The book is based on the lectures, delivered by the author in recent years at the Faculty of Mathematics and Informatics of the `St. Kliment Ohridski'' University of Sofia.

Some of the basic principles of mathematical modelling are introduced in Chapter 1. The dimensional analysis is also included here together with its central result - the Buckingham P-theorem. Several examples of its application are presented as well.

Chapter 2 is devoted to the elementary modelling that stems from `word problems.' The classical population dynamics is of main concern here. The models of Malthus, Ferhulst, Lotka-Volterra are included, as well as D. Bernoulli's model of immunization and Newton's law of cooling.

The final Chapters 3 to 5 represent an elementary introduction to the classical models of the 19th century mathematical physics: Fourier's heat conduction theory, elasticity and hydrodynamics.

First, in Chapter 3, the fundamentals of statics of rigid solids are presented, following Poinsot. The equation of the chain line is derived at the end with the aim to introduce and illustrate Bernoulli's idea of internal tension.

Chapter 4 deals with the simplest models of deformable media: the Hooke body and the Newton-Navier fluid. To avoid the tensorial formalism, one-dimensional cases are only treated. The longitudinal vibrations of an elastic rod are analyzed as an example and D'Alambert's general solution of the simplest wave equations is derived. The hereditary effects and basic viscoelastic models of Maxwell, Voigt and Kelvin are also discussed together with the more general superposition principle of Boltzmann.

In Chapter 5 the classical theory of thermal conduction in solids is presented. To this end, the nabla-formalism is first introduced. The heat equation is then derived. Fourier's idea of separation of variables is illustrated on the simplest initially-boundary value problem for a finite rod. Finally, a more complicated problem is considered at some length. It concerns the modelling of heterogeneous solids and composite materials using, in particular, the idea of homogenization. The definition of the effective properties of a homogenized solid is given in the context of the heat conduction through heterogeneous medium. The classical Maxwell formula for the effective conductivity of a two-phase mixture is derived together with the simplest approximation due to Voigt and Reuss.

The main point that the book tries to convey to the reader is simple: mathematics is not only a beautiful logical construction, but a powerful tool for solving important purely applied problems as well. The students in mathematics, unfortunately, often miss this very important point soon after their first years of studies.

1.  Subject and basic ideas of mathematical modelling   13
2.  Dimensional analysis. Buckingham P-theorem. Examples (including fractal dimensions)   22

3.  The Malthus law and its interpretations    40  
4.  Cooling of solids. Spreading of infectious diseases (Bernoulli model). Kinetics of chemical reactions   48  
5.  Limited populations and the Ferhulst model   56  
6.  Population among predators   63  
7.  Two competing species   71
8.  The Lotka-Volterra (predator-prey) model   75

 9.  The Poinsot definitions and axioms; constraints    86
10.  The simplest systems of forces, applied to a rigid body    93
11.  Theory of force couples   102
12.  Reduction of an arbitrary force system to a force and a couple.  Equilibrium equations. Examples   111
13.  Equilibrium of chain   124

14.  The model of elastic solid. The Hooke law   134
15.  Some applications of the Hooke law to simplest rod systems  140
16.  Longitudinal vibrations of a rod - wave equation (1D) and D'Alambert's solution   145
17.  The model of viscous fluid. The Navier-Stokes law   149
18.  Viscoelastic solids. Maxwell, Voigt and Kelvin's models   153
19.  Hereditary effects in solids. Volterra integral operators. The Boltzmann principle of superposition   162

20.  Elementary vector analysis. Nabla-calculus  172
21.  Cauchy tetrahedron and heat conduction vector. Heat equation. Separation of variables and Fourier series  186
22.  Modelling of composite and heterogeneous media. The idea of homogenization   199
23.  The Maxwell formula for the heat conductivity of a mixture   218