Mathematical Modelling

 

The Basic Ideas Behind Mathematical Modelling

Mathematical modelling is the process of describing a real world problem in mathematical terms, usually in the form of equations, and then using these equations both to help understand the original problem, and also to discover new features about the problem. Modelling both lies at the heart of much of our understanding of the world, and it allows engineers to design the technology of the future. With modelling we can travel to the edge of the universe, peer into the heart of the atom, and understand the future of our climate.

We are all very familiar with one application of mathematical modelling, namely the weather forecast. A modern weather forecast is based on the following steps

  • Start with the laws of physics
  • Encode these as (differential) equations, in particular the Navier-Stokes equations.
  • Take data from satellites and weather stations to determine today’s weather accurately.
  • Using this as an initial condition, (using a super computer) solve the equations for 24 hours to give us the weather tomorrow.
  • Continuously update the forecast.
  • Present the results in a way that all can understand.

Despite rumours to the contrary this process works, and works well. At least for short term weather forecasting. This process is a special case of the more general process of mathematical modelling which can be described simply as:

  1. Identify the problem i.e talk to the people involved.
  2. Clarify the science.
  3. Formulate the science mathematically.
  4. Solve the mathematics possibly using a computer.
  5. Draw conclusions.
  6. Explain your results.

As well as weather forecasting, this process is used to design aeroplanes and cars, develop new drugs, control the electricity supply network and even help establish the cause of the 1987 Kings Cross Fire.


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