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Mathematical modelling with case studies : a differential equation approach using Maple

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Chapter Questions

Solving differential equations. For the differential equation $$ \frac{d y}{d t}=2 y $$ (a) Show that $y(t)=c e^{2 t}$, where $c$ is any real valued constant, satisfies the differential equation by substituting into both sides of the equation. Is there any value of $c$ that isn't a solution? (b) Find the one solution that corresponds to the initial condition $y(0)=5 .$ See Appendix A.1 if you need any revision of the meaning of differential equations.

Srikar Pasumarthy

For the following first-order differential equations, find the general solution, solving for the dependent variable. See Appendix $A .3$ if you need to revise how to solve first-order differential equations. (a) $\frac{d y}{d t}=-3 y$, (b) $\frac{d C}{d t}=3 C-1$, (c) $\frac{d y}{d t}=3 y t^{-1}$.

Raj Bala

The Earth's atmospheric pressure $p$ is often modelled by assuming that $d p / d h$ (the rate at which pressure $p$ changes with altitude $h$ above sea level) is proportional to $p$. Suppose that the pressure at sea level is 1,013 millibars and that the pressure at an altitude of $20 \mathrm{~km}$ is 50 millibars. Answer the following questions and then check your calculations with Maple or MATLAB. (a) Use an exponential decay model $$ \frac{d p}{d h}=-k p $$ to describe the system, and then by solving the equation find an expression for $p$ in terms of h. Determine $k$ and the constant of integration from the initial conditions. (b) What is the atmospheric pressure at an altitude of $50 \mathrm{~km}$ ? (c) At what altitude is the pressure equal to 900 millibars?

Brittany Knowlton

Continuous compounding for invested money can be described by a simple exponential model, $M^{\prime}(t)=0.01 r M(t)$, where $M(t)$ is the amount of money at time $t$ and $r$ is the percent interest compounding. Business managers commonly apply the Rule of 72, which says that the number of years it takes for a sum of money invested at $r \%$ interest to double, can be approximated by $72 / r .$ Show that this rule always overestimates the time required for the investment to double.

If an archaeologist uncovers a seashell which contains $60 \%$ of the ${ }^{14} \mathrm{C}$ of a living shell, how old do you estimate that shell, and thus that site, to be? (You may assume the half-life of ${ }^{14} C$ to be 5,568 years.

Kimberly Waterbury

(From Borelli and Coleman (1996).) Olduvai Gorge, in Kenya, cuts through volcanic flows, tuff (volcanic ash), and sedimentary deposits. It is the site of bones and artefacts of early hominids, considered by some to be precursors of man. In 1959, Mary and Louis Leakey uncovered a fossil hominid skull and primitive stone tools of obviously great age, older by far than any hominid remains found up to that time. Carbon-14 dating methods being inappropriate for a specimen of that age and nature, dating had to be based on the ages of the underlying and overlying volcanic strata. The method used was that of potassium-argon decay. The potassium-argon clock is an accumulation clock, in contrast to the ${ }^{14} \mathrm{C}$ dating method. The potassium-argon method depends on measuring the accumulation of 'daughter' argon atoms, which are decay products of radioactive potassium atoms. Specifically, potassium- $40\left({ }^{40} \mathrm{~K}\right)$ decays to argon $\left({ }^{40} \mathrm{Ar}\right)$ and to Calcium- 40 $\left({ }^{40} \mathrm{Ca}\right)$ by the branching cascade illustrated below in Figure 2.17. Potassium decays to calcium by emitting a $\beta$ particle (i.e. an electron). Some of the potassium atoms, however, decay to argon by capturing an extra-nuclear electron and emitting a $\gamma$ particle. The rate equations for this decay process may be written in terms of $K(t), A(t)$ and $C(t)$, the potassium, argon and calcium in the sample of rock: $$ \begin{aligned} &K^{\prime}=-\left(k_{1}+k_{2}\right) K \\ &A^{\prime}=k_{1} K \\ &C^{\prime}=k_{2} K \end{aligned} $$ where $$ k_{1}=5.76 \times 10^{-11} \text { year }^{-1}, \quad k_{2}=4.85 \times 10^{-10} \text { year }^{-1} \text { . } $$ (a) Solve the system to find $K(t), A(t)$ and $C(t)$ in terms of $k_{1}, k_{2}$, and $k_{3}=k_{1}+k_{2}$, using the initial conditions $K(0)=k_{0}, A(0)=C(0)=0 .$ (b) Show that $K(t)+A(t)+C(t)=k_{0}$ for all $t \geq 0 .$ Why would this be the case? (c) Show that $K(t) \rightarrow 0, A(t) \rightarrow k_{1} k_{0} / k_{3}$ and $C(t) \rightarrow k_{2} k_{0} / k_{3}$ as $t \rightarrow \infty$. (d) The age of the volcanic strata is the current value of the time variable $t$ because the potassiumargon clock started when the volcanic material was laid down. This age is estimated by measuring the ratio of argon to potassium in a sample. Show that this ratio is $$ \frac{A}{K}=\frac{k_{1}}{k_{3}}\left(e^{k_{3} t}-1\right) $$ (e) Now show that the age of the sample in years is $$ \frac{1}{k_{3}} \ell \mathrm{n}\left[\left(\frac{k_{3} A}{k_{1} K}\right)+1\right] $$ (f) When the actual measurements were made at the University of California at Berkeley, the age of the volcanic material (and thus the age of the bones) was estimated to be $1.75$ million years. What was the value of the measured ratio $A / K ?$

