18.311: Principles of Applied Mathematics
Simulation of granular flow in a draining silo (by Chris Rycroft, Dry Fluids Lab)
Lectures: MWF 10-11 in 2-105.
Lecturer: Prof. M. Z. Bazant,
email@example.com, 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.
[H1] R. Haberman, Applied Partial Differential Equations
4th edition, 2003);
[H2] R. Haberman, Mathematical Models:
Mechanical Vibrations, Population Dynamics and Traffic Flow (SIAM,
For further reading:
L. Debnath, Nonlinear Partial Differential Equations for Engineers and
[W] G. B. Whitham, Linear and Nonlinear Waves;
[B] G. I. Barenblatt, Scaling, Self-Similarity, and
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
Grading: Problem sets (35% total), midterm exam
(25%) and a final exam (40%).
Five, due on Wednesdays Feb 21, Mar 7, Mar 21,
Apr 25, May 9.
One, in class on Wed Apr 11.
Final Exam: Thursday May 24, 9-12 in Walker Gym.
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
Linear Waves: Waves in an elastic medium, characteristics,
d'Alembert's solution [H1: 4.2, 12.2-12.5]. (3)
- 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)
- Multicomponent Waves:. Gas dynamics, shallow water
waves [W] [D]. (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)
- Linear Diffusion: Green function for the diffusion
equation, delta function, some Fourier analysis [H1: 10.4] [C].
- Nonlinear Diffusion: Burgers equation and shock structure,
Cole-Hopf transformation [W] [D]; porous medium equation,
dimensional analysis, similarity solutions [B] [C].
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.