Lecture Notes on Mathematical Modelling in Applied Sciences

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Lecture Notes on Mathematical Modelling in Applied Sciences Authors Nicola Bellomo, Elena De Angelis, and Marcello Delitala c ° 2007 N. Bellomo, E. De Angelis, M. Delitala1 An Introduction to the Science of Mathematical Modelling 1.1 An Intuitive Introduction to Modelling The analysis of systems of applied sciences, e.g. technology, economy, biology etc, needs a constantly growing use of methods of mathematics and computer sciences. In fact, once a physical system has been observed and phenomenologically analyzed, it is often useful to use mathematical models suitable to describe its evolution in time and space. Indeed, the interpretationofsystemsandphenomena,whichoccasionallyshowcomplex features,isgenerallydevelopedonthebasisofmethodswhichorganizetheir interpretationtowardsimulation. Whensimulationsrelatedtothebehavior of the real system are available and reliable, it may be possible, in most cases, to reduce time devoted to observation and experiments. Bearing in mind the above reasoning, one can state that there exists a strong link between applied sciences and mathematics represented by mathematical models designed and applied, with the aid of computer sci- ences and devices, to the simulation of systems of real world. The term mathematicalsciences refersto variousaspectsof mathematics, speci¯cally analytic and computational methods, which both cooperate to the design of models and to the development of simulations. Before going on with speci¯c technical aspects, let us pose some prelim- inary questions: ² What is the aim ofmathematical modelling, and what is a mathemat- ical model? ² There exists a link between models and mathematical structures? ²Thereexistsacorrelationbetweenmodelsandmathematicalmethods? 12 Lectures Notes on Mathematical Modelling in Applied Sciences ² Which is the relation between models and computer sciences? Moreover: ²Canmathematicalmodelscontributetoadeeperunderstandingofphys- ical reality? ² Is it possible to reason about a science of mathematical modelling? ² Can education in mathematics take some advantage of the above mentioned science of mathematical modelling ? Additional questions may be posed. However, it is reasonable to stop here considering that one needs speci¯c tools and methods to answer pre- cisely to the above questions. A deeper understanding of the above topics willbeachievedgoingthroughthechaptersoftheseLectureNotesalsotak- ing advantage of the methods which will be developed later. Nevertheless an intuitive reasoning can be developed and some preliminary answers can be given: ² Mathematical models are designed to describe physical systems by equa- tions or, more in general, by logical and computational structures. ² The above issue indicates that mathematical modelling operates as a sciencebymeansofmethodsandmathematicalstructureswithwellde¯ned objectives. ² Intuitively, it can be stated that education in mathematics may take ad- vantage of the science of mathematical modelling. Indeed, linking mathe- maticalstructuresandmethodstotheinterpretationandsimulationofreal physical systems is already a strong motivation related to an inner feature of mathematics, otherwise too much abstract. Still, one has to understand if modelling provides a method for reasoning about mathematics. ² At this preliminary stage, it is di±cult to reason about the possibility that mathematical models may contribute to a deeper understanding of physical reality. At present, we simply trust that this idea will be clari¯ed all along the contents of these Lectures Notes. This chapter has to be regarded as an introduction to the science of mathematical modelling which will be developed through these Lecture Notes with reference to well de¯ned mathematical structures and with the help of several applications intended to clarify the above concepts. Specif- ically it deals with general introduction to mathematical modelling, and is organized into six more sections which follow this introduction: Section1.2dealswiththepresentationofsomesimpleexamplesofmathe- maticalmodelswhichactasapreliminaryreferenceforthevariousconcepts introduced in the following sections. Then, the de¯nition of mathematical model is given as an equation suitable to de¯ne the evolution in time and spaceofthevariablechargedtodescribe(ateachspeci¯cscale)thephysical state of the real system.An