Mathematical and Experimental Modeling of

Through several case study problems, this textbook provides a fundamental understanding of how mathematics is applied in science and engineering. Integrating real-world applications into the traditional mathematics curriculum, the book deals with the formulation and analysis of mathematical models in science and engineering.

H.T. Banks is Distinguished University Professor and Drexel Professor of Mathematics at North Carolina State University (NCSU) and Director of the Center for Research in Scientific Computation (CRSC). His research interests include inverse problems and control in electromagnetics, structures, and biology. He has taught modeling courses in biological and physical systems for the past 30 years at Brown University, University of Southern California, and NCSU in which materials for this book were developed. H.T. Tran is Professor of Mathematics and Associate Head of the Department of Mathematics at NCSU. He is also Director of the CRSC/Math Instructional and Research Laboratory, which was used in a number of the projects in this book. His research interests include mathematical modeling of general biological and physical systems, nonlinear feedback control design and synthesis, filtering, and modeling in cardiovascular physiology.

Introduction: The Iterative Modeling Process	p. 1
Modeling and Inverse Problems	p. 7
Mechanical Vibrations	p. 7
Inverse Problems	p. 11
References	p. 15
Mathematical and Statistical Aspects of Inverse Problems	p. 17
Probability and Statistics Overview	p. 18
Probability	p. 18
Random Variables	p. 20
Statistical Averages of Random Variables	p. 21
Special Probability Distributions	p. 22
Parameter Estimation or Inverse Problems	p. 29
The Mathematical Model	p. 29
The Statistical Model	p. 30
Known Error Processes: Maximum Likelihood Estimators	p. 31
Normally Distributed Errors	p. 31
Unspecified Error Distributions and Asymptotic Theory	p. 33
Ordinary Least Squares (OLS)	p. 34
Numerical Implementation of the Vector OLS Procedure	p. 37
Generalized Least Squares (GLS)	p. 38
GLS Motivation	p. 39
Numerical Implementation of the GLS Procedure	p. 40
Computation of <$$$>, Standard Errors and Confidence Intervals	p. 41
Investigation of Statistical Assumptions	p. 45
Residual Plots	p. 46
An Example Using Residual Plots	p. 47
Statistically Based Model Comparison Techniques	p. 51
RSS Based Statistical Tests	p. 54
P-Values	p. 56
Alternative Statement	p. 57
Application: Cat-Brain Diffusion/Convection Problem	p. 57
References	p. 63
Mass Balance and Mass Transport	p. 65
Introduction	p. 65
Compartmental Concepts	p. 65
Compartment Modeling	p. 67
General Mass Transport Equations	p. 71
Mass Flux Law in a Stationary (Non-Moving) Fluid	p. 73
Mass Flux in a Moving Fluid	p. 75
References	p. 79
Heat Conduction	p. 81
Motivating Problems	p. 81
Radio-Frequency Bonding of Adhesives	p. 81
Thermal Testing of Structures	p. 82
Mathematical Modeling of Heat Transfer	p. 83
Introduction	p. 83
Fourier's Law of Heat Conduction	p. 84
Heat Equation	p. 85
Boundary Conditions and Initial Conditions	p. 89
Properties of Solutions	p. 93
Experimental Modeling of Heat Transfer	p. 94
The Thermocouple as a Temperature Measuring Device	p. 95
Detailed Hardware and Software Lists	p. 98
References	p. 101
Structural Modeling: Force/Moments Balance	p. 103
Motivation: Control of Acoustics/Structural Interactions	p. 103
Introduction to Mechanics of Elastic Solids	p. 104
Normal Stress and Strain	p. 105
Stress and Strain Relationship (Hooke's Law)	p. 106
Shear Stress and Strain	p. 109
Deformations of Beams	p. 112
Differential Equations of Thin Beam Deflections	p. 114
Force Balance	p. 114
Moment Balance	p. 116
Moment Computation	p. 117
Initial Conditions	p. 123
Boundary Conditions	p. 124
Separation of Variables: Modes and Mode Shapes	p. 129
Numerical Approximations: Galerkin's Method	p. 135
Energy Functional Formulation	p. 141
The Finite Element Method	p. 143
Experimental Beam Vibration Analysis	p. 147
References	p. 153
Beam Vibrational Control and Real-Time Implementation	p. 155
Introduction	p. 155
Controllability and Observability of Linear Systems	p. 155
Controllability	p. 156
Time-Varying Case	p. 156
Time-Invariant Case	p. 162
Observability	p. 170
Time-Varying Case	p. 170
Time-Invariant Case	p. 172
Design of State Feedback Control Systems and State Estimators	p. 175
Effect of State Feedback on System Properties	p. 179
Stability	p. 179
Controllability	p. 180
Observability	p. 181
Pole Placement (Relocation) Problem	p. 182
State Estimator (Luenberger Observer)	p. 190
Dynamic Output Feedback Compensator	p. 191
Linear Quadratic Regulator Theory	p. 197
Beam Vibrational Control: Real-Time Feedback Control Implementation	p. 200
References	p. 213
Wave Propagation	p. 215
Fluid Dynamics	p. 215
Newton's Law of Viscosity	p. 216
Derivative in Fluid Flows	p. 220
Equations of Fluid Motion	p. 220
Fluid Waves	p. 229
Terminology	p. 229
Sound Waves	p. 231
Euler's Equation	p. 233
Equation of Continuity	p. 233
Equation of State	p. 233
Wave Equations	p. 234
Experimental Modeling of the Wave Equation	p. 238
References	p. 243
Size-Structured Population Models	p. 245
Introduction: A Motivating Application	p. 245
A Single Species Model (Malthusian Law)	p. 246
The Logistic Model	p. 247
A Predator/Prey Model	p. 249
A Size-Structured Population Model	p. 251
The Sinko-Streifer Model and Inverse Problems	p. 265
Size Structure and Mosquitofish Populations	p. 268
References	p. 277
An Introduction to Fourier Techniques	p. 281
Fourier Series	p. 281
Fourier Transforms	p. 284
Review of Vector Calculus	p. 287
References	p. 293
Index	p. 295
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Mathematical and Experimental Modeling of Physical and Biological Processes

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