Partial Differential Equations

PDEs

MTH2104 Ordinary Differential Equations

3 hours One requisite

1 Definition of a partial differential equation (PDE). Definition of properties such as ‘order’ and ‘linear/nonlinear’. Descriptions of how partial differential equations arise in the context of applications. Specifically, how conservation laws lead to the derivations of Laplace’s equation (elliptic), diffusion equation (parabolic) and the Wave Equation (hyperbolic)

2 First order equations: / • Define the general form of a first order partial differential equation. Find / solution of first order linear equations of the generic type. Cauchy problem / in linear partial differential equation. / • The use of characteristic methods to solve nonlinear first order PDEs

3 Classification of second order linear equation: / Types of second order partial differential equations (examples and solutions). / Classification by reduction to canonical form. Use of change of variable to find the general solution of second order linear partial differential equation in two variables. Determination of particular solutions from given information

4 Fourier Series and applications: / • Description of Fourier series, and its particularizations to half-range sine and cosine series. The Dirichlet conditions for the existence of a Fourier series. / • Solution of linear partial differential equations by the method of separation of variables. Examples of the application of the method to the solution of boundary value problems for Laplace’s equation in two dimensions and initial boundary value problems for the diffusion equation in onedimension

5 Revision

1.1 Mapped to: K1, K2

Be familiar with various concepts and problems of partial differential equations (PDEs), including the order and classification of PDEs, Cauchy problem, and Dirichlet conditions for the existence of a Fourier series.

Teaching Strategy Lectures, problem-based learning