Partial Differential Equations

[Astronuc]

There have been a number of questions on Partial Differential Equations.

Within the Math & Science Tutorials (http://www.physicsforums.com/forumdisplay.php?f=160) are -

http://www.physics.miami.edu/nearing/mathmethods/ - link in thread (Mathematical Tools for Physics) posted by ranger, which leads to -

http://www.physics.miami.edu/nearing/mathmethods/pde.pdf - pdf file on PDE's.


For further information -
http://mathworld.wolfram.com/PartialDifferentialEquation.html
http://mathworld.wolfram.com/EllipticPartialDifferentialEquation.html
http://mathworld.wolfram.com/HyperbolicPartialDifferentialEquation.html
http://mathworld.wolfram.com/ParabolicPartialDifferentialEquation.html

http://www.sst.ph.ic.ac.uk/angus/Lectures/compphys/node24.html

Many thanks to ranger for that link. :smile:

[Astronuc]

From MIT Open Courses
The lecture notes below are courtesy Hristina Hristova, a student in the class. Used with permission.
http://ocw.mit.edu/OcwWeb/Mathematics/18-306Spring2004/LectureNotes/

Topics:

1 Introduction, Theme for the Course, Initial and Boundary Conditions, Well-posed and Ill-posed Problems (PDF)

2 Conservation Laws in (1 + 1) Dimensions (PDF)
Introduction to 1st-order PDEs: Linear and Homogeneous, and Linear, Non-Homogeneous PDEs

3 Theory of 1st-order PDEs (cont.): Quasilinear PDEs, and General Case, Charpit's Equations (PDF)

4 Theory of 1st-order PDEs (cont.): Examples, The Eikonal Equation, and the Monge Cone (PDF)
Introduction to Traffic Flow

5 Solutions for the Traffic-flow Problem, Hyperbolic Waves (PDF)
Breaking of Waves, Introduction to Shocks, Shock Velocity
Weak Solutions

6 Shock Structure (with a Foretaste of Boundary Layers), Introduction to Burgers' Equation (PDF)
Introduction to PDE Systems, The Wave Equation

7 Systematic Theory, and Classification of PDE Systems (PDF)

8 PDE Systems (cont.): Example from Elementary Gas Dynamics, Riemann Invariants (PDF)
More on the Wave Equation, The D'Alembert Solution

9 Remarks on the D'Alembert Solution (PDF)
The Wave Equation in a Semi-infinite Interval
The Diffusion (or Heat) Equation in an Infinite Interval, Fourier Transform and Green's Function

10 Properties of Solutions to the Diffusion Equation (with a Foretaste of Similarity Solutions) (PDF)
Conversion of Nonlinear PDEs to Linear PDEs: Simple Transformations, Parabolic PDE with Quadratic Nonlinearity, Viscous Burgers' Equation and the Cole-Hopf Transformation

11 The Laplace Equation in a Finite Region, Separation of Variables in a Circular Disc (PDF)
Conversion of Nonlinear PDEs to Linear PDEs: Potential Functions

12 Generalities on Separation of Variables for Solving Linear PDEs, The Principle of Linear Superposition (PDF)
Conversion of PDEs to ODEs, Traveling Waves, Fisher's Equation
Conversion of Nonlinear PDEs to linear PDEs: The Hodograph Transform

Quiz 1

13 Conversion of Nonlinear PDEs to Linear PDEs: The Legendre Transform (PDF)
Natural Frequencies and Separation of Variables: Linear PDEs, Fourier Series, Example: Vibrating String
The Sturm-Liouville Problem
About the Question: Can One Hear the Shape of a Drum?

14 Natural Frequencies for Linear PDEs (cont.): Vibrating Circular Membrane, Bessel's Functions, Linear Schrödinger's Equation (PDF)

15 Vibrating Circular Membrane (cont.) (PDF)
Natural Frequencies and Separation of Variables: Nonlinear PDEs, Example: Nonlinear Schrödinger's Equation, Elliptic Integrals and Functions

16 Remarks on the Nonlinear Schrödinger Equation (PDF)
General Eigenvalue Problem for Linear PDEs with Self-adjoint Operators
Classification of 2nd-order Quasilinear PDEs, Initial and Boundary Data

17 Introduction to Green's Functions, The Poisson Equation in 3D, Integral Equation for the "Nonlinear Poisson Equation" (PDF)
Green's Functions for Nonlinear Problems

18 Green's Functions for Nonlinear PDEs: Example: Infinite Vibrating String with Forcing, The Issue of (Classical) Causality, Formulation of the Integral Equation, Analytical Solution by Regular Perturbation (PDF)

19 Conversion of Self-adjoint Problems to Integral Equations (PDF)
Introduction to Dispersive Waves, Dispersion Relations, Uniform Klein-Gordon Equation, Linear Superposition and the Fourier Transform, The Stationary-phase Method for Linear Dispersive Waves

20 Extra Lecture (PDF)
Linear Dispersive Waves (cont.): Phase and Group Velocities, Energy Propagation, Theory of Caustics, Airy Function
Generalizations: Local Wave Number and Frequency, Slowly Varying Wave Amplitudes

21 Asymptotic Expansions for Non-uniform PDEs, Example: Non-uniform Klein-Gordon Equation (PDF)
Kinematic Derivation of Group Velocity

