University of Torino
by
Prof. Jüri Engelbrecht DSc
Tallinn University of Technology / Estonian Academy of Sciences
Spring 2012
Lecture 1: Methodology of mathematical
modelling
The main principles of mathematical modelling are briefly
explained together with a short historical sketch. The main
stages of modelling are described focusing on validation of
results and improvement of a model. For modelling of
physical processes, the importance of conservation laws is
stressed and explained as well as the need for using
thermodynamical considerations. This forms a solid basis
for many well-known and also for newly developed models.
For modelling of processes in biology or social life, the
situation is much more complicated and the emphasis is
more on interpreting the observation data. Several
arguments used for modelling such processes is discussed
and examples given.
Motto: Everything must be made as simple as possible. But not
simpler. A.Einstein
Lecture 2: Complexity in mathematical
modelling
Contemporary understanding of the world is based on the
concept of complexity. Complexity research is a holistic
process and takes many constituents of a whole into account
while the outcome is strongly dependent on interaction
between the constituents. That is why the systems out of
equilibrium, adaptive and self-organising systems, networks,
nonlinear dynamical systems, etc are of importance. The
main properties of complex systems are explained and
several examples given which demonstrate simple models of
complex behaviour in physics, biology, mathematics,
economics and so on.
Motto: The whole is more than the sum of its parts. Aristotle
References I (methodology)
1. A.C.Fowler. Mathematical Models in the Applied Sciences.
Cambridge University Press, 1997.
2. D.Clements. Mathematical Modelling. Cambridge
University Press, 1989.
3. Concise Encyclopedia of Modelling and Simulation.
D.P.Atherton, P.Borne (eds.), Pergamon Press,
Oxford et al., 1992.
4. P.Bak. How Nature Works. Oxford University Press, 1997.
5. G.I.Barenblatt. Scaling, Selfsimilarity, and Intermediate
Asymptotics. Cambridge University Press, 1996.
6. J.Caldwell, D.K.S.Ng. Mathematical Modelling. A Case
Studies Approach. Kluwer, 2004.
7. M.M.Meerschaert. Mathematical Modelling. Elsevier, 2007
(Chapter 1 in Web).
References II (complexity)
1. G. Nicolis, C. Nicolis. Foundations of Complex Systems.
World Scientific, Singapore, 2007.
2. P. Érdi. Complexity Explained. Springer, Berlin, 2008.
3. S. Strogatz. Sync: Rhythms of Nature, Rhythms of
Ourselves. Allen Lane, 2003.
4. J. Gribbin. Deep Simplicity. Allen Lane, 2004.
5. M. Buchanan. Small World. Phoenix, 2003.
6. N. Johnson. Simply Complexity. One World, Oxford, 2007.
7. Encyclopedia of Complexity and Systems Science. Robert
A. Meyers (Ed.): Springer 2009, vol 1-11, (10370pp).
From the web
1. Mathematical model
www.answers.com./topic/mathematical-model.
2. G. de Vries - what is mathematical modelling?
www.math.ualberta.ca/~devries/erc2001/slides.pdf.
3. Teacher package: Mathematical Modelling
http://plus.math.org/issue44/package/index.html.
4. ScienceDaily: Mathematical Modeling News.
5. Wikipedia. Images for mathematical modeling.