Table Of ContentEngineering Mathematics
MATHEMATICAL
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a
MATHEMATICAL n
e MODELING
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MODELING
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Models, Analysis and Applications
Models, Analysis and Applications
Almost every year, a new book on mathematical modeling is published, so, why
another? The answer springs directly from the fact that it is very rare to find a Sandip Banerjee
book that covers modeling with all types of differential equations in one volume.
Until now. Mathematical Modeling: Models, Analysis and Applications covers
modeling with all kinds of differential equations, namely ordinary, partial, delay, and
stochastic. The book also contains a chapter on discrete modeling, consisting of
differential equations, making it a complete textbook on this important skill needed
M
for the study of science, engineering, and social sciences.
A
More than just a textbook, this how-to guide presents tools for mathematical
M
modeling and analysis. It offers a wide-ranging overview of mathematical ideas T
and techniques that provide a number of effective approaches to problem solving.
OH
Topics covered include spatial, delayed, and stochastic modeling. The text provides
real-life examples of discrete and continuous mathematical modeling scenarios.
DE
MATLAB® and Mathematica® are incorporated throughout the text. The examples
and exercises in each chapter can be used as problems in a project. EM
Features
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• Addresses all aspects of mathematical modeling with mathematical tools used
in subsequent analysis IT
N
• Incorporates MATLAB and Mathematica
I
• Covers spatial, delayed, and stochastic models
GC
• Presents real-life examples of discrete and continuous scenarios
• Includes examples and exercises that can be used as problems in a project A
Since mathematical modeling involves a diverse range of skills and tools, the author L
focuses on techniques that will be of particular interest to engineers, scientists, and
others who use models of discrete and continuous systems. He gives students a
foundation for understanding and using the mathematics that is the basis of com-
puters, and therefore a foundation for success in engineering and science streams.
K12528
K12528_Cover.indd 1 12/18/13 12:21 PM
MATHEMATICAL
MODELING
Models, Analysis and Applications
K12528_FM.indd 1 12/10/13 11:42 AM
K12528_FM.indd 2 12/10/13 11:42 AM
MATHEMATICAL
MODELING
Models, Analysis and Applications
Sandip Banerjee
Indian Institute of Technology
Roorkee
K12528_FM.indd 3 12/10/13 11:42 AM
MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not
warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® soft-
ware or related products does not constitute endorsement or sponsorship by The MathWorks of a particular
pedagogical approach or particular use of the MATLAB® software.
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i
To my wife Usha
and
son Aditya
who brought
love and joy
to my life
and to Otso
who taught me
how to do research
Contents
1 About Mathematical Modeling 1
1.1 What is Mathematical Modeling? . . . . . . . . . . . . . . . 1
1.2 History of Mathematical Modeling . . . . . . . . . . . . . . . 2
1.3 Importance of Mathematical Modeling . . . . . . . . . . . . 4
1.4 Latest Developments in Mathematical Modeling . . . . . . . 5
1.5 Limitations of Mathematical Modeling . . . . . . . . . . . . 6
2 Mathematically Modeling Discrete Processes 9
2.1 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Linear Difference Equation with Constant Coefficients 10
2.1.2 Solution of Homogeneous Equations . . . . . . . . . . 10
2.1.3 Difference Equation: Equilibria and Stability . . . . . 14
2.1.3.1 Linear Difference Equation . . . . . . . . . . 14
2.1.3.2 System of Linear Difference Equations . . . . 14
2.1.3.3 Non-Linear Systems . . . . . . . . . . . . . . 16
2.2 Introduction to Discrete Models . . . . . . . . . . . . . . . . 17
2.3 Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Population Model Involving Growth . . . . . . . . . . 18
2.3.2 Newton’s Law of Cooling . . . . . . . . . . . . . . . . 19
2.3.3 Bank Account Problem . . . . . . . . . . . . . . . . . 20
2.3.4 Drug Delivery Problem . . . . . . . . . . . . . . . . . 22
2.3.5 Economic Model (Harrod Model) . . . . . . . . . . . . 23
2.3.6 Arms Race Model . . . . . . . . . . . . . . . . . . . . 24
2.3.7 Linear Prey-PredatorProblem . . . . . . . . . . . . . 24
2.4 Non-Linear Models . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.1 Density Dependent Growth Models . . . . . . . . . . . 27
2.4.2 The Learning Model . . . . . . . . . . . . . . . . . . . 27
2.5 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . 28
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Continuous Models Using Ordinary Differential Equations 47
3.1 Introduction to Continuous Models . . . . . . . . . . . . . . 47
3.2 Formation of Various Continuous Models . . . . . . . . . . . 48
3.2.1 Carbon Dating . . . . . . . . . . . . . . . . . . . . . . 48
3.2.2 Drug Distribution in the Body . . . . . . . . . . . . . 49
3.2.3 Growth and Decay of Current in an L-R Circuit . . . 50
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iv
3.2.4 Rectilinear Motion under Variable Force . . . . . . . . 52
3.2.5 Mechanical Oscillations . . . . . . . . . . . . . . . . . 53
