# 18.311: Principles of Applied Mathematics

## Spring 2007, Prof. Bazant

Simulation of granular flow in a draining silo (by Chris Rycroft, Dry Fluids Lab)

## Logistics

Lectures: MWF 10-11 in 2-105.
Lecturer: Prof. M. Z. Bazant, bazant@math.mit.edu, Office hours: Mon 2-3, Tu 1-2 in 2-363B.
TA: Chris Rycroft, Office hours: Tu 3:45-4:45, Fri 11-12 in 2-331.

Required books: [H1] R. Haberman, Applied Partial Differential Equations (Prentice-Hall, 4th edition, 2003); [H2] R. Haberman, Mathematical Models: Mechanical Vibrations, Population Dynamics and Traffic Flow (SIAM, 1998).
For further reading: [D] L. Debnath, Nonlinear Partial Differential Equations for Engineers and Scientists; [W] G. B. Whitham, Linear and Nonlinear Waves; [B] G. I. Barenblatt, Scaling, Self-Similarity, and Intermediate Asymptotics; C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences; [C] J. Crank, The Mathematics of Diffusion. On reserve at the Science Library and math reading room.

Grading: Problem sets (35% total), midterm exam (25%) and a final exam (40%).
Problem Sets: Five, due on Wednesdays Feb 21, Mar 7, Mar 21, Apr 25, May 9.
Midterm Exam: One, in class on Wed Apr 11.
Final Exam: Thursday May 24, 9-12 in Walker Gym.

## Outline

The class introduces fundamental concepts in ``continuous'' applied mathematics, with an emphasis on nonlinear partial differential equations (PDE). The approximate number of lectures on each topic is given in parentheses, and the assigned reading in brackets. The actual list of lecture topics is also available.
1. Introduction: Continuum models, dense granular flow in a silo. (2)

2. WAVES

1. Linear Waves: Waves in an elastic medium, characteristics, d'Alembert's solution [H1: 4.2, 12.2-12.5]. (3)
2. Nonlinear Waves: Lighthill-Whitham theory of traffic flow, density waves, general method of characteristics, expansion fans, shock formation and dynamics [H1: 12.6; H2: 56-86.]; river waves, glaciers, tsunamis. (20)
3. Multicomponent Waves:. Gas dynamics, shallow water waves [W] [D]. (4)

MIDTERM EXAM

4. Dispersive Waves: Fourier transform, group velocity and caustics [H1: 14.2, 14.6]; KdV equation, solitons [H1: 14.7.1-3] [W] [D]. (4)

3. DIFFUSION

1. Linear Diffusion: Green function for the diffusion equation, delta function, some Fourier analysis [H1: 10.4] [C]. (1)
2. Nonlinear Diffusion: Burgers equation and shock structure, Cole-Hopf transformation [W] [D]; porous medium equation, diffusion fronts, dimensional analysis, similarity solutions [B] [C]. (6)

FINAL EXAM

## Homework Policy

Students are encouraged to work together on the homework, but solutions must be written independently by each student, in his or her own words. Any significant collaborators should be noted on the solutions. It is considered cheating (and not allowed) to consult or copy solutions from prior years for any identical problems assigned this year, unless they have been officially distributed by the instructor for practice. Late homework is not accepted without a Dean's or doctor's note.

bazant@math.mit.edu