School of Mathematics
School of Mathematics
Course 419 - Partial Differential Equations
2008-09 (JS & SS Mathematics
SS Theoretical Physics
)
Lecturer: Dr. John Stalker
Requirements/prerequisites: 216
Duration: 19 weeks
Number of lectures per week: 3
Assessment: Assignments will count for 10% of the mark for 419.
End-of-year Examination: 3-hour annual examination in May/June (will count for 90the mark).
Description:
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.
- Introduction
- Examples
- Terminology
- Order of an Equation
- Ordinary vs. Partial Differential Equations
- Linear vs. Non-linear Equations
- Scalar Equations vs. Systems
- First Order Scalar Equations
- Method of Characteristics
- Examples
- Problems of Existence and Uniqueness
- 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
- Diffusion in 1+1 dimensions
- Maximum Principle
- Symmetries
- Fundamental Solution
- Boundary Problems and Reflection
- Problems on the Half Line
- Diffusion
Dirichlet Condition;
Neumann Condition
- Wave Equation
Dirichlet Condition,
Neumann Condition
- Problems on Finite Intervals
- Dirichlet at Both Ends
- Neumann at Both Ends
- Dirichlet and Neumann
- Inhomogeneous Problems
- 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
- Rectangular Domain
- Circular Domain
- Conformal Symmetry Group
- Laplace Equation in 3 Dimensions
- Wave Equation in Higher Dimensions
- 1+3 dimensions
- 1+2 dimensions
- Burger's Equation
- Solution by Method of Characteristics
- Shocks
- Korteweg-de Vries
- Solitons
- Conservation Laws
Textbooks:
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.
Dec 10, 2008
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On 10 Dec 2008, 18:50.