Introduction to the Science of Mathematical Modelling 3 Section 1.3 deals with a preliminary aspect of the modelling process, that is the identi¯cation of the representation scales, microscopic, macroscopic and statistical, needed to observe and represent a real system. The above concepts are related to a variety of examples of models at each one of the above scales. Section 1.4 deals with the dimensional analysis of mathematical models. It is shown how writing the model in terms of dimensionless variables is usefultowardscomputationalanalysisandallowstoextractsuitablescaling parameterswhichcanbeproperlyusedtowardsaqualitativeunderstanding of the properties of the model. Section 1.5 analyzes the various concepts proposed in the preceding sec- tions by means of models of vehicular tra±c °ow. Such a system can be described by di®erent models and scales, all of them are analyzed with reference to the above mentioned de¯nitions and scaling methods. Section 1.6 deals with a classi¯cation of models and mathematical prob- lems still referring to the various aspects of the modelling process dealt with in the preceding sections. Section 1.7 provides a description and critical analysis of the contents of this chapter with special attention to complexity problems. The analysis is also proposed in view of the contents of the next chapters. 1.2 Elementary Examples and De¯nitions Thissectiondealswiththedescriptionofthreesimpleexamplesofmath- ematicalmodelswhichwillbeatechnicalreferenceforthede¯nitionsgiven in the following sections. The models are derived by an intuitive approach, while well de¯ned modelling methods will be developed in the chapters which follow and applied to the design of relatively more sophisticated models. The ¯rst example describes linear oscillations of a mass constrained to move along a line, while the second one refers to modelling heat di®usion phenomena. The third example is a generalization of the second one to a nonlinear case.4 Lectures Notes on Mathematical Modelling in Applied Sciences Example 1.2.1 Linear Elastic Wire-Mass System Consider, with reference to Figure 1.2.1, a mechanical system consti- tuted by a mass m constrained to translate along an horizontal line, say the x-axis. The location of the mass is identi¯ed by the coordinate of its centerofmassP,whichisattachedtoanelasticwirestretchedwithendsin AandP. Theassumptionsde¯ningthemechanicalmodelarethefollowing: Figure 1.2.1 Elastic wiremass system ² The system behaves as a point mass with localization identi¯ed by the variable x. ² The action of the wire is a force directed toward the point A with module: T =kx. ² Friction forces are negligible with respect to the action of the wire. Application of Newton's principles of classical mechanics yields 2 d x m =¡kx: (1:2:1) 2 dt The mathematical model is an evolution equation for the following vec- tor variable: µ ¶ dx u= u =x; u = ¢ (1:2:2) 1 2 dt Using the above variables, the second order ordinary di®erential equa- tion (1.2.1) reads 8 du 1 =u ; 2 dt (1:2:3) du k 2 : =¡ u ; 1 dt mAn Introduction to the Science of Mathematical Modelling 5 which is a linear equation. Example 1.2.2 Linear Heat Di®usion Model Consider the one-dimensional linear heat di®usion model in a rod. The assumptions de¯ning the mechanical model are the following: ² The state of the system is described by the temperature u = u(t;x) along the axis of the rod identi¯ed by the variable x 2 0;1. Variations orthogonal to the axis of the rod are neglected as the walls of the rod are perfectly isolated. ² The heat °ow q per unit area is proportional to the temperature gradient: u , q =¡h (1:2:4) 0 x where h is the heat conduction coe±cient. 0 ² The material properties of the conductor are identi¯ed by the heat conduction coe±cient h and heat capacity c . 