Description
An Invitation to Applied Mathematics: Differential Equations, Modeling, and Computation introduces the reader to the methodology of modern applied mathematics in modeling, analysis, and scientific computing with emphasis on the use of ordinary and partial differential equations. Each topic is introduced with an attractive physical problem, where a mathematical model is constructed using physical and constitutive laws arising from the conservation of mass, conservation of momentum, or Maxwell's electrodynamics.
Relevant mathematical analysis (which might employ vector calculus, Fourier series, nonlinear ODEs, bifurcation theory, perturbation theory, potential theory, control theory, or probability theory) or scientific computing (which might include Newton's method, the method of lines, finite differences, finite elements, finite volumes, boundary elements, projection methods, smoothed particle hydrodynamics, or Lagrangian methods) is developed in context and used to make physically significant predictions. The target audience is advanced undergraduates (who have at least a working knowledge of vector calculus and linear ordinary differential equations) or beginning graduate students.
Readers will gain a solid and exciting introduction to modeling, mathematical analysis, and computation that provides the key ideas and skills needed to enter the wider world of modern applied mathematics.
- Presents an integrated wealth of modeling, analysis, a
Chapter
Chapter 3.
An Environmental Pollutant
Chapter 4.
Acid Dissociation, Buffering, Titration, And Oscillation
4.1 A Model for Dissociation
4.2 Titration with a Base
4.3 An Improved Titration Model
4.4 The Oregonator: An Oscillatory Reaction
Chapter 5. Reaction, Diffusion, and Convection
5.1 Fundamental and Constitutive Model Equations
5.2 Reaction-diffusion in One Spatial Dimension: Heat, Genetic Mutations, and Traveling Waves
5.3 Reaction-diffusion Systems: The Gray–scott Model and Pattern Formation
5.4 Analysis of Reaction-diffusion Models: Qualitative and Numerical Methods
5.5 Beyond Euler’s Method for Reaction-diffusion PDE: Diffusion of Gas in a Tunnel, Gas in Porous Media, Second-order in Time.....
Chapter 6.
Excitable Media: Transport of Electrical Signals on Neurons
6.1 The Fitzhugh–Nagumo Model
6.2 Numerical Traveling Wave Profiles
7.2 Products For Nonlinear Systems
Chapter 8.
Feedback Control
8.1 A Mathematical Model for Heat Control of a Chamber
8.2 A One-dimensional Heated Chamber with PID Control
Chapter 9.
Random Walks And Diffusion
9.1 Basic Probability Theory
9.3 Continuum Limit of the Random Walk
9.4 Random Walk Generalizations and Applications
Chapter 10. Problems And Projects: Concentration Gradients, Convection, Chemotaxis, Cruise Control, Constrained Control, Pearson’s Random Wa.....
Chapter 11.
Equations of Fluid Motion
11.1 Scaling: The Reynolds Number and Froude Number
11.2 The Zero Viscosity Limit
11.3 The Low Reynolds Number Limit
Chapter 12.
Flow in a Pipe
Chapter 13.
Eulerian Flow
13.1 Bernoulli’s Form of Euler’s Equations
13.3 Potential Flow in Two Dimensions
13.4 Circulation, Lift, and Drag
Chapter 14.
Equations of Motion in Moving Coordinate Systems
14.1 Moving Coordinate Systems
14.3 Fluid Motion in Rotating Coordinates
14.4 Water Draining in Sinks Versus Hurricanes
14.5 A Counterintuitive Result: The Proudman–Taylor Theorem
15.1 The Ideal Water Wave Equations
15.2 The Boussinesq Equations
15.4 Boussinesq Steady State Water Waves
Chapter 16. Numerical Methods for Computational Fluid Dynamics
16.1 Approximations of Incompressible Navier–stokes Flows
16.2 A Numerical Method for Water Waves
16.3 The Boundary Element Method (bem)
16.4 Boundary Integral Representation
16.5 Boundary Integral Equation
16.6 Discretization For Bem
16.7 Smoothed Particle Hydrodynamics
16.8 Simulation of a Free-surface Flow
17.1 Conservation of Mass
17.3 Boundary Layer Theory
17.4 Flow in Prismatic Channels with Rectangular Cross Sections of Constant Width
17.6 Saint-venant Model and Systems of Conservation Laws
Chapter 18. Elasticity: Basic Theory and Equations of Motion
18.1 The Taut Wire: Separation of Variables and Fourier Series for the Wave Equation
18.2 Longitudinal Waves in a Rod with Varying Cross Section
18.4 A Three-dimensional Elastostatics Problem: a Copper Block Bolted to a Steel Plate
18.5 A One-dimensional Elasticity Model
18.6 Weak Formulation of One-dimensional Boundary Value Problems
18.7 One-dimensional Finite Element Method Discretization
18.8 Coding for the One-Dimensional Finite Element Method
18.9 Weak Formulation and Finite Element Method for Linear Elasticity
18.10 A Three-dimensional Finite Element Application
Chapter 19. Problems And Projects: Rods, Plates, Panel Flutter, Beams, Convection-diffusion in Tunnels, Gravitational Potential of a Galaxy.....
19.1 Problems: Fountains, Tapered Rods, Elasticity, Thermoelasticity, Convection-Diffusion, and Numerical Stability
19.2 Gravitational Potential of a Galaxy
19.4 Lid-driven Cavity Flow
19.6 Low Reynolds Number Flow
19.7 Fluid Motion in a Cylinder
19.9 Channel Flow Traveling Waves
Chapter 20.
Classical Electromagnetism
20.1 Maxwell’s Laws and the Lorentz Force Law
20.3 An Electromagnetic Boundary Value Problem
20.4 Comments on Maxwell’s Theory
20.5 Time-harmonic Fields
Chapter 21. Transverse Electromagnetic (TEM) Mode
Chapter 22.
Transmission Lines
22.1 Time-domain Reflectometry Model
22.3 Initial Value Problem for the Ideal Transmission Line
22.4 The Initially Dead Ideal Transmission Line with Constant Dielectrics
22.6 Reflected and Transmitted Waves
22.7 A Numerical Method for the Lossless Transmission Line Equation
22.8 The Lossy Transmission Line
Chapter 23.
Problems And Projects: Waveguides, Lord Kelvin’s Model
23.1 TE Modes in Waveguides with Circular Cross Sections
23.2 Rectangular Waveguides and Cavity Resonators
Mathematical and Computational Notes
A.1 Arzela–Ascoli Theorem
A.3 Existence, Uniqueness, and Continuous Dependence
A.4 Green’s Theorem and Integration by Parts
A.5 Gerschgorin’s Theorem
A.6 Gram–Schmidt Procedure
A.7 Grobman–hartman Theorem
A.12 Least Squares and Singular Value Decomposition
A.15 Variation of Parameters Formula
A.16 The Variational Equation
A.17 Linearization and Stability
A.18 Poincaré–bendixson Theorem
A.19 Eigenvalues of Tridiagonal Toeplitz Matrices
A.20 Conjugate Gradient Method
A.21 Numerical Computation and Programming Gems of Wisdom
Answers to Selected Exercises
An Invitation to Applied Mathematics
：Differential Equations, Modeling, and Computation
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