22 Dimensional Analysis for Stationary-phase Method (Linear Dispersive Waves), Characteristic Length and Time of a Dispersive System (PDF)
Introduction to Dimensional Analysis and Similarity for PDEs, Example: The Diffusion Equation

23 Dimensional Analysis and Similarity (cont.): Idea of Stretching Transformations, Example: Nonlinear Diffusion (PDF)

24 Extra Lecture (PDF)
Dimensional Analysis and Similarity (cont.): More on Nonlinear Diffusion, Solutions of Compact Support

25 Comments on the Blasius Problem (PDF)
Introduction to Perturbation Methods for PDEs: Regular Perturbation, Example

26 Regular Perturbation for Linear Schrödinger Equation with a Potential (PDF)
Perturbation Methods for PDEs: Singular Perturbation, Boundary Layers, Elementary Example

27 Singular Perturbation for PDEs (cont.), More Advanced Examples (PDF)

Quiz 2

28 Boundary Layers (cont.): Anatomy of Inner and Outer Solutions (PDF)
Introduction to Solitary Waves and Solitons, Water Waves, Solitary Waves for the KdV Equation, The Sine-Gordon Equation: Kink and Anti-kink Solutions

29 Extra Lecture (PDF)

(Heuristic) Definition of Soliton, Some Nonlinear Evolution PDEs with Soliton Solutions, Solutions to the Sine-Gordon Equation via Separation of Variables, Outline of the Inverse Scattering Transform Idea and Technique

Special Topics: The Painlevé Conjecture, The Painlevé Property, The Painlevé Equations

[Pseudo Statistic]

There have been a number of questions on Partial Differential Equations.
Within the Math & Science Tutorials (http://www.physicsforums.com/forumdisplay.php?f=160) are -
http://www.physics.miami.edu/nearing/mathmethods/ - link in thread (Mathematical Tools for Physics) posted by ranger, which leads to -
http://www.physics.miami.edu/nearing...ethods/pde.pdf - pdf file on PDE's.
For further information -
http://mathworld.wolfram.com/Partial...lEquation.html
http://mathworld.wolfram.com/Hyperbo...lEquation.html
http://mathworld.wolfram.com/Ellipti...lEquation.html
http://www.sst.ph.ic.ac.uk/angus/Lec...ys/node24.html
Many thanks to ranger for that link. :smile:
On here your links are showing up a bit weird for some reason.... with the dots intact in the links because they're so long so that when you click then, it uses those dots in the link. :\
Might be a problem with my browser, but blah, just thought it was worth mentioning...

[Astronuc]

A brief introduction to PDE's -
http://csep1.phy.ornl.gov/pde/pde.html

[Astronuc]

Partial differential equations

Nils Andersson
Department of Mathematics
University of Southampton

http://www.soton.ac.uk/~jhr/MA361/main.html

[Astronuc]

Applied Mathematics: Partial Differential Equations
Professor Alan Hood, School of Mathematics and Statistics, University of St. Andrews, Scotland
http://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/PDE.html

[Astronuc]

These can be encountered in PDE's, or at least are useful to understanding some PDE's.

http://mathworld.wolfram.com/VectorDerivative.html

[PhilosophyofPhysics]

Excellent, I really need to learn this stuff.

[Astronuc]

Numerical Methods for Partial Differential Equations:
an Overview and Applications
André Jaun, NADA, Royal Institute of Technology, Stockholm

http://www.lifelong-learners.com/pde/SYL/pde.php

[FrogPad]

Here is an excellent introduction to PDE's with Maple. They are written very well, and provide graphical progress through solving them.
http://www.maplesoft.com/applications/app_center_view.aspx?AID=1553

you have to be a maple member, but it is free, and seriously worth the few minutes to sign up for it.

[Physics_wiz]

Here is an excellent introduction to PDE's with Maple. They are written very well, and provide graphical progress through solving them.
http://www.maplesoft.com/applications/app_center_view.aspx?AID=1553

you have to be a maple member, but it is free, and seriously worth the few minutes to sign up for it.

Do I have to have maple to view the lessons? I signed up and clicked on the html version, but it seems to only show the first section.

[FrogPad]

Do I have to have maple to view the lessons? I signed up and clicked on the html version, but it seems to only show the first section.

Download the code. I believe it is a zip file containing all the lessons.

[Physics_wiz]

Download the code. I believe it is a zip file containing all the lessons.

Yes, but how do I open the files in the big zip file if I don't have maple?

[andytran]

Does anyone know where i can find some useful sites on reduction to canonical form?

none of the sites above say much about them.

thanks!

[manojsoni]

what is weak solution of on dimensional wave eqation ?

[Astronuc]

On weak solution - see http://en.wikipedia.org/wiki/Weak_solution

http://www.mathphysics.com/pde/ch7wr.html

Abstract - http://www.asp.ucar.edu/gtp/CMG/4Nov04-Abs.pdf



Math 553 Partial Differential Equations
http://www.math.uiuc.edu/~laugesen/553/sec_3_12.html

http://www.math.uiuc.edu/~laugesen/553/homework.html

Problems - http://www.mathematik.uni-karlsruhe.de/mi1weis/lehre/pde2006w/seite/uebung/media/pde14.pdf


Re canonical forms, see -
http://www.math.uiuc.edu/~laugesen/553/canonical_forms.pdf