1.1 Consider the problem of tracking the vertical position,z(t), of a rocket whose mass changes as it consumes its fuel. If the rocket starts from rest atz(0)=0 with initial massm(0)=m0and obeys
d dt
m(t)d z
dt
= −mg+τm, dm
dt = −m,
solve the ODEs to determinez(t)and determine the condition onτ that is necessary for lift-off.
1.2 Use basic solution methods for first order ODEs to solve the elementary reactions (1.8–1.13) forA(t),B(t),C(t)starting from initial conditions A0,B0,C0
respectively.
1.3 Write the four rate equations for chemicalsC,E,P,S governed by the reac- tions4
S+E−k−1
k2
C C −→k3 P+E.
1.4 Write the rate equations forX(t),Y(t),Z(t)describing the reaction system (source)−→A X Y −−B
C X Z −D−
E Y 3Y −→G Z Z −→H (waste).
1.5 Consider the dynamics ofx(t)satisfying the first order ODE, d x
dt =x−x3−k x(0)=x0, for different values of the parameterk(see Fig.1.2).
4We delay solving more complicated systems of reactions to Chap.10. Here we only want to set up the rate equations.
20 1 Rate Equations (a) Fork=0, obtain the exact solutions starting fromx0=1/2 andx0= −2. Show that these solutions match the linearised results from (1.26) at the equilibrium pointsx∗ = {−1,0,1}for the long-time behaviour (t → ∞), and the “pre- history” of the solution (t → −∞). Show that for all initial conditions with x0=0, one of two final states are approached ast→ ∞.
(b) Determine the two values ofkfor which the ODE has only two distinct equilib- rium points. Sometimes calledcritical valuesorbifurcation points, such parame- ter values determine special cases, where different analysis is needed to describe the system; in this case, this will involve second-order degenerate equilibrium points. Determine the coefficientb occurring in the equation for the local be- haviour of solutions near the degenerate equilibrium points,du/dt=bu2with u(t)=x(t)−x∗.
Show that for this problem, these bifurcation values separate ranges ofkwhere there areglobal attractors(unique final states approached by all initial conditions fort → ∞) from cases where two stable equilibrium states co-exist (calledbi- stability).
1.6 Consider the equations for local behaviour at second- and third-order degenerate equilibrium points, (1.27) and (1.28).
(a) Use separation of variables to solve the ODE analytically and subsequently describe the stability of the equilibrium point atu =0 for both choices of the sign ofb.
(b) Using only the sign of the velocity on the phase line consider solutions starting from initial conditions withu0 ≷ 0 to obtain the stability of the equilibrium point without the need for the ODE solutions.
(c) Repeat (a, b) for the dependence of (1.28) onc.
1.7 Consider the second order ODE forx(t), d2x
dt2 =x−12x2.
(a) Lety=d x/dtand write the ODE as a phase plane system. Determine the linear stability properties of the two equilibrium points.
(b) Show that the solutions satisfy the equation
1
2(x)2−12x2+16x3=H,
whereHis a constant of integration, sometimes called theHamiltonian.
(c) For a range of values, 0<x<M, this problem has a continuous set of periodic solutions. Show that the maximum and minimum values ofx(t)of each periodic solution satisfy a polynomial equation involvingH. Define the amplitude of the oscillations asA=xmax−xmin. Show thatA=0 corresponds to an equilibrium point. Show that the largest amplitude solution hasxmin=0; determine its value forxmax(=M). What is the range of values for H?
(d) Uset =
dt= d x
d x/dt to show that the period of oscillation for these solutions is given by
P=2
xmax
xmin
d x
x2−13x3+2H .
(e) Show that the simplerpiecewise-linear model d2x
dt2 = f(x) with f(x)= x x<1 2−x x≥1
has the same equilibrium points and the same linear stability properties at the equilibria.
Solve the two linear problems d2xA
dt2 =xA xA(0)=xmin xA(0)=0 d2xB
dt2 =2−xB xB(P/2)=xmax xB(P/2)=0
and construct a periodic solution of the piecewise-linear model by enforcing smoothness,xA=xBatxA=xB =1.
Chapter 2
Transport Equations
Many systems exhibit evolution over time with properties of interest that vary throughout their spatial domains. Examples of such systems arise in population dynamics, which describes the distribution of individuals in some population and how they interact. “Individuals” could refer to molecules, electrons, particles, ani- mals, people, company stocks, or network messages.
One way to study the overall population is to attempt to track each individual, such approaches are sometimes calledindividual-based models. The tracking process is usually very labour intensive and involves collecting a lot of data on the actions of all individuals. If this level of detail is not crucial and a more ‘large-scale’ view is of interest, thencontinuum theorymay be a better option.
Continuum theories yield evolution equations with respect to properties that are averaged over small intervals of time and small regions of space. In such cases, we implicitly assume acontinuum hypothesiswhich states that appropriately averaged behaviours of individuals can be generally predicted from trends in the local popula- tion. We consequently formulatecontinuum modelsas partial differential equations (PDE) governing the evolution of density functions (f(x,t)) describing properties of the population at a given position and time. Integrating the density over the entire domain can be used to capture the time-dependence of the property on the whole population,F(t)=
f(x,t)dx.
Some specific applications of continuum models include:
• Fluid dynamics: the flow of liquids and gases (density of molecules in space)
• Solid mechanics: the deformation of solids (density of molecules in space)
• Electromagnetics: the flow of electric currents in materials (density of electric charges in space)
• Scattering theory: dynamics due to collisions in high energy particle physics (den- sity of particles having different velocities)
© Springer International Publishing Switzerland 2015
T. Witelski and M. Bowen,Methods of Mathematical Modelling,
Springer Undergraduate Mathematics Series, DOI 10.1007/978-3-319-23042-9_2
23
• Age-structured population dynamics: birth, death, and aging of a population (distribution of individuals having different ages) [28, 74]
• Size-structured population dynamics: growth and decay of physical sizes of indi- viduals in a population (distribution of individuals having different sizes) [8]
Fluid dynamics and solid mechanics are often grouped together under the heading ofcontinuum mechanics. Other population models focus not on change in position, but on properties like stock price in financial models, popularity in social networks or genetic traits in biological systems.
In each of these contexts, PDEs can be used to describe the redistribution of the property of interest over time. Conveying shifts or “motion” in the density, whether with respect to spatial position or with the independent variablexrepresenting other properties (e.g. velocity, age, size), such PDEs are also broadly called transport equations. The common structure shared by these models is having a PDE for the rate of change of the density involving the spatial gradient of aflux function, which characterises transport of the property within the population. Transport equations are fundamental for describing problems in many fields extending from theoretical physics, chemical engineering, and mathematical biology to probability theory.
In this chapter we introduce the fundamental approach for formulating transport models (conservation laws and the Reynolds transport theorem). We then go on to describe the method of characteristics, a methodology for constructing exact solu- tions to basic transport models.