Course Materials 

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Midterm Topics, Practice Problems
Final Topics, Practice Problems

Chapter 7 First Order PDEs

• Method of Characteristics

• Check YouTube Videos

• Conservation Laws and First Order Eqns at Stanford
• Nonlinear Advection: Burgers' Equation
• Method of Characteristics and Conservation Laws
• Rankine-Hugonoit Conditions
• Burger’s Equation: Shocks And Entropy Solutions
• Shock Waves and Traffic, Video Lecture
  • Shockwave Traffic Jam (0:40)
  • Phantom Traffic Jams (1:08)
  • The Phantom Traffic Jam - An Explanation (2:35)
• Maple Files (Right click, download and open in Maple)
  • First Order PDEs - char.mws, char2.mws
  • Shocks and Rarefactions, Characteristics for Sample Shocks and Rarefactions
• Merging of Shocks Fig 1, Fig 2
• MATLAB FOPDE.m Plots of shock and rarefaction solutions

Nonhomogeneous Problems

• Heat equation with Dirichlet BCs

• Heat equation with Neumann BCs

• Nonhomogeneous Heat Equation

• Solution of a Nonhomogeneous Heat Equation

Green's Functions

• Green's Function Cheat Sheet - New

• Poisson Equation in a Disk - New

• Poisson Equation Handout - New

• Variation of Parameters 

• Linear Methods of Applied Math

• For First and Second Order Equations: Site 1, Site 2, 

• NIST Tutorial 

• Introduction to Green's Functions

• Finding Green functions for ODEs.

• Green's Function Library

Applications of Complex Analysis

• Irrotational Flow around objects

• Conformal Mapping Example

• Plotting Functions of a Complex Variable

• Ideal Flow Machine

• Visual Complex Analysis, The Book

• Zipper

• Conformal Mapping and Minimal Surfaces

• Software Collection for Complex Variables

• Visualizing Complex Functions Using Domain Coloring

• Graphics for Complex Analysis

• Visualizing Complex Functions

• Complex Function Viewer - Java

• The Chaos Hypertextbook

• Free Textbook on Complex Analysis

Transforms

• Maple File for Fourier Transforms

• Finite Wave Train - Maple

• Fourier Transform, Fourier Series, Discrete Fourier Transform - Eric Weinstein's World of Math

• Wikipedia Entry

• Fourier Analysis Tutorial

• An Intuitive Explanation of Fourier Theory

• MIT Opencourseware notes

• Laplace Transform Table - MS Word

• Practice Laplace Transform Problems

• Introduction to Laplace Transforms for Engineers

• Mathworld

• Laplace Transform Module

• Tables

  • Collections of Worksheets! http://www.equationsheet.com/ 

  • http://www.vibrationdata.com/Laplace.htm 

  • eFunda: Laplace Transform Table

  • http://cnyack.homestead.com/files/alaplace/laptre.htm 

  • http://www.sosmath.com/tables/laplace/laplace.html 

• The Inverse Laplace Transform

• More Examples - Inverse Laplace Transform

• Bromwich Integral Example

• Using Laplace Transform to Sum Series

Numerical Methods

• Finite Difference Methods - Wikipedia

• Finite Difference Method slides

• Finite Difference Methods in Finance Examples include MATLAB code

• Iterative Methods to Solve Ax = b

• Moler's Chapter on PDEs

• Can one hear the shape of a drum? Mark Kac; and Cleve's Corner

• Eigenmodes of Isospectral Drums

• Solving Equations in Excel

• Note on Pseudo-Spectral Methods - NLS

Nonlinear Equations

• Forced KdV Example

• MAA SE 2008 - Stochastic KdV

• SIAM - Perturbed KdV

• Sample Student Research in PDEs

  • Caylah Retz - Lie Symmetries, 2012

  • Kara Roberson, Sprial Waves. 2010

  • Rebecca Wilkinson, Cake Baking, 2008

  • David Neal, Shark populations, 2007

  • Andrew Rose, Stochastic KdV, 2006

  • Erik Minges, Numerical simulation of air pollution dynamics due to point source emissions from an industrial stack, 2010 - Honors Project

• Numerical Solution of Stochastic Differential Equations

Miscellaneous

• Method of Stationary Phase

• Fresnel integrals - See Cornu spiral

• The Buckingham Pi Theorem

• Lie Symmetries of PDEs, Origins and KdV