Khoobchandra Agrawal

(Adapted from Borelli and Coleman (1996).) In a biochemical laboratory radioactive phosphorus $\left({ }^{32} \mathrm{P}\right)$ was used as a tracer. (A tracer, through its radioactive emission, allows the course followed by a substance through a system to be tracked, which otherwise would not be visible.) ${ }^{32} \mathrm{P}$ decays exponentially with a half-life of $14.5$ days and its quantity is measured in curies (Ci). (Although it is not necessary for the calculations, one curie is the quantity of a radioactive isotope undergoing $3.7 \times 10^{-5}$ disintegrations per second.) After the experiment the biochemists needed to dispose of the contents, but they had to store them until the radioactivity had decreased to the acceptably safe level of $1 \times 10^{-5} \mathrm{Ci} .$ The experiment required 8 Ci of $^{32}$ P. Using a simple model of exponential decay, establish how long they had to store the contents of the experiment before it could be disposed of safely.

Aadit Sharma

Read the case study on Lake Burley Griffin. The average summer flow rate for the water into and out of the lake is $4 \times 10^{6} \mathrm{~m}^{3} /$ month. (a) Using this summer flow, how long will it take to reduce the pollution level to $5 \%$ of its current level? How long would it take for the lake with pollution concentration of $10^{7}$ parts $/ \mathrm{m}^{3}$, to drop below the safety threshold? (Assume in both cases that only fresh water enters the lake.) (b) Use Maple or MATLAB to replicate the results in the case study, for both constant and seasonal flow and constant and seasonal pollution concentrations entering the lake. Comment on the solutions.

Oswaldo Jiménez

Pollution with chemical activity. Consider the concentration, $C(t)$, of some pollutant chemical in a lake. Suppose that polluted water with concentration $c_{i}$ flows into the lake with a flow rate of $F$ and the well-stirred mixture leaves the lake at the same rate $F .$ In addition, suppose some chemical agent is present in the lake that breaks down the pollution at a rate $r \mathrm{~kg} /$ day per $\mathrm{kg}$ of pollutant. Assuming that the volume of mixture in the lake remains constant and the chemical agent is not used up, formulate (but do not solve) a mathematical model as a single differential equation for the pollution concentration $C(t)$.

Odera Egbuonu

North American lake system. Consider the American system of two lakes: Lake Erie feeding into Lake Ontario. What is of interest is how the pollution concentrations change in the lakes over time. You may assume the volume in each lake to remain constant and that Lake Erie is the only source of pollution for Lake Ontario. (a) Write a differential equation describing the concentration of pollution in each of the two lakes, using the variables $V$ for volume, $F$ for flow, $c(t)$ for concentration at time $t$ and subscripts 1 for Lake Erie and 2 for Lake Ontario. (b) Suppose that only unpolluted water flows into Lake Erie. How does this change the model proposed? (c) Solve the system of equations to get expressions for the pollution concentrations $c_{1}(t)$ and $c_{2}(t)$ (d) Set $T_{1}=V_{1} / F_{1}$ and $T_{2}=V_{2} / F_{2}$, and then $T_{1}=k T_{2}$ for some constant $k$ as $V$ and $F$ are constants in the model. Substitute this into the equation describing pollution levels in Lake Ontario to eliminate $T_{1}$. Then show that, with the initial conditions $c_{1,0}$ and $c_{2,0}$, the solution to the differential equation for Lake Ontario is $$ c_{2}(t)=\frac{k}{k-1} c_{1,0}\left(e^{-t /\left(k T_{2}\right)}-e^{-t / T_{2}}\right)+c_{2,0} e^{t / T_{2}} $$ (One way of finding the solution would be to use an integrating factor. See Appendix A.4.) (e) Compare the effects of $c_{1}(0)$ and $c_{2}(0)$ on the solution $c_{2}(t)$ over time.

