8 09:09:56 MEST 2004