Course Content
More or less everything worth describing can be described by partial differential equations. While there is not much useful that one can say about partial differential equations in general, one can say a fair amount about broad classes of partial differential equations, and a great deal about some particular partial differential equations. Luckily these include the differential equation most relevant to physics.
The course will cover standard introductory material from the theory of partial differential equations. The approach will be fairly standard, except that I will emphasise the role of symmetry groups somewhat more than is usual.
A preliminary outline of course follows, but this is subject to change. In particular, I don't yet what fraction of the class took prof. Constantin's sophister course last year. If the proportion is high then I will modify the outline to reduce overlap between the two courses.
- Introduction
- Examples
- Terminology
- Order of an Equation
- Ordinary vs. Partial Differential Equations
- Linear vs. Non-linear Equations
- Scalar Equations vs. Systems
- First Order Equations
- Method of Characteristics
- Examples
- Simple Physical Examples
- Transport
- Diffusion
- Vibration
- Initial and Boundary Value Problems
- Well Posed Problems
- Existence
- Uniqueness
- Stability
- The Wave Equation in 1+1 Dimensions
- Method of Characteristics
- Explicit Solution
- Existence
- Causality and Energy
- Domain of Dependence
- Domain of Influence
- Uniqueness
- Stability
- Weak Solutions
- Method of Characteristics
- Diffusion in 1+1 dimensions
- Maximum Principle
- Uniqueness
- Stability
- Symmetries
- Fundamental Solution
- Existence
- Maximum Principle
- Boundary Problems and Reflection
- Problems on the Half Line
- Diffusion
- Dirichlet Condition
- Neumann Condition
- Robin Condition
- Wave Equation
- Dirichlet Condition
- Neumann Condition
- Robin Condition
- Diffusion
- Problems on Finite Intervals
- Dirichlet at Both Ends
- Neumann at Both Ends
- Dirichlet and Neumann
- Problems on the Half Line
- Inhomogeneous Problems
- Diffusion
- Wave Equation
- Separation of Variables
- Harmonic Oscillator on the Real Line
- Diffusion on a Finite Interval
- Wave Equation on a Finite Interval
- Diffusion on the Real Line
- Wave Equation on the Real Line
- Harmonic Functions
- Laplace Equation in 2 Dimensions
- Maximum Principle
- Uniqueness
- Stability
- Rectangular Domain
- Circular Domain
- Conformal Symmetry Group
- Laplace Equation in 2 Dimensions
- Wave Equation in Higher Dimensions
- 1+3 dimensions
- 1+2 dimensions
- Rays, Characteristics
- Full Symmetry Group of the Wave Equation
- Conserved Quantities
- Scattering Theory
- Burger's Equation
Textbook
The text for the course is Partial Differential Equations, An Introduction by Walter Strauss. You don't need to buy it immediately, but you should do so eventually.
Tutorials
There are no tutorials in the usual sense of the word.
Exams
There will be an annual exam on 24 May 2007 in Luce Hall from 9:30 to 12:30. I have prepared a revision guide which is available in Postscript or PDF formats. There will be 9 problems, of which you are to attempt five.
Assignments
| Questions | Solutions | |||
|---|---|---|---|---|
| PostScript | PostScript | |||
| Due 25 October | a1.ps | a1.pdf | s1.ps | s1.pdf |
| Due 15 November | a2.ps | a2.pdf | s2.ps | s2.pdf |
| Due 8 December | a3.ps | a3.pdf | s3.ps | s3.pdf |
| Due 31 January | a4.ps | a4.pdf | s4.ps | s4.pdf |
| Due 28 February | a5.ps | a5.pdf | s5.ps | s5.pdf |
| Due 18 April | a6.ps | a6.pdf | s6.ps | s6.pdf |
Notes
I had hoped to give detailed notes for the material on symmetries of the Laplace Equation, but these will probably not be finished in time to be useful. I am making a partial version available in PostScript and PDF versions.