# An Invitation to Applied Mathematics

### Differential Equations, Modeling, and Computation

## Description

## Key Features

- Presents an integrated wealth of modeling, analysis, and numerical methods in one volume
- Provides practical and comprehensible introductions to complex subjects, for example, conservation laws, CFD, SPH, BEM, and FEM
- Includes a rich set of applications, with more appealing problems and projects suggested

## Readership

Advanced undergraduates and beginning graduate students. Professionals in mathematics, engineering, or the other sciences looking for a useful introduction to thd subject. The reader should have mathematical maturity at the level of basic ordinary differential equations, vector calculus, and matrix theory. Previous knowledge of PDE and numerical methods is not assumed, but some experience with computers is.

## Table of Contents

- Preface
- Acknowledgments
- To the Professor
- To the Student
- Chapter 1: Applied Mathematics and Mathematical Modeling
- Abstract
- 1.1 What is applied mathematics?
- 1.2 Fundamental and constitutive models
- 1.3 Descriptive models
- 1.4 Applied mathematics in practice

- Chapter 2: Differential Equations
- Abstract
- 2.1 The harmonic oscillator
- 2.2 Exponential and logistic growth
- 2.3 Linear systems
- 2.4 Linear partial differential equations
- 2.5 Nonlinear ordinary differential equations
- 2.6 Numerics

- Part I: Conservation of Mass: Biology, Chemistry, Physics, and Engineering
- Chapter 3: An Environmental Pollutant
- Abstract

- Chapter 4: Acid Dissociation, Buffering, Titration, and Oscillation
- Abstract
- 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
- Abstract
- 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 methods, and unconditional stability

- Chapter 6: Excitable Media: Transport of Electrical Signals on Neurons
- Abstract
- 6.1 The FitzHugh–Nagumo model
- 6.2 Numerical traveling wave profiles

- Chapter 7: Splitting Methods
- Abstract
- 7.1 A product formula
- 7.2 Products for nonlinear systems

- Chapter 8: Feedback Control
- Abstract
- 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
- Abstract
- 9.1 Basic probability theory
- 9.2 Random walk
- 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 Walk, Molecular Dynamics, Pattern Formation
- Abstract

- Chapter 3: An Environmental Pollutant
- Part II: Newton’s Second Law: Fluids and Elastic Solids
- Chapter 11: Equations of Fluid Motion
- Abstract
- 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
- Abstract

- Chapter 13: Eulerian Flow
- Abstract
- 13.1 Bernoulli’s form of Euler’s equations
- 13.2 Potential flow
- 13.3 Potential flow in two dimensions
- 13.4 Circulation, lift, and drag

- Chapter 14: Equations of Motion in Moving Coordinate Systems
- Abstract
- 14.1 Moving coordinate systems
- 14.2 Pure rotation
- 14.3 Fluid motion in rotating coordinates
- 14.4 Water draining in sinks versus hurricanes
- 14.5 A Counterintuitive result: the Proudman–Taylor theorem

- Chapter 15: Water Waves
- Abstract
- 15.1 The ideal water wave equations
- 15.2 The Boussinesq equations
- 15.3 KDV
- 15.4 Boussinesq steady state water waves
- 15.5 A free-surface flow

- Chapter 16: Numerical Methods for Computational Fluid Dynamics
- Abstract
- 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

- Chapter 17: Channel Flow
- Abstract
- 17.1 Conservation of mass
- 17.2 Momentum balance
- 17.3 Boundary layer theory
- 17.4 Flow in prismatic channels with rectangular cross sections of constant width
- 17.5 Hydraulic jump
- 17.6 Saint-Venant model and systems of conservation laws
- 17.7 Surface waves

- Chapter 18: Elasticity: Basic Theory and Equations of Motion
- Abstract
- 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
- 18.3 Ultrasonics
- 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, Taylor Dispersion, Cavity Flow, Drag, Low and High Reynolds Number Flows, Free-Surface Flow, Channel Flow
- Abstract
- 19.1 Problems: fountains, tapered rods, elasticity,thermoelasticity, convection-diffusion, and numerical stability
- 19.2 Gravitational potential of a galaxy
- 19.3 Taylor dispersion
- 19.4 Lid-driven cavity flow
- 19.5 Aerodynamic drag
- 19.6 Low Reynolds number flow
- 19.7 Fluid motion in a cylinder
- 19.8 Free-surface flow
- 19.9 Channel flow traveling waves

- Chapter 11: Equations of Fluid Motion
- Part III: Electromagnetism: Maxwell’s Laws and Transmission Lines
- Chapter 20: Classical Electromagnetism
- Abstract
- 20.1 Maxwell’s laws and the Lorentz force law
- 20.2 Boundary conditions
- 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
- Abstract

- Chapter 22: Transmission Lines
- Abstract
- 22.1 Time-domain reflectometry model
- 22.2 TDR matrix system
- 22.3 Initial value problem for the ideal transmission line
- 22.4 The initially dead ideal transmission line with constant dielectrics
- 22.5 The Riemann problem
- 22.6 Reflected and transmitted waves
- 22.7 A numerical method for the lossless transmission line equation
- 22.8 The lossy transmission line
- 22.9 TDR applications
- 22.10 An inverse problem

- Chapter 23: Problems and Projects: Waveguides, Lord Kelvin’s Model
- Abstract
- 23.1 TE modes in waveguides with circular cross sections
- 23.2 Rectangular waveguides and cavity resonators

- Chapter 20: Classical Electromagnetism
- Mathematical and Computational Notes
- A.1 Arzel–Ascoli theorem
- A.2 C
^{1}convergence - 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.8 Order notation
- A.9 Taylor’s formula
- A.10 Liouville’s theorem
- A.11 Transport theorem
- A.12 Least squares and singular value decomposition
- A.13 The morse lemma
- A.14 Newton’s method
- 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
- References
- Index