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Module MA3426: Partial Differential Equations II

Credit weighting (ECTS)
5 credits
Semester/term taught
Hilary term 2014-15
Contact Hours
11 weeks, 3 lectures including tutorials per week
Lecturer
Prof. John Stalker
Learning Outcomes
On successful completion of this module, students will be able to:
  • Solve, in a higher number of dimensions, problems for the Wave, Heat, and Laplace Equations which were treated in low dimensions in MA3425. In addition to those, students should be able to use Young's inequality to obtain $L^p$ estimates on solutions in terms of data;
  • Demonstrate a familiarity with the definition and main properties of distributions and the principal operations on distributions: addition, multiplication by smooth functions, differentiation and convolution. Give the definition of the term ''fundamental solution'' and verify that a given distribution is a fundamental solution for a given differential equation;
  • Solve, by the method of characteristics, first order linear scalar partial differential equations. Students should also be able to determine when the initial value problem for such an equation has a unique global solution.
  • Solve the initial value problem for Burgers' equation, including cases where shocks are present initially or develop later. Give the definitions of ''weak solution'' and ''shock'' and determine whether the singularity of a given weak solution are shocks.
Module Content
  • Theory of the Wave, Heat and Laplace equations in higher dimensions;
  • $L^p$ distributions, convolution and fundamental solutions;
  • First order linear scalar partial differential equations;
  • Burger's equation, weak solutions, the entropy condition and shocks;
 
Module Prerequisite
MA2223 - Metric Spaces, MA2224 - Lebesgue Integral, MA2326 - Ordinary Differential Equations
Assessment Detail
This module will be examined in a 2 hour examination in Trinity term. Continuous assessment will contribute 10% to the final grade for the module at the annual examination session.