# Math 704: Spring 2001 Methods of Applied Mathematics-2

## Fabian Waleffe

Day  Time  Place
TR  9:30-10:45pm B305 Van Vleck Hall

 Kelvin's ship wave pattern Photo by Ensign John Gay, US Navy, somewhere over the Pacific between Hawaii and Japan, July 7, 1999.

### Handouts

Fourier transform, series and discrete Fourier transform

REFERENCES:

• Partial Differential Equations of Applied Mathematics, Erich Zauderer, 2nd Ed., Wiley-Interscience 1989. [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]
• Partial Differential Equations , R. McOwen, Prentice-Hall 1996. [Not applied math, but nice concise overview]
• Similarity, Self-Similarity and Intermediate Asymptotics, G.I. Barenblatt, Cambridge [Research Monograph]

## Description

This is basically a course on partial differential equations taught from an applied mathematics view point. We'll look briefly at where the equations come from and discuss their validity and limitations. We'll look at the properties and methods of solutions of various types of equations, both linear and nonlinear. The methods will be mostly analytical but we'll also look at asymptotic techniques. There will be little discussion of numerical methods (for that see e.g. CS/Math 712, 713). Students should have had a course on differential equations ( e.g. Math 319) and, ideally, one on partial differential equations ( e.g. Math 322). A basic knowledge of general physics (especially mechanics, thermodynamics and electromagnetism) is expected.

We'll cover Chapters 2 and 3 in Zauderer pretty completely and sample the rest of the book.

• Introduction: conservation laws, constitutive laws, basic PDEs
• First order PDEs, characteristics, traffic flow, eikonal equation, Hamilton-Jacobi.
• Random walks and diffusion
• Classification of PDEs, well-posedness; gas dynamics, shallow-water equations
• Separation of variables, Sturm-Liouville, Fourier series and transforms. Duhamel's principle. Similarity solutions.
• Green's functions, distributions
• Equations of Fluid Mechanics, Electromagnetism and Elasticity: Navier-Stokes, Maxwell, Navier.
• Perturbation and asymptotic methods; boundary layers, geometrical optics