MATH 6620: Introduction to Perturbation Methods Spring 2004

MATH 6620: Introduction to Perturbation Methods
Spring 2004
Perturbation methods involve a systematic construction of approximate solutions to mathematical problems
which are otherwise intractable. These methods rely on there being a parameter in the problem that
is relatively small. Small parameters commonly appear in a broad range of natural and technological applications
including, but not limited to, mechanics, population dynamics, solid mechanics, fluid mechanics,
aerodynamics, electrodynamics, wave propagation, detonation theory, chemical reactors, heat transfer, combustion,
biological sciences, celestial mechanics, optics, shock waves, and nonlinear oscillators. By taking
advantage of a small parameter and/or disparate scales in a system, a complicated problem can be replaced
by a series of simpler solvable ones. The simpler problems can be solved one by one to obtain an
increasingly accurate approximation to the true solution. Though solutions can typically be computed numerically,
analytical approximations of solutions often provide physical insight into the solution behavior
and its dependence on the control parameters.
Instructor: Chris Wahle (Amos Eaton 412, wahlec@rpi.edu)
Office Hours: Thursday 11:00-1:00pm, or by appointment
Webpage: eaton.math.rpi.edu/faculty/Wahle/Courses/6620/website/home.html
Suggested References: The first three textbooks listed should be at the RPI bookstore.
• Advanced Mathematical Methods for Scientists and Engineers, C.M. Bender
and S.A. Orszag, Springer-Verlag, ISBN: 0387989315, Amazon cost: $65.00.
• Introduction to Perturbation Methods, M. Holmes, Springer-Verlag, ISBN: 03879
42033, Amazon cost: $64.95.
• Multiple Scale and Singular Perturbation Theory, J. Kevorkian and J.D. Cole,
Springer-Verlag, ISBN: 0387942025, Amazon cost: $82.95.
• Perturbation Methods in Applied Mathematics, J. Kevorkian and J.D. Cole,
Springer-Verlag, ISBN: 3540905073, Amazon cost: out of stock.
• Perturbation Methods in Fluid Dynamics, M. Van Dyke, Parabolic Press Inc.,
ISBN: 0915760010, Amazon cost: $15.00.
• Perturbation Methods, E.J. Hinch, Cambridge University Press, ISBN: 0521378974,
Amazon cost: $26.00.
• Perturbation Methods, A.H. Nayfeh, Wiley-Interscience, ISBN: 0471399175,
Amazon cost: $69.95.
• Introduction to Perturbation Techniques, A.H. Nayfeh, Wiley-Interscience, ISBN:
0471310131, Amazon cost: $88.50.
• Introduction to Methods of Applied Mathematics, Chapter 24, S. Mauch,
No Copyright, Cost: FREE, download from the course webpage (∼ 1500 pages, 6.5 Mb).
Grading: Grades will be based on homework (40%) and two take-home exams (30% each).

Topics
1. Algebraic/Transcendental Equations
(a) regular perturbation problems
(b) scaling/balancing terms
(c) singular perturbation problems
2. Theory/Definitions/Terminolgy
(a) order symbols, o and O
(b) gauge functions
(c) logarithmic expansions
(d) asymptotic sequences, series, and convergence
3. Ordinary Differential Equations / Applications
(a) regular perturbation problems
(b) singular perturbation problems
4. Matched Asymptotic Expansions / Applications
(a) boundary layer theory
(b) outer/inner expansions
(c) matching
(d) interior layers
(e) general theory
5. Time Dilation and the Duffing equation
(a) Poincaré-Lindstedt method of strained coordinates
(b) Poincaré-Lighthill-Kao method of strained coordinates
(c) renormalization
(d) method of multiple scales (two-timing) for odes
6. Partial Differential Equations
(a) hyperbolic/parabolic/elliptic pdes
(b) method of multiple scales
(c) boundary layers
7. Bifurcation and Stability Theory / Applications
(a) linear stability analysis
(b) bifurcations (transcritical, pitchfork, Hopf)
(c) weakly nonlinear stability analysis
8. Asymptotic Integration
(a) integration by parts
(b) Laplace’s Integral Method
(c) Watson’s Lemma

Other Potential Topics (time permitting)
• homogenization
• integral equations
• method of averaging
• slowly varying coefficients
• eigenvalues/vectors
• WKB/WKJB methods
• Floquet theory and Mathieu’s equation
• asymptotics beyond all orders
• intermediate layers / two-ply boundary layers
• multiple scales: three-timing
• asymptotic integration
– method of steepest decent
– method of stationary phase
• dynamical systems (systems of nonlinear odes)
– equilibrium solutions (classification and stability)
• miscellaneous applications, for example
– correction to Kepler’s laws
– large activation energy asymptotics
– thin films
– nonlinear wave propagation
– shocks
– explosions
– fluid flow
– stability problems
– etc.
+ requests and suggestions