Asymptotic and Perturbation Methods in Applied Asymptotic and Perturbation Methods in Applied Mathematics
ESAM 420-1
Fall 2002
Prof. H. Riecke

Suggested Textbooks

  1. M.H. Holmes, Introduction to Perturbation Methods, Springer, 1995
  2. C.M. Bender & S.A. Orszag Advanced Mathematical Methods for Scientists and Engineers, McGraw Hill, 1978

Other Books

  1. E.J. Hinch, Perturbation Methods, Cambridge University Press, 1991
  2. J. Kevorkian & J.D. Cole, Multiple Scale and Singular Perturbation Methods, Springer, 1996
  3. P.A. Lagerstrom, Matched Asymptotic Expansion: Ideas and Techniques, Springer, 1988
  4. A.H. Nayfeh, Introduction to Perturbation Techniques, Wiley, 1981, 1993
    Perturbation Techniques, Wiley, 1973
  5. M. van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, 1975

Topics to be covered:

  1. Perturbation methods for algebraic equations
    1. Explorative examples
      regular vs. singular perturbations
    2. Asymptotic approximation
      order ( O, o), gauge functions, uniformity, transcendentally small terms, manipulating asymptotic expansions, asymptotic series and convergence
  2. Regular perturbation problems
    1. Projectile problem
    2. Plane Couette flow with variable viscosity
    3. Electrial potential of slightly perturbed sphere
  3. Singular perturbations
    1. Small-mass harmonic oscillator
      regular perturbation (weak damping)
      singular perturbation (small mass), inner and outer expansion, matching
      1. Composite expansion
      2. van Dyke's asymptotic matching principle
    2. Boundary-value problem
    3. General linear equation with variable coefficients
      1. Coefficient non-zero
      2. Coefficient vanishes at boundary
      3. Coefficient vanishes inside domain. Turning point
        internal layers, no unique solution from matching
    4. Boundary-layer problem involving lne
    5. multiple deck
    6. nonlinear problem
    7. a slider bearing
    8. model example for Stokes-Oseen problem
    9. viscous flow past a sphere
    10. a combustion problem
    11. premixed flame propagation
    12. beam string
    13. multiple solutions
  4. Method of strained coordinates
    1. flow past an air foil
    2. weakly nonlinear oscillator
      1. Poincaré-Lindstedt method
      2. Poincaré-Lighthill-Kuo (PLK) method
    3. pendulum
    4. Rayleigh equation. van der Pol oscillator
  5. Floquet theory. 2nd-order linear equations
    1. Mathieu equation

The class will continue with 420-2,3 to be taught by Moshe Matalon.


Solutions for HW 3: 1 2 3 4 5 6 7 8 9 problem 3 PS problem 3 PDF
Solutions Problem 2
Midterm PS PDF
Solutions part 1 part 2 part 3
Solutions Problem 1 Problem 2 Problem 3 Problem 4 Final: PS PDF

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Office Hours: Mo Tu Th 5-6 in L550

There will be a take-home midterm and a take-home final.

Homework will be assigned and handed in. Unfortunately, I cannot grade the assignments in detail.

File translated from TEX by TTH, version 2.60.
On 25 Sep 2002, 22:32.