This is an optional module for second-year students in physical sciences. The module introduces a number of more advanced methods for solving linear matrix equations and ordinary differential equations, as well as introducing Fourier series, and partial differential equations.
Pre-requisites: (MATH1006 OR MATH1008) AND (MATH1007 OR MATH1009)
Aims and Objectives
Having successfully completed this module you will be able to:
- Solve a range of first and second order ordinary differential equations, including initial and boundary value problems, recognising separable and exact equations, using integrating factors and methods of reduction of order and variation of parameters
- Solve a range of first and second order partial differential equations with boundary conditions using methods of characteristic curves and separation of variables
- Perform matrix mathematics techniques, including computing inverses, determinants, eigenvalues and eigenvectors and solving systems of linear equations
- Compute the Fourier series expansion of a given periodic function or its periodic extension
1. Matrix mathematics and linear systems: Properties of matrices, determinants, and inverses. Linear independence and orthogonality of vectors. Matrices and systems of ordinary differential equations. Solution of both homogeneous and inhomogeneous linear systems by Gauss elimination to Hermite form. Calculation of eigenvalues and eigenvectors. Matrix diagonalization.
2. Fourier series: Periodic functions. Orthogonality of sines and cosines. Extension to non-periodic functions (full and half-range expansions). Complex Fourier series expansion. Convergence and Gibbs phenomenon. Introduction to Fourier transforms. Orthogonal functions.
3. Ordinary differential equations: Definition and notation. Initial and boundary value problems. Separable and exact ODEs. Integrating factors. Constant coefficient and equidimensional ODEs. Methods of reduction of order and variation of parameters. Eigenfunction expansion.
4. Introduction to partial differential equations: Examples. Method of characteristic curves. Method of separation of variables. Second order PDEs.
Learning and Teaching
Teaching and learning methods
Lectures, tutorials, private study.
|Total study time||150|
Resources & Reading list
Jordan, D.W and Smith, P. Mathematical Techniques.
Arfken, G.B. Mathematical Methods for Physicists.
Kreyszig, E. Advanced Engineering Mathematics.
McQuarrie, D.A. Mathematical Methods for Scientists and Engineers.
Riley, K.F. Mathematical Methods for Physics and Engineering.
Summative assessment description
Repeat type: Internal & External