Amy Jiang

(Adapted from Fulford et al. (1997).) A public bar opens at $6 \mathrm{p.m}$. and is rapidly filled with clients of whom the majority are smokers. The bar is equipped with ventilators that exchange the smoke-air mixture with fresh air. Cigarette smoke contains $4 \%$ carbon monoxide and a prolonged exposure to a concentration of more than $0.012 \%$ can be fatal. The bar has a floor area of $20 \mathrm{~m}$ by $15 \mathrm{~m}$, and a height of $4 \mathrm{~m} .$ It is estimated that smoke enters the room at a constant rate of $0.006 \mathrm{~m}^{3} / \mathrm{min}$, and that the ventilators remove the mixture of smoke and air at 10 times the rate at which smoke is produced. The problem is to establish a good time to leave the bar, that is, sometime before the concentration of carbon monoxide reaches the lethal limit. (a) Starting from a word equation or a compartmental diagram, formulate the differential equation for the changing concentration of carbon monoxide in the bar over time. (b) By solving the equation above, establish at what time the lethal limit will be reached.

P Krishnamurthy

Detecting art forgeries. Based on methods used in the case study describing the detection of art forgeries (Section 2.3), comment on whether each of the paintings below is a possible forgery, based on the time it was painted: (a) 'Washing of Feet', where the disintegration rate for ${ }^{210} \mathrm{Po}$ is $8.2$ per minute per gram of white lead, and for ${ }^{226} \mathrm{Ra}$ is $0.26$ per minute per gram of white lead. (b) 'Laughing Girl', where the disintegration rate for ${ }^{210} \mathrm{Po}$ is $5.2$ per minute per gram of white lead and for ${ }^{226} \mathrm{Ra}$ is 4 per minute per gram of white lead.

Sandra Lundell

In Section $2.7$, we developed the model $$ \begin{aligned} &\frac{d x}{d t}=-k_{1} x, \quad x(0)=x_{0} \\ &\frac{d y}{d t}=k_{1} x-k_{2} y, \quad y(0)=0 \end{aligned}$$ where $k_{1}, k_{2}>0$ determine the rate at which a drug, antihistamine or decongestant moves between two compartments in the body, the GI-tract and the bloodstream, when a patient takes a single pill. Here $x(t)$ is the level of the drug in the GI-tract and $y(t)$ is the level in the bloodstream at time $t$. (a) Find solution expressions for $x(t)$ and $y(t)$ that satisfy this pair of differential equations, when $k_{1} \neq k_{2}$. Show that this solution is equivalent to that provided in the text. (b) The solution above is invalid at $k_{1}=k_{2} .$ Why is this, and what is the solution in this case? (c) For old and sick people, the clearance coefficient (that is, the rate at which the drug is removed from the bloodstream) is often much lower than that for young, healthy individuals. How does an increase or decrease in $k_{2}$ change the results of the model? Using Maple or MATLAB to generate the time-dependent plots, check your results.

Kush Khamesra

In Section $2.7$, we also developed a model to describe the levels of antihistamine and decongestant in a patient taking a course of cold pills: $$ \begin{aligned} &\frac{d x}{d t}=I-k_{1} x, \quad x(0)=0, \\ &\frac{d y}{d t}=k_{1} x-k_{2} y, \quad y(0)=0 . \end{aligned} $$ Here $k_{1}$ and $k_{2}$ describe rates at which the drugs move between the two sequential compartments (the GI-tract and the bloodstream) and $I$ denotes the amount of drug released into the GI-tract in each time step. The levels of the drug in the GI-tract and bloodstream are $x$ and $y$, respectively. By solving the equations sequentially show that the solution is $$ x(t)=\frac{I}{k_{1}}\left(1-e^{-k_{1} t}\right), \quad y(t)=\frac{I}{k_{2}}\left[1-\frac{1}{k_{2}-k_{1}}\left(k_{2} e^{-k_{1} t}-k_{1} e^{-k_{2} t}\right)\right] . $$

