Introduction to the mathematical modeling ofmulti-scale phenomenaScales & scalingsMCB/MATH 303MCB/MATH 303 Multi-scale modeling - Scales & scalingsScales & scalingsThe first lecture was concerned with multi-scale aspects ofvarious physical or biological systems.We introduced the concept of scales and discussed examplesof scale-free systems.In particular, we looked at examples of fractals and ofself-similar systems found in nature.In this lecture, we consider how ideas of scales appear inmodels that involve differential equations.We start by introducing dimensional analysis. Then, we usethis technique to make models dimensionless. Finally, webriefly discuss self-similar solutions to partial differentialequations.MCB/MATH 303 Multi-scale modeling - Scales & scalingsThe nonlinear pendulumj→ı→θmθ→r→lg→Sketch of a point-mass pendulumThe equation of motion for thenonlinear pendulum is given bym ld2θdt2= −m g sin(θ)−c ldθdt,whereθ and t are variables.m, l, g and c areparameters.Most of these quantitiesare defined on the figure,except c, which measuresfriction.MCB/MATH 303 Multi-scale modeling - Scales & scalingsDimensions and dimensional analysisMost quantities that appear in models have a dimension.T represents time. Quantities which have the dimension of atime may be expressed in different units, such as seconds,minutes, days, years, etc.M represents mass. The corresponding units may be grams,ounces, kilograms, etc.L represents length. Here again, such quantities my havedifferent units, such as meters, feet, yards, kilometers, etc.To indicate that one considers the dimension of a quantity,one writes the symbol of that quantity between squarebrackets.It is thus important to distinguish between dimension and units.MCB/MATH 303 Multi-scale modeling - Scales & scalingsDimensions and dimensional analysis (continued)For example, in the nonlinear pendulum equationm ld2θdt2= −m g sin(θ) − c ldθdt,[m] = M, [t] = T , [l ] = L.[g] = LT−2and [c] = MT−1.θ is dimensionless, i.e. [θ] = 1.Note that all of the terms in the equation have the samedimension.The number of significant parameters in a model can often beconsiderably reduced by introducing dimensionless variables.MCB/MATH 303 Multi-scale modeling - Scales & scalingsRescaling the equation for the nonlinear pendulumDefine the following time scale t0=pl/g, and an associateddimensionless time variable τ , such that τ =tt0.Then, make this change of variable in the nonlinear pendulumequationm ld2θdt2= −m g sin(θ) − c ldθdt,The resulting equation readsd2θdτ2= −sin(θ) − αdθdτ, α =cmslg.One can check that the parameter α is dimensionless.Note that this equation has only one parameter, α.MCB/MATH 303 Multi-scale modeling - Scales & scalingsThe classic SIR modelPenned goats in a village within a regioninvestigated for a Rift Valley fever outbreakin Saudi Arabia. Picture # 8362, PublicHealth Image LibraryThe classic SIR model readsdSdt= −α SIN,dIdt= α SIN− β I ,dRdt= β I,where S, I , and R represent thenumbers of susceptible, infectious, andrecovered (or removed) individuals, in apopulation of size N. The parameter αmeasures the average number ofpositive contacts per susceptible perunit of time, and β measures the rate atwhich individuals recover.MCB/MATH 303 Multi-scale modeling - Scales & scalingsViral infectionsTransmission electron micrograph showinghepatitis virions of an unknown strain.Picture # 8153, Public Health ImageLibraryThe the dynamics of a viral infection,such as hepatitis B or C, may bedescribed by the following m odel (M.A.Nowak et al., Proc. Natl. Acad. Sci.USA 93, 4398-4402 (1996)).dXdt= λ − δ X − b V XdYdt= b V X − a YdVdt= k Y − κ VThe variable X represents the numberof uninfected cells, Y is the number ofinfected cells, and V is the viral load (ornumber of free virions in the body).MCB/MATH 303 Multi-scale modeling - Scales & scalingsSelf-similar functionsAssignment: do the homework problems on scalings anddimensional analysis. They are due in class on Thursday,September 11th.Recall that a self-similar object looks like itself over a range ofscales.Self-similar structures found in nature are typically invariantover a finite number of magnifications.On the other hand, fractals are self-similar over an infinitenumber of magnifications.We now consider another example of self-similarity, this timein functions.MCB/MATH 303 Multi-scale modeling - Scales & scalingsSelf-similar functions (continued)Consider a function of two variables, f (x, t), and assume thatf (x, t) =1tβvxtα, α, β ∈ Rwhere v is a given function of one variable.As t varies, one can think of f (x, t) as describing thetime-evolution of a function of x.The form of f (x, t) defined above is such that if onemultiplies x by tαand f by tβ, then the resulting functionremains unchanged as t varies.This is an example of self-similarity in a function, and isillustrated in the MATLAB GUI Self similar function.MCB/MATH 303 Multi-scale modeling - Scales & scalingsSelf-similar solution of the heat equationThe heat equation is given by∂f∂t= D∂2f∂x2, D > 0.We look for a solution in the form f (x, t) =1tβvxtα,where α, β ∈ R.Note that since z =x√D tis dimensionless, it is natural toexpect that v should be a function of z.Substituting the expression for f (x, t) into the heat equationleads to the condition 2α = 1.Assuming α = β = 1/2, the heat equation simplifies into−v − zdvdz= 2Dd2vdz2, where z =xtα=x√t.MCB/MATH 303 Multi-scale modeling - Scales & scalingsSelf-similar solution of the heat equation (continued)Solving this equation with the boundary conditions that v andits derivatives go to zero as z → ∞ leads tov(z) = κ exp−z24D,where κ is an arbitrary constant.In other words, we have found the following family ofself-similar solutions to the heat equation,f (x, t) =κ√texp−x24D t.This method clearly does not provide all solutions to a partialdifferential equation, but often gives useful insights into thebehavior of the system modeled by the PDE in question.MCB/MATH 303 Multi-scale modeling - Scales &