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;

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.