Hello,

A concise explanation with examples, in 12pt font, will certainly exceed an A4 page. I recommend a look at this book, its very detailed and has examples:

Title: Mathematical Modeling and Simulation: Introduction for Scientists and Engineers

Author: Kai Velten

Publisher: John Wiley & Sons, 2009

ISBN: 3527627618, 9783527627615

Best regards.

Thank u dr Bello for input

Dear Aijt

This very wide and difficult problem. At the start I suppose that You have mathematical modelling in mind. Because genarally modelling is a kind of reflecting reality, sometimes in a very personal way.

For example all art is modelling of reality based of a peronal perception and experience, which is completely subjective.

But always in modelling You have to make assumptions. On the level of art it is also the case and for example You have to choose if You make a painting or a sculpture, if You use colours or Your work is black and white etc.

But in case of mathematical modelling Your aim is to get the objective tools in an abstract mathematical form to reflect some kind of reality - still there are various forms ofreality that may be described in the form of the mathematical model. They are for example: problems of economy, problems of a social nature (demographical problems), problems of a health nature both of groupes of people as well as individuals, agricultural problems etc.

But still I think that we have in mind mothematical modelling of physical or technological objects, for which You can make some experiments concerning their reaction to input signals and to measure their response to that stimulation.

You have to answer the basic question: wheather the behaviour of a given object is of a strictly repetitive kind (deterministic) or rather stochastic kind - the respponses are different within some range for the same input signals.

These are always assumptions based on observations (measurements); the other very important one concerns spatial objects - to use disributed parameters or lumped parameters to describe the behaviour of an object in space. This is very important assumption - a number of variables to model the object depend on that very much.

My personal experience in mathematical modelling is in modelling electronic and electromechanical devices, and these kind of objects are rather easy to model with a large experince gathered and a lot of literature to study. To formutate mathematical models I use mostly lumped parameters, linear approximation of a magnetic characteristics of ferromagnetic cores, simple approximated influence of temperature upon resistances in a system - and finally to formulate the model I employ Lagrange-Euler's equation method - resulting in a set of ordinary differential equations, so the results are fully deterministic. The above method (Lagrange-Euler's equations) belong to so called energy methods that have a number of important positive characteristics.

But in each field of science You have already a number of tested methods and a lot of experience and literature ready, and it is advisible to read and study a little. But it is very nice that in mathematical modelling it is always possible to find new solutions, which could better reflect measured (observed) reality or more efficient model that cost less in their applications, or models that are used in real time in a control of systems (obsertvers), and many more.

So a lot of success in Your field of research!

With kind regards

Piotr Wach, Opole, Poland

The basic steps of the model-building process are:

1.model selection

2.model fitting, and

3.model validation.

See link for details:

https://www.itl.nist.gov/div898/handbook/pmd/section4/pmd41.htm

many thanks Dr Wach for valuable points.