Course Meeting Times
Lectures: 3 lectures / week, 1 hour / lecture
18.311 Principles of Continuum Applied Mathematics covers fundamental concepts in continuous applied mathematics, including applications from traffic flow, fluids, elasticity, granular flows, etc. The class also covers continuum limit; conservation laws, quasi-equilibrium; kinematic waves; characteristics, simple waves, shocks; diffusion (linear and nonlinear); numerical solution of wave equations; finite differences, consistency, stability; discrete and fast Fourier transforms; spectral methods; transforms and series (Fourier, Laplace). Additional topics may include sonic booms, Mach cone, caustics, lattices, dispersion, and group velocity. Uses MATLAB® computing environment.
The material in the course is spread out over several books, listed here. Class notes for some topics are also available in the Lecture Notes.
- Haberman, Richard. Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow. SIAM, 1998. ISBN: 9780898714081. [Preview with Google Books]
Covers many course topics. Problems from this book will be assigned frequently.
- Lin, C. C., and Lee A. Segel. Mathematics Applied to Deterministic Problems. SIAM, 1988. ISBN: 9780898712292. [Preview with Google Books]
An extremely good book to have.
- Wan, Frederic Y. M. Mathematical Models and Their Analysis. Harper & Row, 1989. ISBN: 9780060469023.
Problems from this book may be assigned occasionally.
- Logan, J. David. An Introduction to Nonlinear Partial Differential Equations. Wiley-Interscience, 2008. ISBN: 9780470225950. [Preview with Google Books]
This book covers all the partial differential equation theory that we will see in this course – part I below.
- Richtmyer, Robert D., and K. W. Morton. Difference Methods for Initial-Value Problems. 2nd ed. Interscience Publishers, 1967. ISBN: 9780470720400.
- Fowler, A. C. Mathematical Models in the Applied Sciences. Cambridge University Press, 1997. ISBN: 9780521467032. [Preview with Google Books]
- Stoker, J. J. Nonlinear Vibrations in Mechanical and Electrical Systems. Interscience Publishers, 1950.
- Whitham, G. B. Linear and Nonlinear Waves (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts). Wiley-Interscience, 1974. ISBN: 9780471940906.
- Haberman, Richard. Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems. 4th ed. Prentice Hall, 2003. ISBN: 9780130652430.
There will be about one problem set per week (8 total). Each problem set will have two parts, "regular" and "special." The Teaching Assistant (TA) will grade the regular problems and the instructor the special ones.
The TA will grade a few problems in each set, selected randomly; answers will be provided for all. Please, do all the problems; this is the only way to really learn the course material [besides: If you just happen to miss the ones picked, you will get no credit, even if you did all the others]. The reason for this policy is that I would rather have the TA grade well a few problems than attempt to grade all with "in-a-hurry" sort of grading. There will be stiff penalties for late homework.
|Regular Problem Set Parts||30%|
|Special Problem Set Parts||70%|
I encourage you to collaborate and form study groups with other students. It is permitted to work and exchange information with other students, meaning: hints, general ideas, pitfalls to avoid and so on. You can learn a lot from other students, and vice versa. This, however, has to be done within reason. For example: "let me see / copy the answer" is not within reason. You can also consult books, and other material available in the library, for information relevant to a problem you may have forgotten—but not to see if you can find the problem solution somewhere!
When you write your final answer, you must do so 100% alone, with full understanding of every dot that goes there. Even if you worked together with others, as explained above, the final writing of the answer should be done individually, and in your own words, not copying from a group master. This is the only way that you can be sure that you actually understood everything that goes into the problem answer. You will be held responsible for everything that is in your answers, and the instructor reserves the right to call you for an explanation of your answer. Specifically: You may be asked to do a problem (or part of it) again, in front of the instructor, and to explain your steps. If this process indicates that you do not understand your own answer, you may lose up to all the credit for the entire problem set, depending on how severe the situation is. Note that a few (randomly selected) students may be called up to explain their answers, as a way of gauging how well the class is following the material.
The keyword is: "Reasonable". Reasonable actions are allowed. Reasonable is defined in terms of the purpose. For example: An assigned problem is aimed at both:
- Test your knowledge and understanding.
- Provide a means for you to learn new things and practice your knowledge.
If an action defeats any of these aims, it is not reasonable. Searching for the answers somewhere defeats both purposes. Hence searching the Internet, libraries, bookstores, whatever, for answers to the problems is not allowed. If in doubt, ask the instructor. There may be billions of ways in which this policy (reasonable actions are allowed) can be defeated. I cannot list them, I do not even know them, but you should be able to judge on your own if something violates it.
Outline of the 18.311 Lectures
PART I. Some Basic Topics in Nonlinear Waves
- Shock waves and hydraulic jumps. Description and various physical setups where they occur: Traffic flow, shallow water. What is a wave?
- Fundamental concepts in continuous applied mathematics. Continuum limit. Conservation laws, quasi-equilibrium. Kinematic waves.
- Traffic flow (TF). Continuum hypothesis. Conservation and derivation of the mathematical model. Integral and differential forms. Other examples of systems where conservation is used to derive the model equations (in nonlinear elasticity, fluids, etc.)
- Linearization of equations of TF and solution. Meaning and interpretation. Solution of the fully nonlinear TF problem. Method of characteristics, graphical interpretation of the solution, wave breaking. Weak discontinuities, shock waves and rarefaction fans. Envelope of characteristics. Irreversibility in the model.
- Quasilinear First Order PDE's.
- Shock structure, diffusivity. Burger's equation. The Cole-Hopf transformation. The heat equation: Derivation, solution, and application to the Burger's equation. Inviscid limit and Laplace's method.
- Dimensional analysis. Nonlinear diffusion and fronts.
PART II. Numerical Solutions, Series, and Transforms
- Numerical solution of wave equations. Finite differences. Good and bad numerical schemes: Consistency, stability, von Neumann analysis. Associated equation. Short wave stability analysis.
- Computers and numerical issues. MatLab.
- Fourier series and von Neumann stability. Discrete and Fast Fourier transforms. Spectral methods.
- Transforms and series: Fourier, Laplace.
- Lattices. Fermi-Pasta-Ulam problem.
A few of these topics will be covered if time permits:
- Sonic booms. Mach cone.
- Shallow water waves. Equations. Linearization and solution. Radiation conditions. Characteristics and shocks.
- Random walks, Brownian motion, diffusion.
- Water waves. Derivation of the equations and linearization. Dispersion and group speed. Weak nonlinearity and solitary waves. Perturbation expansions.
- Linear and nonlinear oscillations, relaxation. Phase plane methods and multiple scales. Applications to celestial mechanics and mechanical vibrations.
- Dynamical systems examples from mathematical biology and population dynamics.