This module introduces the concept of partial differential equations, the concept of solution to partial differential equations, with initial and initial boundary value problems for evolution PDE, and steady state boundary value problems for steady state PDE. First order PDE and systems of first order PDE’s are considered (both linear and nonlinear) and solution methods are developed, with specific examples relating to linear and nonlinear simple wave PDE’s. General second order linear PDE are considered, and classified accordingly to canonical form in canonical coordinates. Second order linear hyperbolic, parabolic and elliptic PDEs are examined in general, and then in detail for the wave equation, the diffusion equation and Laplace’s equation. More general linear evolution PDE will also be considered. The solution methods with involve Fourier integral, Fourier series and Green’s Function approaches. Uniqueness will be established in all cases. Specific nonlinear evolution PDE such as the Burger’s equation and Korteweg de Vries equation will be introduced and discussed.

Learning Outcomes

By the end of the module students should be able to:

Appreciate PDE problems and understand the concept of classical solution to PDE problems.

Students should be able to effectively employ the method of characteristics to construct solutions to PDE problems.

For linear second order PDE problems, students should be able to construct the canonical coordinates and the canonical form and classify the PDE.

For the wave equation, the diffusion equation and Laplace’s equation, students should be able to construct solutions to the Cauchy, Neumann and Dirichlety problems using Fourier Integral, Fourier series and Green’s Function methods.

Students should also appreciate how to apply these methods to more general linear evolution PDE

Assessment

27714-01 : Raw Module Mark : Coursework (100%)

Assessment Methods & Exceptions

2 hour Written Unseen January Examination (80%); In-course Assessment (20%).