This accessible and practical text is designed to nurture a "modeling
intuition" for a wide range of disciplines, including mathematics, science,
engineering, and economics. The numerous examples and mathematical
techniques it includes demonstrate that mathematical modeling can be an
important tool for revealing the underlying links between apparently
disparate phenomena. Its flexible approach also reinforces the idea that
there is no fixed set of tools for modeling.
Includes many real-world problems related to industry,
medicine, and engineering.
Provides self-testing questions and exercises on almost
every page allowing professor to devote class time to discussion of projects.
Presents important core ideas at the beginning of each
chapter (starred (*) sections present optional material.)
Each chapter includes a comprehensive problem section with
extensive references that provides a number of
opportunities for class projects.
Includes a guide to the relevant literature at the end of each
1. The Modeling Art.
What is Modeling? Multiple Models; Shopping Around. An Example of the Modeling Process. Free Radical Formation by Ultrasound. Approximations. Curve Fitting and Parameter Estimating. The O ( Ã) and 0( Ã) notation. Fourier Transform. Problems and Recommended Reading.
2. Stability and Bifurcation.
Potentials. Bifurcation. Catastrophe. Problems and Recommended Reading.
Dimensions. Dimensions in Electricity and Magnetism. Scaling and Life. Dimensional Analysis and the Pi Procedure. The Pi Theorem. Limitations and Extensions. Scale Modeling. Problems and Recommended Reading.
4. Growth and Relaxation.
Exponential Growth. The Relaxation Response. Self-Limiting Growth. Autoregulation. Economic Growth. Problems and Recommended Reading.
Free Vibrations. Mechanical Vibrations. Other Harmonic Oscillators. A Splash of Reality: Nonlinear Oscillations. Forced Vibrations. Linear Response. The Energy Cycle and the Power Absorption Curve. General Resonance. Nonlinear Response. Problems and Recommended Reading.
6. Random Thinking.
Probabilities. The Law of Averages. Drunkard's Walk. Counting on Probabilities. Aside on Entropy and Information. Conditional Probabilities. Random Variables. Continuous Random Variables. Time between Random Events. Simulation of Random Variables. Problems and Recommended Reading.
7. Random Processes.
Processes: Poisson Points and Random Walks. Poisson Points. Stochastic Processes. Markov Chain Models. The Parking Lot Problem. Orbital Debris: Distribution. Continuous-Time Processes. Service Facilities. Orbital Debris: Population Growth. Beyond Markov. General Point Processes. Queues. Cell-cycle Modeling. Simulation and the Monte-Carlo Method. Problems and Recommended Reading.
8. Complex Systems.
Coupled Oscillators. Biological Rhythms. Swaying Smokestacks. Dynamo Theory. Problems and Recommended Reading.
9. Snakes and Chains.
Snakes and Chains. A Line of Cars. Crystal Vibrations. Fixed-end Boundary Condition. Forced Vibrations. Filters and Ladders. Modeling the Ear. Earthquakes. Snakes versus Chains. Problems and Recommended Reading.
Waves Here and There. Conservation Laws. An Example of the Method of Characteristics. Constitutive Laws. A nonlinear conservation law. Shocks. Conservation Laws in Higher Dimensions. Population Models. Birth-Death Processes. First-order Quasilinear Equations: General Theory. Cauchy Problem. Linear Systems. Blood Flow. Acoustics. Transmission Lines. Solving Linear Hyperbolic Systems. Solving the Transmission Line Equations. Systems of Nonlinear Conservation Laws. Momentum Equation Using Pull-back Method. Sound Speed in Gases. Shock as a Dissipative Structure. Conservation of Energy. Quasilinear Hyperbolic Systems: Simple Waves. Water Waves. A Big Bore. Surge on Deep Water. Problems and Recommended Reading.
How It Goes at Small Scales. The Diffusion Equation. The Einstein-Smoluchowski Relation. Conservation Laws: Heat Conduction. Newton's Law of Cooling. Reactions. Solving the Diffusion Equation. Signal Distortion. Steady-state Diffusion Boundary Conditions. Melting and Freezing: Moving Boundaries. Problems and Recommended Reading.