| LEC # | TOPICS | READINGS |
|---|---|---|
| 1 | Introduction: Dense Granular Flow in a Silo. | LS, Sections 1.1 and 3.3. |
| 2-3 | Linear Waves: PDE for Waves in an Elastic Medium, Characteristics, d'Alembert's Solution | H1, Sections 4.2 and 12.2-12.5. |
| 4-13 | Nonlinear Kinematic Waves: Lighthill-Whitham Theory of Traffic Flow, Density Waves, General Method of Characteristics, Hodograph Method, Expansion Fans, Wave Breaking, Shocks River Waves, Gas Compression Waves, Shallow Water Waves | H1, Section 12.6. H2, pp. 56-86. |
| 14-16 | Dispersive Waves: Fourier transform, group velocity and caustics Envelope Equations, KdV Equation, Solitons | H1, Sections 14.2 and 14.6. H1, Sections 14.7.1-3. |
| 17-18 | Linear Diffusion: Green function for the Diffusion Equation, Some Fourier Analysis Similarity Solutions | H1, Section 10.4. |
| 19-25 | Nonlinear Diffusion: Burgers Equation and Shock Structure, Cole-Hopf Transformation Porous Medium Equation, Similarity Solution Regular and Singular Perturbations, Boundary Layers, Matched Asymptotic Expansions Nernst-Planck Equations | LS, Section 9. |
Table of Contents
Study Materials
Readings
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Readings are assigned in the following textbooks:
Required
[H1] Haberman, Richard. Applied Partial Differential Equations. 4th ed. Upper Saddle River, NJ: Prentice Hall, March 24, 2003. ISBN: 0130652431.
[H2] ———. Mathematical Models: Mechanical Vibrations, Population Dynamics and Traffic Flow: An Introduction to Applied Mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1998. ISBN: 0898714087.
For Further Reading
[D] Debnath, Lokenath. Nonlinear Partial Differential Equations for Scientists and Engineers. Boston, MA: Birkhäuser, 1997. ISBN: 0817639020.
[W] Whitham, Gerald Beresford. Linear and Nonlinear Waves. New York, NY: Wiley, 1974.
[B] Barenblatt, Grigory Isaakovich. Scaling, Self-Similarity, and Intermediate Asymptotics. New York, NY: Cambridge University Press, 1996. ISBN: 0521435161 (hardcover), 0521435226 (pbk).
[LS] Lin, Chia-Ch'iao, and L. A. Segel. Mathematics Applied to Deterministic Problems in the Natural Sciences. Material on elasticity by G. H. Handelman. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1988. ISBN: 0898712297.
[C] Crank, John. The Mathematics of Diffusion. 2nd ed. Oxford, UK: Clarendon Press, 1979, c1975, (1986 printing). ISBN: 0198534116. On reserve at the Science Library and math reading room.
Readings by Session