Bobby Barnes

(Adapted from Borelli and Coleman (1996).) Tetracycline is an antibiotic prescribed for a range of problems, from acne to acute infections. A course is taken orally and the drug moves from the GI-tract through the bloodstream, from which it is removed by the kidneys and excreted in the urine. (a) Write word equations to describe the movement of a drug through the body, using three compartments: the GI-tract, the bloodstream and the urinary tract. Note that the urinary tract can be considered as an absorbing compartment, that is, the drug enters but is not removed from the urinary tract. (b) From the word equations develop the differential equation system that describes this process, defining all variables and parameters as required. (c) The constants of proportionality associated with the rates at which tetracycline (measured in milligrams) diffuses from the GI-tract into the bloodstream, and then is removed, are $0.72$ hour $^{-1}$ and $0.15$ hour $^{-1}$, respectively (Borelli and Coleman (1996)). Suppose, initially, the amount of tetracycline in the GI-tract is $0.0001$ milligrams, while there is none in the bloodstream or urinary tract. Use Maple or MATLAB(with symbolic toolbox) to solve this system analytically, and thus establish how the levels of tetracycline change with time in each of the compartments. In the case of a single dose, establish the maximum level reached by the drug in the bloodstream and how long it takes to reach this level with the initial conditions as given above. (d) Suppose that, initially, the body is free from the drug and then the patient takes a course of antibiotics: 1 unit per hour. Use Maple or MATLAB to examine the level of tetracycline (expressed as units) in each of the compartments over a 24-hour period. Use the constants as given above.

Sana Riaz

Use the model from the case study on alcohol consumption (Dull, dizzy or dead, Section 2.8), to establish, for the case of drinking on an empty stomach, the following: (a) Use Maple or MATLAB to generate graphs to investigate the effects of alcohol on a woman of $55 \mathrm{~kg}$, over a period of time. (b) Compare these results with those for a man of the same weight. (c) Assuming the legal limit to be $0.05$ BAL (the Australian limit), establish roughly how much alcohol the man and woman above can consume each hour and remain below this limit. (d) Repeat (a)-(c) for the case of drinking together with a meal.

Alcohol is unusual in that it is removed (that is, metabolised through the liver) from the bloodstream by a constant amount each time period, independent of the amount in the bloodstream. This removal can be modelled by a Michaelis-Menten type function $y^{\prime}=-k_{3} y /(y+M)$, where $y(t)$ is the 'amount' (BAL) of alcohol in the bloodstream at time $t, k_{3}$ is a positive constant and $M$ a small positive constant. (a) If $y$ is large compared with $M$, then show that $y^{\prime} \simeq-k_{3} .$ Solve for $y$ in this case. (b) Alternatively, as $y$ decreases and becomes small compared with $M$, show that then $y^{\prime} \simeq$ $-k_{3} y / M .$ Solve for $y$ in this case. (c) Now sketch the solution function for $y^{\prime}=-k_{3} y /(y+M)$ assuming that, initially, $y$ is much greater than $M .$ Indicate clearly how the graph changes in character when $y$ is small compared with $M$, compared with when $y$ is large compared with $M .$ Show how the solution behaves as $t \rightarrow \infty$. (d) When and why would this function be more suitable than simply using $y^{\prime}=-k_{3}$ to model the removal rate?

Carlos Pinilla

Consider the differential equations $$t \frac{d x}{d t}=x, \quad x\left(t_{0}\right)=x_{0},$$ and $$ y^{2} \frac{d x}{d y}+x y=2 y^{2}+1, \quad x\left(y_{0}\right)=x_{0} $$ Put each equation into normal form and then use the integrating factor technique to find the solutions. Establish whether these solutions are unique, and which part of each solution is a response to the initial data and which part a response to the input or forcing.

Wasim Sher

Read over the case study in Section 2.8. Consider two compartments, one for the GI-tract and one for the blood. Let $C_{1}(t)$ be the concentration of alcohol in the GI tract and $C_{2}(t)$ be the concentration in the blood, with both concentrations measured in BAL (g per $100 \mathrm{ml}$ ). Also let $F_{1}$ be the flow rate of fluid from the GI-tract and let $F_{2}$ be the flow rate of fluid from the blood to the tissues. Finally, we let $i_{0}$ be the rate of ingestion of alcohol (in $\mathrm{g} / \mathrm{hr}$ ). Use conservation of mass of alcohol to deduce the equations in the form $$\begin{aligned} &\frac{d C_{1}}{d t}=I-k_{1} C_{1} \\ &\frac{d C_{2}}{d t}=k_{2} C_{2}-k_{4} C_{2} \end{aligned}$$ and determine $I, k_{1}, k_{2}$ and $k_{3}$ all in terms of $i_{0}, F_{1}, F_{2}$ and $V_{g}$, the volume of the fluid in the GI-tract, $V_{b}$ the volume of fluid in the blood, and $\alpha$, where $\alpha$ is the proportion of the alcohol leaving the GI-tract goes into the bloodstream. Note: In the case study we let the rate constant $k_{4}$ depend on the blood alcohol concentration $$k_{4}=\frac{k_{3}}{M+C_{2}}$$ where $k_{3}$ and $M$ are positive constants, $k_{3}$ with the same units as $k_{1}$ and $k_{2}$, namely hours $^{-1}$ and $M$ with the same units as $C_{2}$, namely BAL.

