Lecture Notes on Mathematical Modelling in Applied Sciences

1 Lecture Noteson Mathematical Modellingin Applied SciencesAuthorsNicola Bellomo, Elena De Angelis, and Marcello Delitalac 2007 N. Bellomo, E. De Angelis, M. DelitalaNicola BellomoDepartment of MathematicsPolitecnico TorinoCorso Duca Degli Abruzzi 2410129 Torino, De AngelisDepartment of MathematicsPolitecnico TorinoCorso Duca Degli Abruzzi 2410129 Torino, DelitalaDepartment of MathematicsPolitecnico TorinoCorso Duca Degli Abruzzi 2410129 Torino, .. vChapter 1. An Introduction to the Science of Mathemati-cal Modeling .. An Intuitive Introduction to Modeling .. Elementary Examples and Definitions .. Modelling Scales and Representation.

2 Dimensional Analysis for Mathematical Models Traffic Flow Modelling .. Classification of Models and Problems .. Critical Analysis .. 29 Chapter 2. Microscopic Scale Models and Ordinary Diffe-rential Equations.. Introduction .. On the Derivation of Mathematical Models .. Classification of Models and Mathematical Pro-blems .. Solution Schemes and Time Discretization .. Stability methods .. Regular and Singular Perturbation methods .. Bifurcation and Chaotic Motions .. 90iiiivLectures Notes on Mathematical Modelling in Applied Critical Analysis .. 94 Chapter 3. Macroscopic Scale Models and Partial Differen-tial Equations.

3 Introduction .. Modelling methods and Applications .. Classification of Models and Equations .. Mathematical Formulation of Problems .. An Introduction to Analytic methods for Linear Prob-lems .. Discretization of Nonlinear Mathematical Models Critical Analysis .. 133 Chapter 4. Mathematical Modelling by methods of Kine-tic Theory .. Introduction .. The Boltzmann Equation .. Mean Field Models .. Mathematical Problems .. Active Particles .. Evolution Equations for Active Particles .. Discretization Schemes .. Critical Analysis .. 156 Chapter 5. Bibliography.. 159 PrefaceThe Lecture Notes collected in this book refer to a university course deli-vered at the Politecnico of Torino to students attending the Lectures of themaster Graduation in Mathematical Lectures Notes correspond to the first part of the course devotedto Modelling issues to show how the application of models to describe realworld phenomena generates Mathematical problems to be solved by ap-propriate Mathematical methods .

4 The models dealt with in these LectureNotes are quite simple, proposed with tutorial aims, while relatively moresophisticated models are dealt with in the second part of the contents are developed through four chapters. The first one pro-poses an introduction to the science of Mathematical Modelling and focuson the three representation scales of physical reality: microscopic, macro-scopic and statistical over the microscopic states. Then, the three chapterswhich follow deal with the derivation and applications of models related toeach of the afore-mentioned it is shown, already in Chapter 1, different Mathematical structurescorrespond to each scale.

5 Specifically models at the microscopic scale aregenerally stated in terms of ordinary differential equations, while models atthe macroscopic scale are stated in terms of partial differential of the Mathematical kinetic theory, dealt with in Chapter 4, arestated in terms of integro-differential above different structures generate a variety of analytic and com-putational problems. The contents are devoted to understand how compu-tational methods can be developed starting from an appropriate discretiza-tion of the dependent Lecture Notes look at application focussing on Modelling andvviPrefacecomputational issues, while the pertinent literature on analytic methods isbrought to the attention of the interested reader for additional the above introduction to the contents and aims of the LectureNotes, a few remarks are stated to make a little more precise a few issuesthat have guided their redaction.

6 All real systems can be observed and represented at different scales bymathematical equations. The selection of a scale with respect to othersbelong, on one side, to the strategy of the scientists in charge of derivingmathematical models, and on the other hand to the specific application ofthe model. Systems of the real world are generally nonlinear. Linearity has to beregarded either as a very special case, or as an approximation of physicalreality. Then methods of nonlinear analysis need to be developed to dealwith the application of models. Computational methods are necessary tosolve Mathematical problems generated by the application of models to theanalysis and interpretation of systems of real world.

7 Computational methods can be developed only after a deep analysis of thequalitative properties of a model and of the related Mathematical methods may correspond to different models. Modelling is a science which needs creative ability linked to a deep know-ledge of the whole variety of methods offered by Applied mathematics . In-deed, the design of a model has to be precisely related to the methods to beused to deal with the Mathematical problems generated by the applicationof the Lectures Notes attempt to provide an introduction to the aboveissues and will exploit the use of electronic diffusion to update periodicallythe contents also on the basis of interactions with students, taking advan-tage of suggestions generally useful from those who are involved pursuingthe objective of a master graduation in mathematics for engineering Bellomo, Elena De Angelis.

8 Marcello Delitala1An Introduction to the Scienceof Mathematical An Intuitive Introduction to ModellingThe analysis of systems of Applied Sciences , technology, economy,biology etc, needs a constantly growing use of methods of mathematicsand computer Sciences . In fact, once a physical system has been observedand phenomenologically analyzed, it is often useful to use mathematicalmodels suitable to describe its evolution in time and space. Indeed, theinterpretation of systems and phenomena, which occasionally show complexfeatures, is generally developed on the basis of methods which organize theirinterpretation toward simulation. When simulations related to the behaviorof the real system are available and reliable, it may be possible, in mostcases, to reduce time devoted to observation and in mind the above reasoning, one can state that there existsa strong link between Applied Sciences and mathematics represented bymathematical models designed and Applied , with the aid of computer sci-ences and devices, to the simulation of systems of real world.

9 The termmathematical Sciences refers to various aspects of mathematics , specificallyanalytic and computational methods , which both cooperate to the designof models and to the development of going on with specific technical aspects, let us pose some prelim-inary questions: What is the aim ofmathematical Modelling , and what is a mathemat-ical model? There exists a link betweenmodelsandmathematical structures? There exists a correlation betweenmodelsandmathematical methods ?12 Lectures Notes on Mathematical Modelling in Applied Sciences Which is the relation betweenmodelsandcomputer Sciences ?Moreover: Can mathematicalmodelscontribute to a deeper understanding ofphys-ical reality?

10 Is it possible to reason about ascience of Mathematical Modelling ? Caneducation in mathematicstake some advantage of the abovementioned science of Mathematical Modelling ?Additional questions may be posed. However, it is reasonable to stophere considering that one needs specific tools and methods to answer pre-cisely to the above questions. A deeper understanding of the above topicswill be achieved going through the chapters of these Lecture Notes also tak-ing advantage of the methods which will be developed later. Neverthelessan intuitive reasoning can be developed and some preliminary answers canbe given: Mathematical models are designed to describe physical systems by equa-tions or, more in general, by logical and computational structures.