0 0 The mathematical model can be obtained equating the net heat °ux in avolumeelementtotherateofincreaseoftheheatcapacityinthevolume. + ¡ Let q and q be, respectively, the ingoing and outgoing heat °uxes for unit area, see Figure 1.2.2 The above balance writes u q + ¡ , c A dx=¡A(q ¡q )=¡A dx (1:2:5) 0 t x where A is the cross section of the rod. Figure 1.2.2 Di®usion in one space dimension6 Lectures Notes on Mathematical Modelling in Applied Sciences Using Eq. 1.2.4 yields: 2 u u h 0 , =k k = ¢ (1:2:6) 0 0 2 t x c 0 The above model can also be used to describe the steady temperature distribution, which is obtained equating to zero the right-hand side term 2 d u k =0; (1:2:7) 0 2 dx which can also be written as a system of two coupled equations in normal form 8 du , =v dx (1:2:8) dv : =0: dx Example 1.2.3 Nonlinear Heat Di®usion Model Nonlinearity may be related to the modelling of the heat °ux phe- nomenon. For instance, if the heat °ux coe±cient depends on the tem- perature, say h = h(u), the same balance equation generates the following model: · ¸ u u h(u) = k(u) ; k(u)= : (1:2:9) t x x c 0 The reader with a basic knowledge of elementary theory of di®erential equations will be soon aware that the above two simple models generate interesting mathematical problems. In fact, Model 1.2.1 needs initial con- ditions for t = t both for u = u (t) and u = u (t), while Models 1.2.2 0 1 1 2 2 and1.2.3needinitialconditionsatt=t andboundaryconditionsatx=0 0 and x=1 for u=u(t;x). The solution of the above mathematical problems ends up with sim- ulations which visualize the behavior of the real system according to the description of the mathematical model. After the above examples, a de¯nition ofmathematical model can be introduced. This concept needs the preliminary de¯nition of two elements:An Introduction to the Science of Mathematical Modelling 7 ² Independent variables, generally time and space; ² State variables which are thedependent variables, that take val- ues corresponding to the independent variables; Then the following concept can be introduced: ² Mathematical model, that is a set of equations which de¯ne the evolution of the state variable over the dependent variables. The general idea is to observe the phenomenology of a real system in ordertoextractitsmainfeaturesandtoprovideamodelsuitabletodescribe theevolutionintimeandspaceofitsrelevantaspects. Bearingthisinmind, the following de¯nitions are proposed: Independent variables The evolution of the real system is referred to the indepen- dent variables which, unless di®erently speci¯ed, are time t, de¯nedinaninterval(t2t ;T),whichreferstheobservation 0 period; and space x, related to the volume V, (x2V) which contains the system. State variable The state variable is the ¯nite dimensional vector variable n u=u(t;x) : t ;T£V IR ; (1:2:10) 0 where u = fu ;:::;u ;:::;u g is deemed as su±cient to de- 1 i n scribe the evolution of the physical state of the real system in terms of the independent variables. Mathematical model Amathematical model of a real physical system is anevo- lution equation suitable to de¯ne the evolution of the state variable u in charge to describe the physical state of the sys- tem itself. Inordertohandleproperlyamathematicalmodel, thenumberofequa- tions and the dimension of the state variable must be the same. In this case the model is de¯ned consistent: Consistency The mathematical model is said to beconsistent if the num- ber of unknown dependent variables is equal to the number of independent equations.8 Lectures Notes on Mathematical Modelling in Applied Sciences Thismeans that one has to verifywhether an equation belonging to the model can be obtained combining the remaining ones. If this is the case, that equation must be eliminated. The space variable may be referred to a suitable system of orthogonal axes, 0(x;y;z) with unit vectors i, j, k, so that a point P is identi¯ed by its coordinates P=P(x)=xi+yj+zk: (1:2:11) Therealphysicalsystemmaybeinteractingwiththeouterenvironment or may be isolated. In the ¯rst case the interactions has to be modelled. Closed and Open Systems Areal physical system isclosed if it does not interact with the outer environment, while it is open if it does. The above de¯nitions can be applied to real systems in all ¯elds of ap- plied sciences: engineering, natural sciences, economy, and so on. Actually, almostallsystemshaveacontinuousdistributioninspace. Therefore, their discretization,thatamountstothefactthatuisa¯nitedimensionalvector, can be regarded as an approximation of physical reality. In principle, one can always hope to develop a model which can repro- duce exactly physical reality. On the other hand, this idealistic program cannot be practically obtained considering that real systems are character- ized by an enormous number of physical variables. This reasoning applies to Example 1.2.1, where it is plain that translational dynamics in absence offrictionalforcesisonlyacrudeapproximationofreality. Theobservation of the real behavior of the system will de¯nitively bring to identify a gap betweentheobservedvaluesofu andu andthosepredictedbythemodel. 