摘要
Written for students, Mathematical Modeling covers modeling with all kinds of differential equations, namely ordinary, partial, delay, and stochastic. The book addresses all aspects of mathematical modeling with mathematical tools used in subsequent analysis. Topics include difference equations, discrete models, and steady state solutions.
查看更多
目录
1About Mathematical Modeling 1
1.1What is Mathematical Modeling? 1
1.2History of Mathematical Modeling 2
1.3Importance of Mathematical Modeling 4
1.4Latest Developments in Mathematical Modeling 5
1.5Limitations of Mathematical Modeling 6
2Mathematically Modeling Discrete Processes 9
2.1Difference Equations 9
2.1.1Linear Difference Equation with Constant Coeff cients 10
2.1.2Solution of Homogeneous Equations 10
2.1.3Difference Equation：Equilibria and Stability 14
2.1.3.1Linear Difference Equation 14
2.1.3.2System of Linear Difference Equations 14
2.1.3.3Non-Linear Systems 16
2.2Introduction to Discrete Models 17
2.3Linear Models 18
2.3.1Population Model Involving Growth 18
2.3.2Newton'sLaw of Coolingm 19
2.3.3BankAccount Problem 20
2.3.4Drug Delivery Problem 22
2.3.5Economic Model(Harrod Model) 23
2.3.6Arms Race Model 24
2.3.7Linear Prey-Predator Problem 24
2.4Non-Linear Models 27
2.4.1Density Dependent Growth Models 27
2.4.2The Learning Model 27
2.5Miscellaneous Examples 28
2.6Exercises 38
3Continuous Models Using Ordinary Differential Equations 47
3.1Introduction to Continuous Models 47
3.2Formation of Various Continuous Models 48
3.2.1Carbon Dating 48
3.2.2Drug Distribution in the Body 49
3.2.3Growth and Decay of Current in anL-R Circuit 50
3.2.4Rectilinear Motion under Variable Force 52
3.2.5Mechanical Oscillations 53
3.2.5.1Horizontal Oscillations 53
3.2.5.2Vertical Oscillations 54
3.2.5.3Damped Force Oscillation 55
3.2.6Dynamics of Rowing 57
3.2.7Arms Race Models 58
3.2.8Mathematical Model of Influenza Infection(within
Host) 60
3.2.9Epidemic Models 61
3.3Steady State Solutions 65
3.4Linearization and Local Stability Analysis 66
3.5Phase Plane Diagrams of Linear Systems 68
3.6Bifurcations 72
3.6.1Saddle-Node Bifurcation 73
3.6.2Trans critical Bifurcation 75
3.6.3Pitchfork Bifurcation 77
3.6.4Hopf Bifurcation 79
3.7Miscellaneous Examples 80
3.8Exercises 98
4Spatial Models Using Partial Differential Equations 111
4.1Introduction 111
4.2Different Mathematical Models Using Diffusion 112
4.2.1Fluid Flow through a Porous Medium 112
4.2.2Heat Flow through a Small Thin Rod(One Dimen-
sional) 113
4.2.3Wave Equation 115
4.2.4Vibrating String 117
4.2.5Traffic Flow 119
4.2.6Theory of Car-Following 124
4.2.7Crimes Model 126
4.3Linear Stability Analysis 127
4.3.1One Species with Diffusion 127
4.3.2Two Species with Diffusion 128
4.4A Research Problem:Spatiotemporal Aspect of a Mathematical 132
4.4.1Background of the Problem 132
4.4.2Spatiotemporal Model Formulation 132
4.4.3Qualitative Analysis 133
4.4.4Numerical Results 137
4.4.5Conclusion 138
4.5Miscellaneous Examples 139
4.6Exercises 148
5Modeling with Delay Differential Equations 153
5.1Introduction 153
5.2Different Models Using Delay Differential Equations 154
5.2.1Delayed Protein Degradation 154
5.2.2Football Team Performance Model 155
5.2.3Breathing Model 156
5.2.4Housefly Model 157
5.2.5Shower Problem 158
5.2.6Two-Neuron System 159
5.3Linear Stability Analysis 160
5.3.1Linear Stability Criteria 161
5.4Miscellaneous Examples 163
5.4.1A Research Problem:Immunotherapy with Interleukin-2,a Study Based on Mathematical Modeling[8] 171
5.4.1.1Background of the Problem 171
5.4.1.2The Model 173
5.4.1.3Positivity of the Solution 175
5.4.1.4Linear Stability Analysis with Delay 175
5.4.1.5Estimation of the Length of Delay to Preserve Stability 178
5.4.1.6Numerical Results 181
5.4.1.7Conclusion 182
5.5Exercises 184
6Modeling with Stochastic Differential Equations 191
6.1Introduction 191
6.1.1Probability Space 192
6.1.2Stochastic Process 193
6.1.2.1Wiener Process(Brownian Motion) 194
6.1.3Stochastic Differential Equation(SDE) 195
6.1.4Gaussian White Noise 195
6.1.5Stochastic Stability 195
6.2Some Stochastic Models 196
6.2.1Stochastic Logistic Growth 196
6.2.2Heston Model 197
6.2.3Resistor-Inductor-Capacitor(RLC) Electric Circuit with Randomness 199
6.2.4Two Species Competition Model 200
6.3AResearchProblem:CancerSelf-RemissionandTumorStabil-
ity-A Stochastic Approach[116] 200
6.3.1Background of the Problem 200
6.3.2The Deterministic Model 202
6.3.3Equilibria and Local Stability Analysis 203
6.3.4Biological Implications 205
6.3.5The Stochastic Model 206
6.3.6Stochastic Stability of the Positive Equilibrium 207
6.3.7Numerical Results and Explanations 211
6.3.8Concluding Remarks 211
6.4Exercises 214
7Hints and Solutions 219
Bibliography 243
Index 253
查看PDF
查看更多
馆藏单位
中科院文献情报中心
