- Dr Andrew Hazel (Reader - University of Manchester)
|Delivery Type||Delivery length / details|
|Lecture||33 x 1 Hour Lectures|
|Assessment Type||Assessment length / details||Proportion|
|Semester Exam||2 Hours||100%|
|Supplementary Exam||2 Hours||100%|
On completion of this module, a student should be able to:
1. solve simple linear partial differential equations;
2. illustrate with suitable examples the occurrence of such equations in physics and industry;
3. interpret the meaning of mathematical solutions of partial differential equations in the appropriate context.
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.
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. Fourier transform methods: definition, properties, derivative theorem, convolution theorem, consideration of applicable functional spaces, application to heat equation Cauchy problem.
This module is at CQFW Level 6