Department of Physics, Unit Catalogue 2006/07

PH30025 Equations of science

Credits: 6

Level: Honours

Semester: 2

Assessment: EX80CW20

Requisites:

Before taking this unit you must take PH10007 and take PH10008 and take PH20019 and take PH20020

Aims & Learning Objectives:
The aims of this unit are to introduce concepts and methods used in solving some of the most important equations, both linear and non-linear, which arise in the natural sciences, and to introduce students to a broad range of examples and applications. After taking this unit the student should be able to:
* distinguish linear and non-linear equations and contrast the different forms of solution which arise;
* recognise some of the key equations which arise in the natural sciences;
* apply the separation of variables method to linear partial differential equations, and solve the resulting ordinary differential equations by series solution;
* use superposition methods for inhomogeneous equations;
* determine solutions to some of the key non-linear equations, and analyse non-linear ordinary differential equations;
* analyse one-dimensional difference equations.
Content:
Linear equations of science (12 hours): Derivation of the diffusion equation as an example of how partial differential equations arise in the natural sciences. Introduction to Laplace's equation, Poisson's equation, wave equation, Schrodinger's equation. Linearity and superposition. Boundary conditions. Solution by separation of variables; examples showing separation in Cartesian, cylindrical and spherical coordinate systems. Series solutions of differential equations; examples including Legendre polynomials, spherical harmonics and Bessel functions. Solution of inhomogeneous ODE's. Examples from the natural sciences. Non-linearity and chaos (12 hours): Examples of non-linearity in the natural sciences; Non-linear wave equations, solitary waves, physical examples. Nonlinear differential equations: phase space, trajectories, fixed points, bifurcation. Examples from the natural sciences. Non-linear difference equations: orbits, cobwebs, fixed points, bifurcations, chaos. Examples from the natural sciences.

University | Catalogues for 2006/07