Partial Differential Equations

On successful completion of this module students should be able to:

1. 1. classify partial differential equations and identify appropriate solution techniques; 2. Solve first order linear partial differential equations using the method of characteristics; 3. demonstrate an ability to use the method of separation of variables to solve second order linear partial differential equations on rectangular domains; 4. solve classical second order partial differential equations (wave, heat, Laplace’s equation) in infinite and semi-infinite domains and interpret their solutions; 5. prove results concerning uniqueness of wave equation and heat equation solutions.

Many mathematical problems arising in the physical sciences, engineering, and technology, may be formulated in terms of partial differential equations. In attempting to solve such problems, one must be aware of the various types of partial differential equation which exist, and of the different boundary conditions associated with each type. These factors determine which method of solution one should use.

To teach the student how to recognise the type of a partial differential equation, and how to choose and implement an appropriate method of solution.

1. Fundamentals: definitions and examples, simple partial differential equations.
2. First order equations: the method of characteristics.
3. Boundary conditions: Dirichlet, Neumann, Robin, well-posedness and ill-posedness.
4. Second order equations: classification, reduction to canonical forms.
5. The wave equation: general solution, Cauchy problem, reflection principle, Duhamel principle, bounded string, energy and uniqueness.
6. The heat equation: maximum principle, uniqueness, separation of variables, properties of solutions, the fundamential solution.
7. Green's functions

This module is at CQFW Level 6