18.311: Principles of Applied Mathematics
Simulation of granular flow in a draining silo (by Chris Rycroft, Dry Fluids Lab)
Lectures
Handouts
Practice Problems
Logistics
Lectures: MWF 1011 in 2105.
Lecturer: Prof. M. Z. Bazant,
bazant@math.mit.edu, Office hours:
Mon 23, Tu 12 in 2363B.
TA: Chris Rycroft, Office hours: Tu 3:454:45,
Fri 1112 in 2331.
Required books:
[H1] R. Haberman, Applied Partial Differential Equations
(PrenticeHall,
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, SelfSimilarity, 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, 912 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.
 Introduction: Continuum models, dense granular flow in a
silo. (2)
 WAVES

Linear Waves: Waves in an elastic medium, characteristics,
d'Alembert's solution [H1: 4.2, 12.212.5]. (3)
 Nonlinear Waves: LighthillWhitham theory of
traffic flow, density waves, general method of characteristics,
expansion fans, shock formation and dynamics [H1: 12.6; H2:
5686.]; river waves, glaciers, tsunamis. (20)
 Multicomponent Waves:. Gas dynamics, shallow water
waves [W] [D]. (4)
MIDTERM EXAM
 Dispersive Waves: Fourier transform, group velocity and
caustics [H1: 14.2, 14.6]; KdV equation, solitons [H1: 14.7.13] [W] [D]. (4)
 DIFFUSION
 Linear Diffusion: Green function for the diffusion
equation, delta function, some Fourier analysis [H1: 10.4] [C].
(1)
 Nonlinear Diffusion: Burgers equation and shock structure,
ColeHopf 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