Unit of study_

# MATH3078: PDEs and Waves

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 MATH3078 Mathematics and Statistics Academic Operations 6
 Prerequisites: ? 6cp from (MATH2X21 or MATH2X65 or MATH2067) and 6cp from (MATH2X22 or MATH2X61) None MATH3978 or MATH4078 [MATH2X61 and MATH2X65] or [MATH2X21 and MATH2X22]

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

• LO1. demonstrating the ability to recognize different types of partial differential equations: "linear" or "nonlinear", "order of the given equation", "homogeneous" or "inhomogeneous", and if it concerns 2nd-order equations, whether they are of "elliptic", "parabolic", or "hyperbolic" type
• LO2. demonstrating the conceptional understanding of how to apply different methods for solving different types of partial differential equations. Those methods include the use of classical ODE-concepts to solve PDEs
• LO3. understanding the definitions, main theorem, and corollaries of Green's functions and Poisson kernel
• LO4. be fluent with "change of variable" into polar, cylindrical and spherically coordinates and to be able to compute partial derivatives in these coordinates
• LO5. develop an appreciation and strong working knowledge of the theory and application of elementary partial differential equations
• LO6. be fluent in using generalized Fourier transforms to solve parabolic and hyperbolic initial boundary value problems where the spatial variable might be of more than one variable

## Unit outlines

Unit outlines will be available 2 weeks before the first day of teaching for the relevant session.