# Syllabus

## Course Meeting Times

Lectures: 3 lectures / week, 1 hour / lecture

## Prerequisites

The prerequisites for the course are 18.02 Multivariable Calculus and 18.03 Differential Equations.

## Course Description

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.

## Textbooks

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.

## Problem Sets

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.

ACTIVITIES PERCENTAGES
Regular Problem Set Parts 30%
Special Problem Set Parts 70%

## Collaboration

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!

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.
• Caustics.
• 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.