Unit of study_

# MATH3978: PDEs and Waves (Advanced)

The aim of this unit is to introduce some fundamental concepts of the theory of partial differential equations (PDEs) arising in Physics, Chemistry, Biology and Mathematical Finance. The focus is mainly on linear equations but some important examples of nonlinear equations and related phenomena re introduced as well. After an introductory lecture, we proceed with first-order PDEs and the method of characteristics. Here, we also nonlinear transport equations and shock waves are discussed. Then the theory of the elliptic equations is presented with an emphasis on eigenvalue problems and their application to solve parabolic and hyperbolic initial boundary-value problems. The Maximum principle and Harnack's inequality will be discussed and the theory of Green's functions.

Code MATH3978 Mathematics and Statistics Academic Operations 6
 Prerequisites: ? A mark of 65 or greater in 6cp from (MATH2X21 or MATH2X65 or MATH2067) and a mark of 65 or greater 6cp from (MATH2X22 or MATH2X61) None MATH3078 or MATH4078 [MATH2X61 and MATH2X65] or [MATH2X21 and MATH2X22]

At the completion of this unit, you should be able to:

• LO1. employ foundational techniques to analyze and solve a range of different types of partial differential equations
• LO2. explain how classical PDEs are derived and their application in different types of problems
• LO3. solve classical 2nd-order linear differential equations and apply appropriate boundary and initial conditions
• LO4. apply the theory of orthogonal polynomials to solve a range of structured PDEs
• LO5. calculate solutions or perform analysis of classical nonlinear PDEs
• LO6. synthesise solution methods and equations analysis to classify solutions of nonlinear PDEs
• LO7. communicate mathematical analysis accurately, completely and correctly using algebraic, computational, or graphical methods

## Unit outlines

Unit outlines will be available 1 week before the first day of teaching for the relevant session.