SS Theoretical Physics )

**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

- 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 - Wave Equation

Dirichlet Condition, Neumann 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 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

Dec 10, 2008

File translated from T

On 10 Dec 2008, 18:50.