1 2 Uncertainty may be related also to the mathematical problem. Re- ferring again to the above example, it was shown that the statement of mathematical problems need u and u , i.e. the initial position and ve- 10 20 locity of P, respectively. Their measurements are a®ected by errors so that their knowledge may be uncertain. Insomecasesthisaspectcanbedealtwithbyusinginthemodeland/or in the mathematical problems randomness modelled by suitable stochastic variables. The solution of the problem will also be represented by random variables, and methods of probability theory will have to be used. As we have seen, mathematical models are stated in terms of evolution equations. Examples have been given for ordinary and partial di®erential equations. The above equations cannot be solved without complementing themwithsuitableinformationonthebehaviorofthesystemcorresponding to some values of the independent variables. In other words the solution referstothemathematicalproblemobtainedlinkingthemodeltotheabove mentioned conditions. Once a problem is stated suitable mathematicalAn Introduction to the Science of Mathematical Modelling 9 methods have to be developed to obtain solutions and simulations, which are the prediction provided by the model. The analysis of the above crucial problems, which is a fundamental step of applied mathematics, will be dealt with in the next chapters with reference to speci¯c classes of equations. 1.3 Modelling Scales and Representation As we have seen by the examples and de¯nitions proposed in Section 1.2, the design of a mathematical model consists in deriving an evolution equation for the dependent variable, which may be called state variable, which describes the physical state of the real system, that is the object of the modelling process. The selection of the state variable and the derivation of the evolution equation starts from the phenomenological and experimental observation of the real system. This means that the ¯rst stage of the whole modelling method is the selection of the observation scale. For instance one may look atthesystembydistinguishingallitsmicroscopiccomponents,oraveraging locally the dynamics of all microscopic components, or even looking at the systemasawhole byaveragingtheirdynamics inthe wholespaceoccupied by the system. For instance, if the system is a gas of particles inside a container, one may either model the dynamics of each single particle, or consider some macroscopic quantities, such as mass density, momentum and energy, ob- tainedaveraginglocally(inasmallvolumetobeproperlyde¯ned: possibly an in¯nitesimal volume) the behavior of the particles. Moreover, one may average the physical variables related to the microscopic state of the par- ticles and/or the local macroscopic variables over the whole domain of the container thus obtaining gross quantities which represent the system as a whole. Speci¯cally, let us concentrate the attention to the energy and let us assume that energy may be related to temperature. In the ¯rst case one hastostudythedynamicsoftheparticlesandthenobtainthetemperature by a suitable averaging locally or globally. On the other hand, in the other twocasestheaveragingisdevelopedbeforederivingamodel,thenthemodel shouldprovidetheevolutionofalreadyaveragedquantities. Itisplainthat the above di®erent way of observing the system generates di®erent models corresponding to di®erent choices of the state variable. Discussing the validity of one approach with respect to the other is de¯nitively a di±cult, howevercrucial,problemtodealwith. Theaboveapproacheswillbecalled, respectively, microscopic modelling andmacroscopic modelling.10 Lectures Notes on Mathematical Modelling in Applied Sciences As an alternative, one may consider the microscopic state of each mi- croscopic component and then model the evolution of the statistical distri- bution over each microscopic description. Then one deals with the kinetic type(mesoscopic)modellingwhichwillbeintroducedinthischapterand then properly dealt with later in Chapter 4. Modelling by methods of the mathematical kinetic theory requires a detailed analysis of microscopic modelsforthedynamicsoftheinteractingcomponentsofthesystem,while macroscopic quantities are obtained, as we shall see, by suitable moments weighted by the above distribution function. Thissectiondealswithapreliminaryderivationofmathematicalframe- work related to the