Mathematical Methods for Scientists

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Solve a range of first and second order partial differential equations with boundary conditions using methods of characteristic curves and separation of variables
• 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
• 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.

Lectures, tutorials, private study.

Summative

This is how we’ll formally assess what you have learned in this module.