Part 1–A: lecture Part 1–B: examples

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Part 1A: lecturePart 1B: examplesLecture 1: Methodology of mathematical modellingC NSE

A mathematicalmodel is a description of aprocesssystemphenomena usingmathematicalconcepts:functionsalgebraic equations differential equationsdifference equationsintegral equations

Part 1A: lecture

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Methodology of mathematical modellingProf. J.Engelbrecht2Methodology:C NSE

Methodology of mathematical modellingProf. J.Engelbrechtformulatemathematicalstatephysicalmodelexperimentssolvemathematicalinterpreteand validateobservationsimprovementsreal world:real problem

3Where did modelling startancient: astronomyagricultureengineering. . . . .

a need:to countto measureto sortto arrangeto eliminate. . . . .C NSE

Methodology of mathematical modellingProf. J.Engelbrecht4Some rules:LEONARDO DA VINCIMotion is an accident born from the inequality of weight or force. Force is the cause of motion, motion is the cause of force.Observe the phenomenon and list quantities having numerical magnitude that seems to influence it.

Set up linear relations among pairs of these quantities as are not obviously contradicted by experience.

Propose these rules of three trial by experiment.

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Methodology of mathematical modellingProf. J.Engelbrecht5

ISAAC NEWTON PRINCIPIA, 1687

IEvery body continues in its state of rest, or of uniform motion straight ahead, unless it be compelled to change that state by forces impressed upon it.

IIThe change of motion is proportional to the motive force impressed, and it takes place along the right line in which that force is impressed.

IIITo an action there is always a contrary and equal reaction; or, the mutual actions of two bodies upon each other are always equal and directed to contrary parts

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Methodology of mathematical modellingProf. J.Engelbrecht6Physical systemsBiological systemsSocial systemsC NSE

Methodology of mathematical modellingProf. J.Engelbrecht7

Physical systems

Balance lawsmodel Constitutive equations

Main stages:

Classical dynamics Newton; Hamilton . . .Thermodynamics Carnot, Boltzmann . . .Dissipative structures Prigogine, . . .

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Methodology of mathematical modellingProf. J.Engelbrecht8kinetic energy motionpotential energy Helmholz free energy

balance: amount at given time (t) = amount in the past (t - t) + amount which is created (at t) - amount which is destroyed (at t) + amount given by influx (at t)

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Methodology of mathematical modellingProf. J.Engelbrecht9C NSE

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Biological systems

specific features: energy exchange with the environment many chemical reactions, transfer mechanisms non-equilibrium systemsC NSE

Methodology of mathematical modellingProf. J.Engelbrecht11BIOLOGICAL SYSTEMSinformation densespatially extendedorganized in interacting hierarchiesBIOLOGICAL STRUCTURESintracellularcellulartissueorganMATHEMATICALLYcoupling of different types of equationscomputational difficultiesPHYSICALLYdissipative characteractivity/excitabilityspatio-temporal patterningcoupling

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Methodology of mathematical modellingProf. J.Engelbrecht12Social systemsPhenomenological laws are absentTime-seriesThreat : wishful thinking causality? (post hoc, propter hoc)

Nevertheless:economical modelspolitical modelssocial coexistence models. . . .

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Methodology of mathematical modellingProf. J.Engelbrecht13RequirementsRequirements to models:flexibilityparsimony (Occams razor)equipresenceinvariancecausalityImportant notions:static versus dynamiccontinuous versus discretelinear versus nonlinear

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Methodology of mathematical modellingProf. J.Engelbrecht14Plethora of problems:climate changeelementary particlesneuronsearthquakes turbulencefractureuniverse expansion. . . .and questions:cause and effect?symmetry?predictability?. . . .

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Methodology of mathematical modellingProf. J.Engelbrecht15Some useful thoughts:One of the principal objects of theoretical research in any department of knowledge is to find the point of view from which the subject appears in the greatest simplicity.J.W.Gibbs, 1881

Everything should be made as simple as possible, but not simpler.A.Einstein

The research worker, in his effort to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty. It often happens that the requirements of simplicity and beauty are the same, but where they clash the latter must take precedence.P.Dirac 1939C NSE

Methodology of mathematical modellingProf. J.Engelbrecht16Part 1B: examplesPhysical systems

Modelling of wave motion

homogeneous solids models well-knownmicrostructures solids new models developedC NSE

Methodology of mathematical modellingProf. J.Engelbrecht17Materials

FGM

Microstructured materialsSurface relief in Fe-0.44C-0.34Si-0.70Mn-01.10Cr-0.16Ni-0.18Mo wt\% sample. Reproduced from: H. Pantsar. Use of DIC Imaging in Examining Phase Transformations in Diode Laser Transformation Hardening of Steels. 23rd Int. Congress on Applications of Lasers and Electro-optics, San Francisco, CA, USA, 2004Reproduced from: B. Ilschner. Processing-microstructure-property relationships in graded materials. J. Mech. Phys. Solids 44, (1996) 647-656.

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Methodology of mathematical modellingProf. J.Engelbrecht18Balance lawsC NSE

Methodology of mathematical modellingProf. J.Engelbrecht19Energy conservation the second lawS the entropy flux, S the entropy density per unit reference volume, absolute temperature, K extra entropy flux, T the first Piola Kirchhoff tensor, F deformation gradient.

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Methodology of mathematical modellingProf. J.Engelbrecht20Hierarchy of wavesSystem of two 2nd order equations one 4 th order equation

Simplified (slaving principle)

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Methodology of mathematical modellingProf. J.Engelbrecht21Biological systems

Contraction of the heart muscleHuxley modelC NSE

Methodology of mathematical modellingProf. J.Engelbrecht22Structural hierarchy of heart muscle

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Methodology of mathematical modellingProf. J.Engelbrecht23

Sliding filaments

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Prof. J.Engelbrecht24Methodology of mathematical modelling

Cross-bridge

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Methodology of mathematical modellingProf. J.Engelbrecht25Kinetics

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Active stress

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Methodology of mathematical modellingProf. J.Engelbrecht27Model (1)

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Methodology of mathematical modellingProf. J.Engelbrecht28Model (2)C NSE

Methodology of mathematical modellingProf. J.Engelbrecht29Social systems

EconomyElections

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Methodology of mathematical modellingProf. J.Engelbrecht30The Black-Scholes model (1)Assumptionsthese is no way to make a riskless profitit is possible to borrow and lend cash at a constant risk-free interest rateit is possible to buy and sell any amountthe above transactions do not incure any feesthe stock price follows a geometric Brownian motionthe underlying security does not pay a dividend

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Methodology of mathematical modellingProf. J.Engelbrecht31The Black-Scholes model (2)C NSE

Methodology of mathematical modellingProf. J.Engelbrecht32Election of the doge in Venice, 1268Ca 80030approval91291140254541lotslotsDogeelectionelectionlotslotslotselectionelectionelectionC NSE

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