scaling process which has been described above. This processwillendsupwithaclassi¯cationbothofstatevariablesandmathe- matical equations. Simple examples will be given for each class of observa- tion scales and models. The whole topic will be specialized in the following chapters with the aim of a deeper understanding on the aforementioned structures. Both observation and simulation of system of real world need the def- inition of suitable observation and modelling scales. Di®erent models and descriptions may correspond to di®erent scales. For instance, if the mo- tion of a °uid in a duct is observed at a microscopic scale, each particle is singularly observed. Consequently the motion can be described within the framework of Newtonian mechanics, namely by ordinary di®erential equations which relate the force applied to each particle to its mass times acceleration. Applied forces are generated by the external ¯eld and by interactions with the other particles. Ontheotherhand, thesamesystemcanbeobservedanddescribedata largerscaleconsideringsuitableaveragesofthemechanicalquantitieslinked to a large number of particles, the model refers to macroscopic quantities such as mass density and velocity of the °uid. A similar de¯nition can be givenforthemassvelocity, namelytheratiobetweenthemomentumofthe particles in the reference volume and their mass. Both quantities can be measured by suitable experimental devices operating at a scale of a greater order than the one of the single particle. This class of models is generally stated by partial di®erential equations. Actually, the de¯nition of small or large scale has a meaning which has to be related to the size of the object and of the volume containing them. For instance, a planet observed as a rigid homogeneous whole is a single object which is small with respect to the galaxy containing the planet, but large with respect to the particles constituting its matter. So that the galaxy can be regarded as a system of a large number of planets, or as a °uid where distances between planets are neglected with respect to the size of the galaxy. Bearing all above in mind, the following de¯nitions are given:An Introduction to the Science of Mathematical Modelling 11 Microscopic scale Arealsystemcanbeobserved,measured,andmodelledatthe microscopic scale if all single objects composing the system are individually considered, each as a whole. Macroscopic scale Arealsystemcanbeobserved,measured,andmodelledatthe macroscopic scale if suitable averaged quantities related to the physical state of the objects composing the system are considered. Mesoscopic scale A real system can be observed, measured, and modelled at themesoscopic (kinetic) scale if it is composed by a large number of interacting objects and the macroscopic observable quantities related to the system can be recovered from mo- ments weighted by the distribution function of the state of the system. Asalreadymentioned, microscopicmodelsaregenerallystatedinterms of ordinary di®erential equations, while macroscopic models are generally stated in terms of partial di®erential equations. This is the case of the ¯rst two examples proposed in the section which follows. The contents will generally be developed, unless otherwise speci¯ed, within the framework of deterministic causality principles. This means that once a cause is given, the e®ect is deterministically identi¯ed, however, even in the case of deter- ministicbehavior,themeasurementofquantitiesneededtoassessthemodel or the mathematical problem may be a®ected by errors and uncertainty. The above reasoning and de¯nitions can be referred to some simple examples of models, this also anticipating a few additional concepts which will be dealt in a relatively deeper way in the chapters which follow. Example 1.3.1 Elastic Wire-Mass System with Friction Following Example 1.2.1, let us consider a mechanical system consti- tuted by a mass m constrained to translate along a horizontal line, say the x-axis. The location of the mass is identi¯ed by the coordinate of its center of mass P, which is attached to an elastic wire stretched with ends in A andP. ThefollowingassumptionneedstobeaddedtothoseofModel1.2.1 de¯ning the mechanical model: ²Frictionforcesdependonthe p-thpowerofthevelocityandaredirect in opposition with it.12 Lectures Notes on Mathematical Modelling in Applied Sciences Application of Newton's model yields µ ¶ p 2 d x dx m =¡kx¡c : (1:3:1) 2 dt dt The