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Durham University

Department of Mathematical Sciences

MATH2637 Mathematical Modelling II

The aim of the module is to apply mathematics to solve real-world problems. You will learn, by example, how various systems can be modelled mathematically. As most of the mathematical problems arising from modelling are too complex to solve analytically, you will learn to solve them using computer programs written in Python. The systems modelled will cover a wide range of topics, taken from physics, biology and chemistry.

The module is project-based and centered around self-learning. There are 3 hours of practical classes per week. One of these practical classes is a mix of lecturing and practical work on the computer, where you are introduced to the mathematical problems that you solve, as well as some techniques necessary to do so. We also provide an introduction to LaTeX. The other two hours of practicals consist of solving modelling problems in Python on the computer by working through problem sheets. You also learn some Python skills not covered in depth in the first year Programming module, such as using matrices and classes.

The module is assessed entirely by course work, which consists of solving problems related to a specific topic and presenting the results in two summative mini-projects (3000 to 4000 words each) written in LaTeX, as well as writing Python programs. These mini-projects are not just an assessment of your performance but form an integral part of the learning process, and therefore take a substantial fraction (about 40%, so 20 hours on each mini-project) of the time allocated to the module. The first mini-project is to be handed in during the first term and a second one is to be handed in early in the second term. The two mini-projects each count for 50% of your final mark.

Homework problems are set every week and cover material in the practical problem sheets. Their aim is to prepare you for the mini-projects by covering, in a step-by-step manner, the skills necessary to complete them.

Students taking this module will need to spend 20 hours, so about 2 hours per day, in weeks 7 and 8 to work on the first mini-project, and again 20 hours in total over the Christmas break, to work on the second mini-project. Because there is no end-of-year exam, time normally spent on revision after Epiphany needs to be spent partly during term and partly over the Christmas break.

Outline of Course

Aim: To provide an introduction to the mathematical modelling of various systems.

Mathematical skills:

  • Finding roots using the shooting method.
  • Solve ordinary differential equations numerically.
  • Monte-Carlo integration.
  • Solving partial differential equations using relaxation.
  • Reducing the number of parameters in an equation using symmetries.

Programming skills:

  • Use matrices (numpy).
  • Generate simple graphics (matplotlib).
  • Manipulate data sets (pandas).
  • Use classes.

Examples of problems that may be considered are:

  • Compute the trajectory of a cannon ball subjected to friction, or an electron in an electromagnetic field.
  • Compute the trajectory of a satellite in orbit to reach a target in a different orbit.
  • Use the Fast Fourier Transform to extract the frequency content of a recorded note out of your favourite recorded music.
  • Solve the equation describing a system of many masses connected by small springs.
  • Solve a classical diffraction problem using non-relativistic path integrals.
  • Model the diffusion of a small protein in a living cell.
  • Simulate simple chemical reactions.
  • Simulate the time evolution of simple spin systems and their phase transitions.

Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.

Reading List

Please see the Library Catalogue for the MATH2637 reading list.

Examination Information

For information about use of calculators and dictionaries in exams please see the Examination Information page in the Degree Programme Handbook.