Math 704: Spring 2003
Methods of Applied Mathematics-2

Fabian Waleffe

Day	Time	Place
TR	9:30-10:45pm	B231 Van Vleck Hall


Kelvin's ship wave pattern (From "Why are special functions special?" Physics Today, Apr 2001)	Photo by Ensign John Gay, US Navy, somewhere over the Pacific between Hawaii and Japan, July 7, 1999. BOOM!: An example of transition from ELLIPTIC to HYPERBOLIC PDEs

Catalog Description

Derivation, nature and solution of canonical partial differential equations of applied mathematics. Conservation laws, advection, diffusion. First order PDEs, characteristics, shocks. Traffic flow, eikonal and Hamilton-Jacobi equations. Higher order PDEs: Classification, Fourier analysis, well-posedness. Series solutions and integral transforms. Green's functions and distributions. Similarity solutions. Asymptotics of Fourier integrals. Laplace's method, stationary phase. Ship waves. Perturbation methods.

Outline

Introduction: conservation laws, constitutive laws; basic PDEs, advection, diffusion, Poisson, Laplace.
Brief review of Separation of variables, Sturm-Liouville, Fourier series and Fourier transforms.
Fourier Analysis of PDEs. Diffusion, dispersive waves. Ill-posedness. Similarity solutions. Generalized functions.
Asymptotics of Fourier integrals: Laplace's method, Stationary phase (steepest descents). Ship waves.
Brief review of linear systems of ODEs: fundamental matrix, variation of parameters, Duhamel's principle, Green's functions.
Green's functions for initial-boundary value problems.
First order quasilinear PDEs, characteristics, traffic flow, shocks. Nonlinear 1st order PDEs: eikonal equation, Hamilton-Jacobi.
Classification of PDEs, elliptic, parabolic, hyperbolic. Hyperbolic systems. Derivation and study of shallow-water equations; gas dynamics.
Porous medium equation. Fisher's equation, KdV, Nonlinear Shrodinger, Traveling waves. Kuramoto-Sivashinsky.
Equations of Fluid Mechanics, Electromagnetism and Elasticity: Navier-Stokes, Maxwell, Navier.
Perturbation and asymptotic methods; boundary layers, geometrical optics

Grading

Two in-class exams, one or two take-homes. NO FINAL. Exams: TBA

Office hours:

By appointment

Take Home Exam 1 due Thur Mar 13

Take Home Exam 2 due Thur May 8

References

Linear and Nonlinear waves , G.B. Whitham, Wiley 1974. [Research Monograph, a classic]
Elementary Applied Partial Differential Equations, Richard Haberman, THIRD Edition, Prentice-Hall 1998. [Undergrad-Grad Textbook, Separation, Green's functions background, Chap 12-14: graduate applied Math]
Partial Differential Equations of Applied Mathematics, Erich Zauderer, 2nd Ed., Wiley-Interscience 1989. [Graduate, Textbook]
Similarity, Self-Similarity and Intermediate Asymptotics, G.I. Barenblatt, Cambridge [Research Monograph]

Less applied references (some would say `purer')

Partial Differential Equations , F. John, Springer-Verlag, 4th edition, 1986. [Chaps 1, 2]
Partial Differential Equations , L.C. Evans, American Mathematical Society 1999. [Chaps 1-4, Textbook for Math 819, not applied]
Partial Differential Equations , R. McOwen, Prentice-Hall 1996. [Not applied math, but nice concise overview]

Math 704: Spring 2003 Methods of Applied Mathematics-2