mathematical model, according to the de¯nitions proposed in Sec- tion 1.2, is an evolution equation for the variable u de¯ned as follows: µ ¶ dx u= u =x; u = ¢ (1:3:2) 1 2 dt Using the above variables, the second order ordinary di®erential equa- tion (1.3.1) can be written as a system of two ¯rst order equations: 8 du 1 =u ; 2 dt (1:3:3) du k c 2 p : =¡ u ¡ u ¢ 1 2 dt m m The above example has shown a simple model that can be represented by an ordinary di®erential equation, Eq. (1.3.3), which is nonlinear for values of p di®erent from zero or one. Observing Eq. (1.3.3), one may state that the model is consistent, namelytherearetwoindependentequationscorrespondingtothetwocom- ponents of the state variable. The physical system is observed singularly, i.e. atamicroscopicscale,whileitcanbeobservedthatthemodelisstated in terms of ordinary di®erential equations. Linearity of the model is obtained if c = 0. On the other hand, if k is q not a constant, but depends on the elongation of the wire, say k = k x a 0 nonlinear model is obtained 8 du 1 =u ; 2 dt (1:3:4) du k 2 0 q+1 : =¡ u ¢ 1 dt m Independently of linearity properties, which will be properly discussed in the next Chapter 2, the system is isolated, namely it is a closed system. For open system one should add to the second equation the action of theAn Introduction to the Science of Mathematical Modelling 13 outerenvironmentovertheinnersystem. Asimpleexampleisthefollowing: 8 du 1 =u ; 2 dt (1:3:5) du k 1 2 0 q+1 : , =¡ u + F(t) 1 dt m m where F =F(t) models the above mentioned action. The above models, both linear and nonlinear, have been obtained link- ing a general background model valid for large variety of mechanical sys- tems, that is the fundamental principles of Newtonian mechanics, to a phenomenological model suitable to describe, by simple analytic expres- sions, the elastic behavior of the wire. Such models can be re¯ned for each particular system by relatively more precise empirical data obtained by experiments. The example which follows is developed at the macroscopic scale and it is related to the heat di®usion model we have seen in Section 1.2. Here, we consider a mathematical model suitable to describe the di®usion of a pollutant of a °uid in one space dimension. As we shall see, an evolution equation analogous to the one of Example 1.2.2 will be obtained. First the linear case is dealt with, then some gen- eralizations, i.e. non linear models and di®usion in more than one space dimensions, are described. Example 1.3.2 Linear Pollutant Di®usion Model Consider a duct ¯lled with a °uid at rest and a pollutant di®using in the duct in the direction x of the axis of the duct. The assumptions which de¯ne the mechanical model are the following: ² The physical quantity which de¯nes the state of the system is the concentration of pollutant: c=c(t;x) : t ;T£0;`IR ; (1:3:6) 0 + variationsofcalongcoordinatesorthogonaltothex-axisarenegligible. The mass per unit volume of the pollutant is indicated by ½ and is assumed to 0 be constant. ² There is no dispersion or immersion of pollutant at the walls. ² The °uid is steady, while the velocity of di®usion of the pollutant is described by a phenomenological model which states that the di®usion ve- locityisdirectlyproportionaltothegradientofcandinverselyproportional to c.14 Lectures Notes on Mathematical Modelling in Applied Sciences The evolution model, i.e. an evolution equation for c, can be obtained exploiting mass conservation equation. In order to derive such equation let consider, with reference to Figure 1.2.2, the °ux q =q(t;x) along the duct + ¡ and let q and q be the inlet and outlet °uxes, respectively. Under suit- able regularity conditions, which are certainly consistent with the physical system we are dealing with, the relation between the above °uxes is given by: q + ¡ q =q + dx: (1:3:7) x A balance equation can be written equating the net °ux rate to the increase of mass in the volume element Adx, where A is the section of the duct. The following equation is obtained: c (cv) ½ A dx+A dx=0; (1:3:8) 0 t x wherev isthedi®usionvelocitywhich,accordingtotheaboveassumptions, can be written as follows: h c 0 v =¡ ; (1:3:9) c x and h is the di®usion coe±cient. 