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Practice Problems for Midterm Hints
Midterm Review

Final Exam Review
Old Exam

Relevant Class Notes:

• Fourier and Complex Analysis Book
• A Second Course in Differential Equations Book
• An Introduction to Mathematical Physics via Oscillations - Has recent notes including special functions and higher dimensional PDEs

Chapter 1 Heat Equation

• The Diffusion Equation
• Derivation of the Heat/Diffuion Equation
• Introduction to Mathematical Physics
• Vector and Cartesian Tensor Analysis
• Orthogonal Curvilinear Coordinates
  • MathWorld links
    • Curvilinear Coordinates
    • Orthogonal Coordinate Systems
      1. Cartesian coordinates,
      2. spherical coordinates,
      3. cylindrical coordinates,
      4. elliptic cylindrical coordinates,
      5. parabolic cylindrical coordinates,
      6. bipolar cylindrical coordinates
      7. prolate spheroidal coordinates,
      8. oblate spheroidal coordinates,
      9. parabolic coordinates,
      10. toroidal coordinates.
      11. bispherical coordinates
      12. confocal ellipsoidal coordinates,
      13. conical coordinates,
      14. confocal paraboloidal coordinates,
  • Dr. Lugo's Notes
    • Differential Forms Approach:
      • PDE Notes,
  • A Graduate Student Take on Scale Factor Memorization

Chapter 2 Method of Separation of Variables

• Solution of the Heat Equation
• Heat Equation Module

• Special Trigonometric Bases - Harvey Mudd College

• Products of Trigonometric Functions, also at ThinkQuest

• Lectures on Laplace's Equation at Herod's Asynchronous site
  •  Laplace's Equation on a Rectangle
  • Insulated Boundaries
  •  The Structure of Solutions
  • The Laplacian in Polar Coordinates
  • On a Ring and On a Half-Disk
• More on Laplace's Equation by Alan S. Jones part of PDE site

Chapter 3 Fourier Series

• Fourier Series - Eric Weinstein's World of Math

• SOS Math - Fourier Series

• Listen to Fourier Series

• Fourier Series and Convergence

• Fourier Series and Waves

• Maple Worksheet - fseries.mws, EvenOdd.mws

• Gibbs Phenomenon - SOS Math

• The Dirichlet Kernel

• Least Squares, Bessel's Inequality and Parseval's Equality

• Convergence of Trigonometric Fourier Series

• Fourier Series Applet

Chapter 4 Wave Equation

• Derivation of Wave Equation for Vibrating String 1, 2
• Vibrating String
• d'Alembert's Solution - See Dr. Herman's Analysis below
• d'Alembert's Solution of the Wave Equation 1, 2
• Chladni Figures and Vibrating Plates

Chapter 5 Sturm-Liouville Eigenvalue Problems

• Sturm-Liouville Eigenvalue Problem
• The Sturm Liouville Problem

Chapter 6 Finite Difference Methods

• Numerical Solution of Heat Equation: Maple, Matlab
• Class Examples: HeatEq1, HeatEq2, HeatEq3

Chapter 7 Higher Dimensional PDEs

• Rectangular Membrane, Circular Membrane, Annular Membrane
• Circular Membrane Modes

• Vibrations and Waves - Animations

• Maple Files annulus, vibdrum

• Plots of Bessel Functions

• Essentials of Bessel Functions

• Bessel Functions and a Vibrating Drum 

• Automatic Computation of Zeros of Bessel Functions and Other Special Functions
• Frequency Content of Drums
• Acoustic Properties of Drums
• Synthesizing Percussion
• You Can't Always Hear the Shape of a Drum and more
•  

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Miscellaneous Materials from My Past Course

Handouts/Worksheets

• Nonhomogeneous Heat Equation
• PDE Review - Chapters 3 and 4

Maple Files

• Maple Basics: HTML, Basic.mws (Release 5.1)
• Maple Version of Knobel's Visualization, WaveRep.mws (Maple 6)
• d'Alembert's Solution Fixed ends, One Free End
• Examples of Solving Differential Equations in Maple
  • First Order PDEs - char.mws, char2.mws
  • Shocks and Rarefactions, Characteristics for Sample Shocks and Rarefactions
• Fourier Series fseries.mws (Maple 6)
• Uniform Convergence Example: uniform.mws (Maple 6)
• Series Solution of Wave Equation wave_eq_fs.mws (Maple 6)
• Reflection of Pulse off a wall wall.mws

MathCAD

• Mathcad Basics: HTML, Basics.mcd (Version 6.0)
• KdV Two soltiton solution, Twosol.mcd, Twosol.avi

MATLAB

• MATLAB Basics: HTML
• Using MATLAB
• Knobel's Matlab Files or here
  To use: Put these files in the Work subfolder in the MATLABR11 Directory
  In the MATLAB Command Window type one of: wvmovie, wvslice, wvsurf. 
  
  Or, enter the following commands

Enter the function and grid:
x=-10:0.1:10; 
t=0:0.3:6;
[X,T]=meshgrid(x,t);
u = exp(-(X-T).^2);

For Animation:
M = moviein(length(t));
for j=1:length(t),
    plot(x,u(j,:)),
    M(:,j)=getframe;
end;
movie(M)

For slices
waterfall(x,t,u)

For surface plots
surf(x,t,u)

For density plots
pcolor(x,t,u)

Or do them all at once from an m-file

d'Alembert's Solution

• Here is a summary and some animations showing how symmetries are used to construct solutions to the wave equation. 

  • Explanation Html File,  Postscript file

  • Example 1 HTML version, Maple version (Examples for two fixed ends) 
    Example 2 HTML version, Maple version (Examples for one end fixed and the other free)
    Example 3 HTML version, Maple version (Presentation of the odd extensions needed to solve the problem of two fixed ends with g(x)=0.)

• Other Links are above.