David Collins

Read over the case study on a model of economic growth in Section $2.12 .$ In this model the Cobb-Douglas function was used to model production. An alternative model is the Harrod-Domar model of fixed proportions, $Y=\min \{\mathrm{K} / \mathrm{a}, \mathrm{L} / \mathrm{b}\}$ is the minimum of the two values, with $a$ units of capital and $b$ units of labour required to produce a unit of output. The expression for $Y$ describes the 'bottlenecks' for the system, that is, whether it is limitations in capital or labour that determine the outcome for production. (a) For the case $r / a<1 / b$, show that $$ \frac{d r}{d t}=\left(\frac{s}{a}-n\right) r $$ and solve this to obtain $$ r(t)=r_{0} e^{(s / a-n) t} $$ where $r(0)=r_{0}$. (b) Consider the case when $n>s / a$ and $r_{0}>a / b .$ Provide an interpretation of what this scenario means in terms of capital and the demand for labour.

Ahmed Kamel

Show that the Cobb-Douglas function, from Section 2.12, $$ Y=F(K, L)=K^{a} L^{1-a} $$ has the return to scale property.

Linh Vu

Consider Figure $2.16(\mathrm{~b})$, in Section 2.12. Establish the stability of each of the equilibrium points, $r_{e}^{(1)}, r_{e}^{(2)}$ and $r_{e}^{(3)}$, from the underlying equation.

Adam Deaton

Each of the following differential equations has only one equilibrium solution. Find that equilibrium solution and determine if it is stable or unstable? (a) $\frac{d y}{d t}=y-1$. (b) $\frac{d C}{d t}=\frac{F}{V} c_{i}-\frac{F}{V} C$, where $F, V, c_{i}$ are positive constants.

Harshita Goel

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  • © 2004

Mathematical Modelling

Case Studies and Projects

  • Jim Caldwell 0 ,
  • Douglas K. S. Ng 1

City University of Hong Kong, Kowloon, Hong Kong, P.R. China

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Part of the book series: Texts in the Mathematical Sciences (TMS, volume 28)

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Table of contents (19 chapters)

Front matter, introduction, ordinary differential equations, deterministic model in contagious disease, electromagnetic forces in high field magnet coils, mass balance of a reactor in steady state, the free and forced vibration of an automobile, cantilever beam subjected to an end load, ode problems, partial differential equations, cylindrical and spherical solidification in heat transfer, elastic analysis of a square plate with circular holes, motion of fluid layers, mass balance of a reactor with time dependency, flow through porous media, pde problems, optimization, linear programming problem involving wine production, transportation problem involving breweries and hotels, profit from an engineering plant.

  • linear optimization
  • mathematics
  • optimization
  • programming

Jim Caldwell, Douglas K. S. Ng

Book Title : Mathematical Modelling

Book Subtitle : Case Studies and Projects

Editors : Jim Caldwell, Douglas K. S. Ng

Series Title : Texts in the Mathematical Sciences

DOI : https://doi.org/10.1007/1-4020-1993-9

Publisher : Springer Dordrecht

eBook Packages : Springer Book Archive

Copyright Information : Springer Science+Business Media Dordrecht 2004

Hardcover ISBN : 978-1-4020-1991-3 Published: 31 March 2004

Softcover ISBN : 978-90-481-6566-7 Published: 04 December 2010

eBook ISBN : 978-1-4020-1993-7 Published: 10 April 2006

Series ISSN : 0927-4529

Edition Number : 1

Number of Pages : XI, 253

Topics : Mathematical Modeling and Industrial Mathematics , Engineering, general , Classical Mechanics , Numeric Computing , Algorithms

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    Mathematical Modelling with Case Studies: Using Maple™ and MATLAB®, Third Edition provides students with hands-on modelling skills for a wide variety of problems involving differential equations that describe rates of change. While the book focuses on growth and decay processes, interacting populations, and heating/cooling problems, the mathematical techniques presented can be applied to ...

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    xiv, 428 pages : 26 cm Includes bibliographical references (pages 415-419) and index Introduction to mathematical modelling -- Compartmental models -- Models of single populations -- Numerical solution of differential equations -- Formulating interacting population models -- Phase-plane analysis -- Linearisation analysis -- Some improved population models -- Formulating basic heat models ...

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