The University of Southampton

MATH2015 Mathematical Methods for Scientists

Module Overview

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.

Aims and Objectives

Module Aims

This module re-examines linear systems and ordinary differential equations (ODEs) from a more formal point of view, and introduces more advanced ODE methods such as reduction of order and variation of parameters. We introduce Fourier series and Fourier transforms, both from a theoretical and a practical point of view. Lastly, we give an introduction to partial differential equations and such general solution methods such as characteristic curves and separation of variables.

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

  • 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
  • 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


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.

Independent Study102
Total study time150

Resources & Reading list

Jordan, D.W and Smith, P. Mathematical Techniques. 

Arfken, G.B. Mathematical Methods for Physicists. 

McQuarrie, D.A. Mathematical Methods for Scientists and Engineers. 

Riley, K.F. Mathematical Methods for Physics and Engineering. 

Kreyszig, E. Advanced Engineering Mathematics. 



MethodPercentage contribution
Closed book Examination  (120 minutes) 80%
In-class Test  (40 minutes) 20%

Repeat Information

Repeat type: Internal & External

Linked modules

Pre-requisites: (MATH1006 OR MATH1008) AND (MATH1007 OR MATH1009)

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