Synopsis for Fourier Series and PDEs

Overview

The course begins by introducing students to Fourier series, concentrating on their practical application rather than proofs of convergence. Students will then be shown how the heat equation, the wave equation and Laplace's equation arise in physical models. They will learn basic techniques for solving each of these equations in several independent variables, and will be introduced to elementary uniqueness theorems.

Learning Outcomes

Students will be familiar with Fourier series and their applications and be notionally aware of their convergence. Students will know how to derive the heat, wave and Laplace's equations in several independent variables and to solve them. They will begin the study of uniqueness of solution of these important PDEs.

Synopsis

Fourier series. Periodic, odd and even functions. Calculation of sine and cosine series. Simple applications concentrating on imparting familiarity with the calculation of Fourier coefficients and the use of Fourier series. The issue of convergence is discussed informally with examples. The link between convergence and smoothness is mentioned, together with its consquences for approximation purposes.

Partial Differential Equations: Introduction in descriptive mode on partial differential equations and how they arise. Derivation of (i) the wave equation of a string, (ii) the heat equation in one dimension (box argument only). Examples of solutions and their interpretation. Boundary conditions. Solution by separation of variables. Use of Fourier series to solve the wave equation, Laplace's equation and the heat equation (all with two independent variables). (Laplace's equation in Cartesian and in plane polar coordinates). Applications. Transformations in the independent variables. D'Alembert's solution of the wave equation and applications. Characteristic diagrams (excluding reflection and transmission).

Conservation laws: Uniqueness of solutions of heat equation and wave equation.

Reading

1. D. W. Jordan and P. Smith, Mathematical Techniques (Oxford University Press, 3rd Edition, 2003)
2. Erwin Kreyszig, Advanced Engineering Mathematics (Wiley, 8th Edition, 1999)
3. G.F.Carrier and C.E.Pearson, Partial Differential Equations — Theory and Technique (Academic Press, 1988)