Math 704: Methods of Applied Mathematics-2

Fabian Waleffe


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 equations.
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

References

Partial Differential Equations of Applied Mathematics, Erich Zauderer, 2nd Ed., Wiley-Interscience 1989. [Graduate, Textbook]
Partial Differential Equations of Applied Mathematics, J. Kevorkian [Graduate, Textbook]
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]
Linear and Nonlinear waves , G.B. Whitham, Wiley 1974. [Research Monograph, a classic]
Similarity, Self-Similarity and Intermediate Asymptotics, G.I. Barenblatt, Cambridge [Research Monograph]

More abstract references:

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]