Mathematical methods, models and modelling
Solve real problems by finding out how they are transformed into mathematical models and learning the methods of solution. This module covers classical mechanical models as well as some non-mechanical models such as population dynamics; and methods including vector algebra, differential equations, calculus (including several variables and vector calculus), matrices, methods for three-dimensional problems, and numerical methods. Teaching is supported and enhanced by use of a computer algebra package. This module is essential for higher level study of applied mathematics. To study this module you’ll need a sound knowledge of mathematics as developed in Essential mathematics 1 (MST124) and Essential mathematics 2 (MST125) or equivalent.
What you will study
This module will be of particular interest to you if you use mathematics or mathematical reasoning in your work and feel that you need a firmer grounding in it, or if you think you might find it useful to extend your application of mathematics to a wider range of problems. The module is also very suitable for those planning to teach applied mathematics.
Around half of this module is about using mathematical models to represent suitable aspects of the real world; the other half is about mathematical methods that are useful in working with such models. The work on models is devoted mainly to the study of classical mechanics, although non-mechanical models – such as those used in population dynamics – are also studied. The process of mathematical modelling, based on simplifying assumptions about the real world, is outlined. You will work in groups to create a mathematical model and to produce a mini-report. The work on methods comprises topics chosen for their usefulness in dealing with the models; the main emphasis is on solving the problems arising in the real world, rather than on axiom systems or rigorous proofs. These methods include differential equations, linear algebra, advanced calculus and numerical methods.
You’ll begin the mechanics part of the module with statics, where there are forces but no motion, and then you’ll be introduced to the fundamental laws governing the motions of bodies acted on by forces – Newton's laws of motion. These are applied to model:
- the motion of a particle moving in a straight line under the influence of known forces
- undamped oscillations
- the motion of a particle in space
- the motions of systems of particles
- the damped and forced vibrations of a single particle
- the motion (and vibrations) of several particles.
In the methods part of the module you’ll cover both analytic and numerical methods. You’ll explore the analytical (as opposed to numerical) solution of first-order and of linear, constant-coefficient, second-order ordinary differential equations, followed by systems of linear and non-linear differential equations and an introduction to methods for solving partial differential equations. The topics in algebra are vector algebra, the theory of matrices and determinants, and eigenvalues and eigenvectors. You’ll develop the elements of the calculus of functions of several variables, including vector calculus and multiple integrals, and make a start on the study of Fourier analysis. Finally, the study of numerical techniques covers the solution of systems of linear algebraic equations, methods for finding eigenvalues and eigenvectors of matrices, and methods for approximating the solution of differential equations.
Read the full content list here.
You will learn
Successful study of this module should improve your skills in being able to think logically, express ideas and problems in mathematical language, communicate mathematical arguments clearly, interpret mathematical results in real-world terms and find solutions to problems.
This module may help you to gain membership of the Institute of Mathematics and its Applications (IMA). For further information, see the IMA website.
To study this module, normally you should have completed Essential mathematics 2 (MST125) or the discontinued module MS221.
There may be circumstances in which you can study this module without having first studied MST125, but you should speak to an adviser to discuss this before registering on this module.
Knowledge of mechanics is not needed, but we do not recommend the module if you have little mathematical experience. You need a good basic working knowledge of:
- algebra – you must be able to solve linear and quadratic equations with one unknown, to multiply and add polynomials, to factorise quadratic polynomials and to work with complex numbers
- geometry – you must know Pythagoras' theorem and how to use Cartesian coordinates, e.g. the equations of straight lines and circles
- trigonometry – you need to know the basic properties of the three trigonometric ratios sine, cosine and tangent, and the definitions of the corresponding inverse functions
- calculus – you must be able to differentiate and integrate a variety of functions, though great facility in integration is not necessary
- mechanics – you should have some basic knowledge of Newtonian mechanics.
You can try our diagnostic quiz to help you determine whether you are adequately prepared for this module.
Module books, other printed materials, algebra software, and module website.
You will need
A calculator – you may wish to use this during the module, but you are not allowed to take a calculator into the examination.
A computing device with a browser and broadband internet access is required for this module. Any modern browser will be suitable for most computer activities. Functionality may be limited on mobile devices.
Any additional software will be provided, or is generally freely available. However, some activities may have more specific requirements. For this reason, you will need to be able to install and run additional software on a device that meets the requirements below.
A desktop or laptop computer with either:
- Windows 7 or higher
- Mac OS X 10.7 or higher
The screen of the device must have a resolution of at least 1024 pixels horizontally and 768 pixels vertically.
To join in the spoken conversation in our online rooms we recommend a headset (headphones or earphones with an integrated microphone).
Our Skills for OU study website has further information including computing skills for study, computer security, acquiring a computer and Microsoft software offers for students.