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Review Material

• Midterm Review, Sample Exams MAT 418 MAT 518
• Review for Final, Sample Final Exams MAT 418   MAT 518

Chapter 1 Second Order PDEs

• PDEs in Epidemics - Example

• VisualPDE - Interesting selection and simulations of PDEs,Example - Swift–Hohenberg

• Heat Equation

  • Derivation of Heat Equation

  • Heat Equation Module

  • Other Notes
  • Fourier's Heat Conduction Equation - A short history
  • The dichotomous history of diffusion

• Wave Equation

  • The Vibrating String Controversy

    • Unpublished resolution

    • Controversy in science
    • Bernoulli on Vibrating Strings.
    • Vibrations and Fourier Analysis - Herman

  • Derivation of Wave Equation for Vibrating String

  • Vibrating String

  • Summary and Animations showing how symmetries are used to construct solutions to the wave equation.  Explanation Html File,  PDF

    • 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.)

  • d'Alembert Solutions using MATLAB

    • dAlembert.m, dAlembert2.m, dAlembert3.m, stringsol.m

  • Desmos.com Examples of d'Alembert's solutions at Desmos
  • Semi-infinite String - Latest example
  • Simple Plucked String
  • Finite String
  • Finite Plucked String
  • Problem 11
  • Problem 12

  • Chladni Figures and Vibrating Plates

• Laplace's Equation

• Other Lectures on PDEs

• Classification and Cauchy Data

• Nonhomogeneous Boundary Conditions

  • Class Handout on Heat Equation PDF

  • Solution of a Nonhomogeneous Heat Equation - doc file, now in textbook

  • Flowchart PDF

• Duhamel's Principle - wave equation, video1 and video2

• Some History of PDEs
• Partial Differential Equations in the 20th Century - Includes 18th-19th centuries
• The Early History of Partial Differential Equations and of Partial Differentiation and Integration
• Development of Modern Mathematics in China

Chapter 2 Fourier Series

• Fourier Series - Eric Weinstein's World of Math

• Fourier Series and Waves

• Fourier Series and Convergence

• Products of Trigonometric Functions

• Computer Computation

  • Maple Worksheets - fseries.mws, PlotFS.mws, EvenOdd.mws

  • Wolframalpha.com - Fourier series examples

    • plot sum_(k=1)^10(4/pi sin((2 k-1)*x)/(2 k-1))

    • Try FourierSeries[abs(x), x, 10]

  • SageMath http://www.sagemath.org/  - Example, Fourier Series

  • MATLAB plotfs.m, plotfs0.m, FScoeff.m

  • GNU Octave - MATLAB clone

  • Python fsplot.py, How to do FS in Python,

  • Desmos Example, Sum of Series

• The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet ...- Google Books

• Note on Gibbs Phenomenon, 1925, Overshoot video (12:28)

• Historical Notes
• The Gibbs-Wilbraham Phenomenon: An Episode in Fourier Analysis - E. Hewitt, R. E. Hewitt, 1979
• Henry Wilbraham (1848) discovered overshoot on Page 198. Paper forgotten until noted by Burkhardt in 1908.
• Michelson's Oct 6 note started a series of comments by A. E. H. Love and Gibbs based on his device that could compute and re-synthesize the Fourier series, A New Harmonic Analyzer. See Michelson's Harmonic Analyzer
• It was coined Gibbs Phenomenon by Bocher in 1906 according to him in his 1914 paper.

Chapter 3 Numerical Solutions of PDEs:

• Numerical Solution of Heat Equation:

  • Maple: HeatEqn.mws, Matlab: heatex5.m

  • MATLAB Class Examples: HeatEq1, HeatEq2,

  • MATLAB - Matrix Version HeatEq3

  • Introduction to MATLAB - Spencer and Ware
• Group Project - Description, Group Problems
  • HeatEqn1D.m,
  • Modified file for 1D: HeatEqn1Db.m,
  • Find 2D Numerical Scheme for Part III: HeatEqn2D.m,
  • Plot 2D Analytical Solution for Part III: HeatEqn2Dexact.m,
  • AVI Movie heat2d.avi
  • Group Problem Solutions Maple PDF

