Principles of Applied Mathematics

Undergraduate · Historical course

Spring 2014

Topics

Overview of Class
General Framework. Discrete to Continuum Limits.
Sources for Conservation Laws
Conservation Laws in 2-D or 3-D, Dimensional Analysis, Fick’s Law
Conservation Laws and Partial Differential Equations (PDEs), Method of Characteristics, Traffic Flow
Linearized Traffic Flow
Characteristic Curves, Nonlinear Problems
Nonlinear Kinematic Waves and Wave-Breaking
Traffic Flow Examples
Shock Jump Condition
Domain of Dependence, Characteristics, Shocks
Flood Waves
Ordinary Differential Equations (ODEs), Shock and Rarefaction
Arrow of Time
Extended Theory
EIKO, Waves and Caustics
Whole Curves, Envelopes, Mach Cone
Domain of Dependence, 1st Order PDEs
Review of Theory: Expansion Fans, Shocks
Review of Theory: Entropy Increase at Shocks
Entropy, Quadratic Flow
Envelopes and Relationship Between Conservation Laws and Shocks
Shock Structure Due to Higher Order Effects
Gas Dynamics, Acoustics, and Strings
Gas Dynamics, Polytropic Gas, Hyperbolic
Linear Algebra for Gas Dynamics
Linear Gas Dynamics: Acoustic in a Pipe
Gauss / Stokes Theorem
General Solution for the Wave Equation
Multivariable Calculus, Differentiability, Properties of the Gradient, Taylor Expansion, Implicit Function Theorem
Characteristics for Klein-Gordon
Characteristics, Simple Waves, and Shocks for Gas Dynamics, Numerical Method by Characteristics
Separation of Variables and Normal Modes
Scalar Product Definition of Self-Adjoint
Generalize to Operators, Finite Differences Schemes for PDEs
FD Schemes
Introduction to the vNSA Problem Series
Artificial Viscosity
von Neumann Stability Analysis for Numerical Schemes

Readings

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. ↗
Listed reading
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. ↗
Listed reading
Wan, Frederic Y. M. Mathematical Models and Their Analysis . Harper & Row, 1989. ISBN: 9780060469023. Problems from this book may be assigned occasionally.
Listed reading
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. ↗
Listed reading
Richtmyer, Robert D., and K. W. Morton. Difference Methods for Initial-Value Problems . 2nd ed. Interscience Publishers, 1967. ISBN: 9780470720400.
Listed reading
Fowler, A. C. Mathematical Models in the Applied Sciences . Cambridge University Press, 1997. ISBN: 9780521467032. [Preview with Google Books ] ↗
Listed reading
Stoker, J. J. Nonlinear Vibrations in Mechanical and Electrical Systems . Interscience Publishers, 1950.
Listed reading
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.
Listed reading
Haberman, Richard. Applied Partial Differential Equations: With Fourier Series and Boundary Value Problems . 4th ed. Prentice Hall, 2003. ISBN: 9780130652430.
Listed reading

Sources

Course catalogue ↗Historical OpenCourseWare
MIT OCW calendar ↗
MIT OCW syllabus ↗
Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.