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Module Title LH Methods in Partial Differential Equations
SchoolMathematics
Department Mathematics
Module Code 06 27714
Module Lead Dr Chris Good
Level Honours Level
Credits 20
Semester Full Term
Pre-requisites Linear Algebra - (06 15552) LI Multivariable & Vector Analysis - (06 25667) LI Differential Equations - (06 25670)
Co-requisites
Restrictions None
Contact Hours Lecture-46 hours
Seminar-0 hours
Tutorial-10 hours
Project supervision-0 hours
Demonstration-0 hours
Practical Classes and workshops-0 hours
Supervised time in studio/workshop-0 hours
Fieldwork-0 hours
External Visits-0 hours
Work based learning-0 hours
Guided independent study-144 hours
Placement-0 hours
Year Abroad-0 hours
Exclusions
Description 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 (0%)
27714-03 : Final Module Mark : Coursework (100%)
Assessment Methods & Exceptions 90% on one three hour examination; 10% from coursework and/or class tests.
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