• Finite Difference Methods - Wikipedia

• Finite Difference Methods in Finance Examples include MATLAB code

• Iterative Methods to Solve Ax = b

• Moler's Chapter on PDEs - part of Numerical Computing with MATLAB

• "Can One Hear the Shape of a Drum?," Dr. Herman's talk

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

• Article about Moler and MATLAB, 2008.
• The L-shaped Membrane

Chapter 4 Sturm-Liouville Eigenvalue Problems

• Sturm-Liouville Eigenvalue Problem a set of slides

• Wikipedia Entry
• Examples
• Sturm-Liouville History J. Lutzen
Least Squares, Bessel's Inequality and Parseval's Equality - Word Document

Chapter 5 Special Functions

• Classical Orthogonal Polynomials
• Legendre Polynomials
• Gamma Function
• Bessel Functions

Chapter 6 Higher Dimensional PDEs

• Rectangular Membrane, Circular Membrane, Annular Membrane

  • Newest membrane files: membrane.mws, cmembrane.mws

  • Desmos - degenerate Modes - Similar to Chladni Plates below
  • See Vibrational Modeshapes

• Circular Membrane Modes

• Maple Files annulus, vibdrum

• Plots of Bessel Functions

• Spherical Harmonics

  • Associated Legendre functions

  • Spherical Harmonics 3D

  • Spherical Harmonics Nodes

  • Spherical Harmonics Contours

• MATLAB Files for plotting modes of vibrating membranes

  • Rectangular membrane: membrane2Dxy.m

  • Circular membrane: cmembrane.m, (Uses besjroot.m)

  • Circular membrane with ICs: cmembrane2.m, (Uses besjroot.m)

  • Vibrations of Spherical Membrane Animations

  • Cleve Moler's L-Shaped Membrane - MATLAB Logo

• Homework 11 Notes: Hints, Solutions

• Drums

  • "Can One Hear the Shape of a Drum?," Dr. Herman's talk

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

  • You Can't Always Hear the Shape of a Drum, Article

  • Eigenmodes of Isospectral Drums - Driscoll

  • Frequency Content of Drums

  • Synthesizing Percussion

  • Drumhead vibrations - Discussion and link to a good Falstad applet.

  • Eigenmodes of Planar Drums and Nobel Prizes

  • Elliptical Drums
  • Mémoire sur le mouvement vibratoire d'une membrane de forme elliptique by Mathieu and translation
  • Visualization of Special Eigenmode Shapes of a Vibrating Elliptical Membrane
  • Eigenfrequencies of an elliptic membrane
  • Computation and applications of Mathieu functions: A historical perspective
  • Pure tones 5:3 Drum
  • Mathieu Perturbation Method

• Turkeys

  • How to cook a turkey. Musing of an Energy Nerd

  • The Panofsky Turkey Constant

  • Wolframalpha - how long should you cook a turkey

  • Consider a Spherical Turkey Blog

  • "Spherical Turkeys and Vibrating Balloons," MAA-SE 2015, Dr. Herman's talk

  • Dimensional Analysis for Cooking a Turkey at Maplesoft

• Chladni Plates

  • Some History and actual problem with moving boundary - http://www.unige.ch/~gander/Preprints/Ritz.pdf

  • an attempt - https://thelig.ht/chladni/

  • desmos rendition with sliders - https://www.desmos.com/calculator/rdpbran7og

  • Muller paper - https://arxiv.org/pdf/1308.5523.pdf

  • Wolfram DEmo - http://demonstrations.wolfram.com/ChladniFigures/

  •  Experiment - https://www.nature.com/articles/ncomms12764

  • Kundt tube https://en.wikipedia.org/wiki/Kundt%27s_tube

    •  http://nalab.mind.meiji.ac.jp/~mk/chladni/hirano-programs/

    • http://nalab.mind.meiji.ac.jp/~mk/chladni/

    • https://epubs.siam.org/doi/pdf/10.1137/10081931X,

      Chladni Figures and the Tacoma Bridge:
      Motivating PDE Eigenvalue Problems via Vibrating Plates

