Abstract
Apparently the word model does not raise much confidence among general public or journalists. The terms “model” and “modelling” are in fact relatively new, therefore it is perhaps not surprising that their meaning is not very well understood. Of course, scientists have always made models also in the modern sense of the word, but maybe they used some other words like law rather than model. Would the above journalist have written in this case: “Researchers only have various laws”? Anyway, models and modelling have become increasingly popular. On the next few pages, we will consider models which can be expressed with the help of (systems of) differential equations.
Nature of air pollution fallout of Kola peninsula still unknown. Researchers only have various models.
Headline in newspaper “Pohjolan Sanomat”, October 18 1988
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Notes
- 1.
There was some controversy on how much pollution that was generated in the Kola peninsula crossed the border to the Finnish side.
- 2.
In quantum mechanics, the term “quantized” is used instead of discrete.
- 3.
In differential geometry, a curve is defined as a map, although often the image of this map is also called a curve. Similarly in books on differential equations, orbits are sometimes also called solutions.
- 4.
Incidentally, in old literature “to integrate a differential equation” means “to solve a differential equation”. The reason for this is probably that the main technique for obtaining explicit solutions was a separation of variables, which in effect reduces the problem to the computation of certain integrals.
- 5.
The set of all solutions is called the general solution.
- 6.
Incidentally, these works of Liouville and others became again popular when the era of computer algebra systems dawned. The problem there is to precisely characterize which type of differential equations has solutions in a certain function class and how the solution can actually be computed in practice.
- 7.
The minus sign is for historical reasons.
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Tuomela, J. (2016). Modelling with Differential Equations. In: Pohjolainen, S. (eds) Mathematical Modelling. Springer, Cham. https://doi.org/10.1007/978-3-319-27836-0_8
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