3.2.5.1 Horizontal Oscillations . . . . . . . . . . . . 53
3.2.5.2 Vertical Oscillations . . . . . . . . . . . . . . 54
3.2.5.3 Damped Force Oscillation . . . . . . . . . . . 55
3.2.6 Dynamics of Rowing . . . . . . . . . . . . . . . . . . . 57
3.2.7 Arms Race Models . . . . . . . . . . . . . . . . . . . . 58
3.2.8 Mathematical Model of Influenza Infection (within
Host) . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.9 Epidemic Models . . . . . . . . . . . . . . . . . . . . . 61
3.3 Steady State Solutions . . . . . . . . . . . . . . . . . . . . . 65
3.4 Linearization and Local Stability Analysis . . . . . . . . . . 66
3.5 Phase Plane Diagrams of Linear Systems . . . . . . . . . . . 68
3.6 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.6.1 Saddle-Node Bifurcation . . . . . . . . . . . . . . . . . 73
3.6.2 Transcritical Bifurcation . . . . . . . . . . . . . . . . . 75
3.6.3 Pitchfork Bifurcation. . . . . . . . . . . . . . . . . . . 77
3.6.4 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . 79
3.7 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . 80
3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4 Spatial Models Using Partial Differential Equations 111
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2 Different Mathematical Models Using Diffusion . . . . . . . . 112
4.2.1 Fluid Flow through a Porous Medium . . . . . . . . . 112
4.2.2 Heat Flow through a Small Thin Rod (One Dimen-
sional) . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2.3 Wave Equation . . . . . . . . . . . . . . . . . . . . . . 115
4.2.4 Vibrating String . . . . . . . . . . . . . . . . . . . . . 117
4.2.5 Traffic Flow . . . . . . . . . . . . . . . . . . . . . . . . 119
4.2.6 Theory of Car-Following . . . . . . . . . . . . . . . . 124
4.2.7 Crimes Model . . . . . . . . . . . . . . . . . . . . . . . 126
4.3 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . 127
4.3.1 One Species with Diffusion . . . . . . . . . . . . . . . 127
4.3.2 Two Species with Diffusion . . . . . . . . . . . . . . . 128
4.4 AResearchProblem:SpatiotemporalAspectofaMathematical
Model of Cancer Immune Interaction Considering the Role of
Antibodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.4.1 Backgroundof the Problem . . . . . . . . . . . . . . . 132
4.4.2 Spatiotemporal Model Formulation . . . . . . . . . . . 132
4.4.3 Qualitative Analysis . . . . . . . . . . . . . . . . . . . 133
4.4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . 137
4.4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 138
4.5 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . 139
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
v
5 Modeling with Delay Differential Equations 153
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.2 Different Models Using Delay Differential Equations . . . . . 154
5.2.1 Delayed Protein Degradation . . . . . . . . . . . . . . 154
5.2.2 Football Team Performance Model . . . . . . . . . . . 155
5.2.3 Breathing Model . . . . . . . . . . . . . . . . . . . . . 156
5.2.4 Housefly Model . . . . . . . . . . . . . . . . . . . . . . 157
5.2.5 Shower Problem . . . . . . . . . . . . . . . . . . . . . 158
5.2.6 Two-Neuron System . . . . . . . . . . . . . . . . . . . 159
5.3 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . 160
5.3.1 Linear Stability Criteria . . . . . . . . . . . . . . . . . 161
5.4 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . 163
5.4.1 A ResearchProblem:Immunotherapywith Interleukin-
2, a Study Based on Mathematical Modeling [8]. . . . 171
5.4.1.1 Backgroundof the Problem . . . . . . . . . . 171
5.4.1.2 The Model . . . . . . . . . . . . . . . . . . . 173
5.4.1.3 Positivity of the Solution . . . . . . . . . . . 175
5.4.1.4 Linear Stability Analysis with Delay . . . . . 175
5.4.1.5 Estimationofthe LengthofDelaytoPreserve
Stability . . . . . . . . . . . . . . . . . . . . 178
5.4.1.6 Numerical Results . . . . . . . . . . . . . . . 181
5.4.1.7 Conclusion . . . . . . . . . . . . . . . . . . . 182
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6 Modeling with Stochastic Differential Equations 191
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.1.1 Probability Space. . . . . . . . . . . . . . . . . . . . . 192
6.1.2 Stochastic Process . . . . . . . . . . . . . . . . . . . . 193
6.1.2.1 Wiener Process (Brownian Motion) . . . . . 194
6.1.3 Stochastic Differential Equation (SDE). . . . . . . . . 195
6.1.4 Gaussian White Noise . . . . . . . . . . . . . . . . . . 195
6.1.5 Stochastic Stability . . . . . . . . . . . . . . . . . . . . 195
6.2 Some Stochastic Models . . . . . . . . . . . . . . . . . . . . . 196
6.2.1 Stochastic Logistic Growth . . . . . . . . . . . . . . . 196
6.2.2 Heston Model . . . . . . . . . . . . . . . . . . . . . . . 197
6.2.3 Resistor-Inductor-Capacitor(RLC)ElectricCircuitwith
Randomness . . . . . . . . . . . . . . . . . . . . . . . 197
6.2.4 Two Species Competition Model . . . . . . . . . . . . 199
6.3 AResearchProblem:CancerSelf-RemissionandTumorStabil-
ity - A Stochastic Approach [116] . . . . . . . . . . . . . . . 200
6.3.1 Backgroundof the Problem . . . . . . . . . . . . . . . 200
6.3.2 The Deterministic Model . . . . . . . . . . . . . . . . 202
6.3.3 Equilibria and Local Stability Analysis . . . . . . . . . 203
6.3.4 Biological Implications . . . . . . . . . . . . . . . . . . 205
6.3.5 The Stochastic Model . . . . . . . . . . . . . . . . . . 206