  • GIF - Vibrating Plates

  • Sophie Germain,

    • https://hackaday.com/2018/03/20/sophie-germain-the-mathematics-of-elasticity/

    • from PBS: http://www.pbs.org/wgbh/nova/physics/sophie-germain.html

• Mary Waller Publications

  • Vibrations of free circular plates. Part 1: Normal modes

    Proceedings of the Physical Society,  Volume 50,  Number 1

    http://iopscience.iop.org/article/10.1088/0959-5309/50/1/306

  • Vibrations of free circular plates. Part 2: Compounded normal modes

    Proceedings of the Physical Society,  Volume 50,  Number 1

    http://iopscience.iop.org/article/10.1088/0959-5309/50/1/307

  • Vibrations of free circular plates. Part 3: A study of Chladni's original figures

    Proceedings of the Physical Society,  Volume 50,  Number 1

    http://iopscience.iop.org/article/10.1088/0959-5309/50/1/308

    Mary D Waller 1939 Proc. Phys. Soc. 51 831

  • Vibrations of free square plates: part I. Normal vibrating modes

    http://iopscience.iop.org/article/10.1088/0959-5309/51/5/312/pdf

  • Vibrations of free square plates: part II, compounded normal modes

    Proceedings of the Physical Society,  Volume 52,  Number 4

    http://iopscience.iop.org/article/10.1088/0959-5309/52/4/304

  • Vibrations of free plates: isosceles right-angled triangles

    Proceedings of the Physical Society,  Volume 53,  Number 1

    http://iopscience.iop.org/article/10.1088/0959-5309/53/1/304

  • Vibrations of Free Rectangular Plates

    Proceedings of the Physical Society. Section B,  Volume 62,  Number 5

    http://iopscience.iop.org/article/10.1088/0370-1301/62/5/301

  • Vibrations of Free Elliptical Plates

    Proceedings of the Physical Society. Section B,  Volume 63,  Number 6

    http://iopscience.iop.org/article/10.1088/0370-1301/63/6/307

Chapter 7 First Order PDEs

• Method of Characteristics

• Check YouTube Videos

• Nonlinear Flow and Conservation Laws
• Nonlinear Advection: Burgers' Equation
• Desmos.com characteristics and Traffic Light examples, Updated Example
• Method of Characteristics and Conservation Laws
• Rankine-Hugonoit Conditions
• Merging of Shocks Fig 1, Fig 2
• The Study of Partial Differential Equations of the First Order in the 18th and 19th Centuries

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

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Software - Maple, MATLAB

General Programs

• VM Horizon - horizon.uncw.edu - UNCWs access to campus software
• CAS - Computer Algebra Systems
• Maple from Mathsoft
• Mathematica from Wolfram, which also has WolframAlpha
• SageMath is a collection of open-source products
• Maxima a descendent of Macsyma, precursor to Maple and Mathematica
• SymPy Python library for doing symbolic math
• List of other CAS systems
• MATLAB
MATLAB is a MATrix LABoratory and has developed to do computations although it can do symbolic computation
• MATLAB is a product of MathWorks
• Simulink is a block diagram environment from MathWorks. See Solving Differential Equations Using Simulink
• GNU Octave is a MATLAB Clone which can use MATLAB files and is open-source.
• Octave Online
• SciLab - similar to MATLAB/Simulink
• Other Computational Systems
• Python has a number of packages for doing computations
• Computational Math Software pages from my course featuring MATLAB, Python, Julia and Jupyter Notebooks as well as samples implementing numerical algorithms.
• Online IDE has Octave, Python, and other IDEs for running code online

Maple Files

• Maple Basics: HTML, Basic.mws (Release 5.1)

• 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

• KdV Two soltiton solution, Twosol.mcd, Twosol.avi

MATLAB

• MATLAB Basics: HTML or MATLAB Wiki

Other

• Formula Generator for Word

 

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