0 Substituting the above equation into (1.3.8) yields 2 c c h 0 =k ; k = ; (1:3:10) 0 0 2 t x ½ 0 which is a linear model. Nonlinearity related to the above model may occur when the di®usion coe±cient depends on the concentration. This phenomenon generates the nonlinear model described in the following example. Example 1.3.3 Nonlinear Pollutant Di®usion Model Consider the same phenomenological model where, however, the dif- fusion velocity depends on the concentration according to the following phenomenological model: h(c) c v =¡h ; (1:3:11) 0 c xAn Introduction to the Science of Mathematical Modelling 15 where h(c) describes the behavior of the di®usion coe±cient with c. The model writes as follows µ ¶ c c h(c) = k(c) ; k(c)= : (1:3:12) t x x ½ 0 Phenomenological interpretations suggest: k(0)=k(c )=0; (1:3:13) M where c is the maximum admissible concentration. For instance M k(c)=c(c ¡c); (1:3:14) M so that the model reads: µ ¶ 2 2 c c c =c(c ¡c) + (c ¡2c) : (1:3:15) M M 2 t x x The above di®usion model can be written in several space dimensions. For instance, technical calculations generate the following linear model: Example 1.3.4 Linear Pollutant Di®usion in Space Let us consider the linear di®usion model related to Example 1.3.2, and assume that di®usion is isotropic in all space dimensions, and that the di®usion coe±cient does not depend on c. In this particular case, simple technical calculations yield: µ ¶ 2 2 2 c =k + + c=k ¢c: (1:3:16) 0 0 2 2 2 t x y z The steady model is obtained equating to zero the right-hand side of (1.3.16): µ ¶ 2 2 2 + + c=k ¢c=0: (1:3:17) 0 2 2 2 x y z16 Lectures Notes on Mathematical Modelling in Applied Sciences The above (simple) examples have given an idea of the microscopic and macroscopic modelling. A simple model based on the mesoscopic descrip- tion will be now given and critically analyzed. Speci¯cally, we consider an exampleofmodellingsocialbehaviorssuchthatthemicroscopicstateisde- ¯ned by the social state of a certain population, while the model describes the evolution of the probability density distribution over such a state. The above distribution is modi¯ed by binary interactions between individuals. Example 1.3.5 Population Dynamics with Stochastic Interaction Considerapopulationconstitutedbyinteractingindividuals, suchthat: ² The microscopic state of each individual is described by a real variable u20;1, that is a variable describing its main physical properties and/or social behaviors. As examples, in the case of a population of tumor cells this state may have the meaning of maturation or progression stage, for a population of immune cells we may consider the state u as their level of activation. ² The statistical description of the system is described by the number density functions N =N(t;u); (1:3:18) which is such that N(t;u)du denotes the number of cells per unit volume whose state is, at time t, in the interval u;u+du. Ifn isthenumberperunitvolumeofindividualsatt=0,thefollowing 0 normalization of N with respect to n can be applied: 0 1 f =f(t;u)= N(t;u): (1:3:19) n 0 If f (which will be called distribution function) is given, it is pos- sible to compute, under suitable integrability properties, the size of the population still referred to n : 0 Z 1 n(t)= f(t;u)du: (1:3:20) 0 The evolution model refers to f(t;u) and is determined by the interac- tionsbetweenpairsofindividuals,whichmodifytheprobabilitydistribution over the state variable and/or the size of the population. The above ideas can be stated in the following framework: ² Interactions between pairs of individuals are homogeneous in space and instantaneous, i.e. without space structure and delay time. They mayAn Introduction to the Science of Mathematical Modelling 17 changethestateoftheindividualsaswellasthepopulationsizebyshifting individualsintoanotherstateorbydestroyingorcreatingindividuals. Only binary encounters are signi¯cant for the evolution of the system. ²Therateofinteractionsbetweenindividualsofthepopulationismod- elled by the encounter rate which may depend on the state of the inter- acting individuals ´ =´(v;w); (1:3:21) which describes the rate of interaction between pairs of individuals. It is the number of encounters per unit time of individuals with state v with individuals with state w. ² The interaction-transition function A=A(v;w;u); (1:3:22) gives the probability density distribution of the transition, due to binary encounters,oftheindividualswhichhavestatevwiththeindividualshaving state w that, after the interaction, manufacture individuals with state u. The product between ´ and A is the transition rate T(v;w;u)=´(v;w)A(v;w;u): (1:3:23) ² The evolution equations