### Lecture Notes on Mathematical Modelling From Applied Sciences to Complex Systems

#### Abstract

The monograph corresponds to the first part of a course delivered at the Politecnico of Torino to students of the Master Graduation in Mathematical Engineering, devoted to modelling issues to show how the application of models to describe real world phenomena generates mathematical problems to be solved by appropriate mathematical methods. Mathematical models are quite simple being proposed with tutorial aims.

The contents are developed through four chapters. The first one proposes an introduction to the science of mathematical modelling focusing on the three representation scales of physical reality: microscopic, macroscopic and statistical over the microscopic states. The three chapters which follow deal with the derivation and applications of models related to each of the afore-mentioned scales. Different mathematical structures correspond to each scale. Specifically models at the microscopic scale are generally stated in terms of ordinary dierential equations, while models at the macroscopic scale are stated in terms of partial differential equations.

Models of the mathematical kinetic theory, presented in Chapter 4, are stated in terms of integro-differential equations.

The above different structures generate a variety of analytic and computational problems. The Lecture Notes look at applications focussing on modelling and computational issue.

ISBN-13: 978-88-905708-7-2

ISBN-A: 10.978.88905708/72

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