for the density f can be derived by balance equation which equates the time derivative of f to the di®erence between thegain and theloss terms. The gain term models the rate of increase of the distribution function due to individuals which fall into the state u due to uncorrelated pair interactions. The loss term models the rate of loss in the distribution function of u-individuals due to transition to another state or due to death. Combining the above ideas yields the following model Z Z 1 1 f(t;u)= ´(v;w)A(v;w;u)f(t;v)f(t;w)dvdw t 0 0 Z 1 ¡f(t;u) ´(u;v)f(t;v)dv: (1:3:24) 0 The above example, as simple as it may appear, gives a preliminary idea of the way a kinetic type modelling can be derived. This topic will be properly revisited in Chapter 4. At present we limit our analysis to18 Lectures Notes on Mathematical Modelling in Applied Sciences observing that a crucial role is de¯ned by the modelling of interactions at the microscopic scale which allows the application of suitable balance equation to obtain the evolution of the probability distribution. 1.4 Dimensional Analysis for Mathematical Models Examples 1.2.1 and 1.3.2 can be properly rewritten using dimensionless variables. This procedure should be generally, may be always, applied. In fact, it is always useful, and in some cases necessary, to write models with all independent and dependent variables written in a dimensionless form by referring them to suitable reference variables. These should be properly chosen in a way that the new variables take value in the domains 0;1 or ¡1;1. The above reference variables can be selected by geometrical and/or physical arguments related to the particular system which is modelled. Technically, let w be acertain variable(either independentor dependent), v and suppose that the smallest and largest value of w , respectively w v m and w , are identi¯ed by geometrical or physical measurements; then the M dimensionless variable is obtained as follows: w ¡w v m , w = w20;1: (1:4:1) w ¡w M m For instance, if w represents the temperature in a solid material, then v one can assume w = 0, and w = w , where w is the melting tempera- m M c c ture for the solid. In principle, the description of the model should de¯ne the evolution within the domain 0;1. When this does not occur, then the model should be critically analyzed. If w corresponds to one of the independent space variables, say it cor- v respond to x , y , and z for a system with ¯nite dimension, then the said v v v variable can be referred to the smallest and to the largest values of each variable, respectively, x , y , z , and x , y , and z . m m m M M M In some cases, it may be useful referring all variables with respect to only one space variable, generally the largest one. For instance, suppose that x = y = z = 0, and that y = ax , and z = bx , with m m m M M M M a;b1, one has x y z v v v , , , x= y = z = (1:4:2) x x x M M M with x20;1; y20;a; z20;b.An Introduction to the Science of Mathematical Modelling 19 Somehow more delicate is the choice of the reference time. Technically, if the initial time is t and t is the real time, one may use the following: 0 v t ¡t v 0 , t= t¸0; (1:4:3) T ¡t c 0 where generally one may have t = 0. The choice of T has to be related 0 c to the actual analytic structure of the model trying to bring to the same order the cause and the e®ect as both of them are identi¯ed in the model. For instance, looking at models in Example 1.2.1, the cause is identi¯ed by the right-hand side term, while the e®ect is the left-hand term. The modelshouldbereferredtotheobservationtimeduringwhichthesystem should be observed. This time should be compared with T . c Bearing all above in mind let us apply the above concepts to the state- ment in terms of dimensionless variables of the two models described in Examples 1.2.1 and 1.3.2. Example 1.4.1 Dimensionless Linear Elastic Wire-Mass System LetusconsiderthemodeldescribedinExample1.2.1,withtheaddition of the following assumption: ² A constant force F is directed along the x-axis. Therefore, the model can written as follows: 2 d x v m =F ¡kx : (1:4:4) v 2 dt v It is natural assuming ` = F=k, t = t =T , and x = x =`. Then the v c v model writes 2 m d x =1¡x: (1:4:5) 2 2 dt kT c Assuming: m m 2 =1 ) T = c 2 k kT c yields 2 d x =1¡x; (1:4:6) 2 dt which is a second order model. The evolution can be analyzed in terms of unit of T . c

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