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Numerical and Analytical Methods for Scientists and Engineers Using Mathematica

ISBN-10: 0471266108
ISBN-13: 9780471266105
Edition: 2003
Authors: Daniel Dubin
List price: $218.00
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Description: This work is written from the perspective of a physicist, not a mathematician, emphasising modern practical applications in the physical and engineering sciences. The book itself is essentially software, written in the language of Mathematica, which  More...

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Book details

List price: $218.00
Copyright year: 2003
Publisher: John Wiley & Sons, Incorporated
Publication date: 5/5/2003
Binding: Hardcover
Pages: 656
Size: 7.25" wide x 14.00" long x 1.50" tall
Weight: 2.992
Language: English

This work is written from the perspective of a physicist, not a mathematician, emphasising modern practical applications in the physical and engineering sciences. The book itself is essentially software, written in the language of Mathematica, which is widely used in engineering and physics.

Preface
Ordinary Differential Equations in the Physical Sciences
Introduction
Definitions
Exercises for Sec. 1.1
Graphical Solution of Initial-Value Problems
Direction Fields; Existence and Uniqueness of Solutions
Direction Fields for Second-Order ODEs: Phase-Space Portraits
Exercises for Sec. 1.2
Analytic Solution of Initial-Value Problems via DSolve
DSolve
Exercises for Sec. 1.3
Numerical Solution of Initial-Value Problems
NDSolve
Error in Chaotic Systems
Euler's Method
The Predictor-Corrector Method of Order 2
Euler's Method for Systems of ODEs
The Numerical N-Body Problem: An Introduction to Molecular Dynamics
Exercises for Sec. 1.4
Boundary-Value Problems
Introduction
Numerical Solution of Boundary-Value Problems: The Shooting Method
Exercises for Sec. 1.5
Linear ODEs
The Principle of Superposition
The General Solution to the Homogeneous Equation
Linear Differential Operators and Linear Algebra
Inhomogeneous Linear ODEs
Exercises for Sec. 1.6
References
Fourier Series and Transforms
Fourier Representation of Periodic Functions
Introduction
Fourier Coefficients and Orthogonality Relations
Triangle Wave
Square Wave
Uniform and Nonuniform Convergence
Gibbs Phenomenon for the Square Wave
Exponential Notation for Fourier Series
Response of a Damped Oscillator to Periodic Forcing
Fourier Analysis, Sound, and Hearing
Exercises for Sec. 2.1
Fourier Representation of Functions Defined on a Finite Interval
Periodic Extension of a Function
Even Periodic Extension
Odd Periodic Extension
Solution of Boundary-Value Problems Using Fourier Series
Exercises for Sec. 2.2
Fourier Transforms
Fourier Representation of Functions on the Real Line
Fourier sine and cosine Transforms
Some Properties of Fourier Transforms
The Dirac [delta]-Function
Fast Fourier Transforms
Response of a Damped Oscillator to General Forcing. Green's Function for the Oscillator
Exercises for Sec. 2.3
Green's Functions
Introduction
Constructing the Green's Function from Homogeneous Solutions
Discretized Green's Function I: Initial-Value Problems by Matrix Inversion
Green's Function for Boundary-Value Problems
Discretized Green's Functions II: Boundary-Value Problems by Matrix Inversion
Exercises for Sec. 2.4
References
Introduction to Linear Partial Differential Equations
Separation of Variables and Fourier Series Methods in Solutions of the Wave and Heat Equations
Derivation of the Wave Equation
Solution of the Wave Equation Using Separation of Variables
Derivation of the Heat Equation
Solution of the Heat Equation Using Separation of Variables
Exercises for Sec. 3.1
Laplace's Equation in Some Separable Geometries
Existence and Uniqueness of the Solution
Rectangular Geometry
2D Cylindrical Geometry
Spherical Geometry
3D Cylindrical Geometry
Exercises for Sec. 3.2
References
Eigenmode Analysis
Generalized Fourier Series
Inner Products and Orthogonal Functions
Series of Orthogonal Functions
Eigenmodes of Hermitian Operators
Eigenmodes of Non-Hermitian Operators
Exercises for Sec. 4.1
Beyond Separation of Variables: The General Solution of the 1D Wave and Heat Equations
Standard Form for the PDE
Generalized Fourier Series Expansion for the Solution
Exercises for Sec. 4.2
Poisson's Equation in Two and Three Dimensions
Introduction. Uniqueness and Standard Form
Green's Function
Expansion of g and 0 in Eigenmodes of the Laplacian Operator
Eigenmodes of [down triangle, open superscript 2] in Separable Geometries
Exercises for Sec. 4.3
The Wave and Heat Equations in Two and Three Dimensions
Oscillations of a Circular Drumhead
Large-Scale Ocean Modes
The Rate of Cooling of the Earth
Exercises for Sec. 4.4
References
Partial Differential Equations in Infinite Domains
Fourier Transform Methods
The Wave Equation in One Dimension
Dispersion; Phase and Group Velocities
Waves in Two and Three Dimensions
Exercises for Sec. 5.1
The WKB Method
WKB Analysis without Dispersion
WKB with Dispersion: Geometrical Optics
Exercises for Sec. 5.2
Wave Action (Electronic Version Only)
The Eikonal Equation
Conservation of Wave Action
Exercises for Sec. 5.3
References
Numerical Solution of Linear Partial Differential Equations
The Galerkin Method
Introduction
Boundary-Value Problems
Time-Dependent Problems
Exercises for Sec. 6.1
Grid Methods
Time-Dependent Problems
Boundary-Value Problems
Exercises for Sec. 6.2
Numerical Eigenmode Methods (Electronic Version Only)
Introduction
Grid-Method Eigenmodes
Galerkin-Method Eigenmodes
WKB Eigenmodes
Exercises for Sec. 6.3
References
Nonlinear Partial Differential Equations
The Method of Characteristics for First-Order PDEs
Characteristics
Linear Cases
Nonlinear Waves
Exercises for Sec. 7.1
The KdV Equation
Shallow-Water Waves with Dispersion
Steady Solutions: Cnoidal Waves and Solitons
Time-Dependent Solutions: The Galerkin Method
Shock Waves: Burgers' Equation
Exercises for Sec. 7.2
The Particle-in-Cell Method (Electronic Version Only)
Galactic Dynamics
Strategy of the PIC Method
Leapfrog Method
Force
Examples
Exercises for Sec. 7.3
References
Introduction to Random Processes
Random Walks
Introduction
The Statistics of Random Walks
Exercises for Sec. 8.1
Thermal Equilibrium
Random Walks with Arbitrary Steps
Simulations
Thermal Equilibrium
Exercises for Sec. 8.2
The Rosenbluth-Teller-Metropolis Monte Carlo Method (Electronic Version Only)
Theory
Simulations
Exercises for Sec. 8.3
References
An Introduction to Mathematica (Electronic Version Only)
Starting Mathematica
Mathematica Calculations
Arithmetic
Exact vs. Approximate Results
Some Intrinsic Functions
Special Numbers
Complex Arithmetic
The Function N and Arbitrary-Precision Numbers
Exercises for Sec. 9.2
The Mathematica Front End and Kernel
Using Previous Results
The % Symbol
Variables
Pallets and Keyboard Equivalents
Lists, Vectors, and Matrices
Defining Lists, Vectors, and Matrices
Vectors and Matrix Operations
Creating Lists, Vectors, and Matrices with the Table Command
Operations on Lists
Exercises for Sec. 9.5
Plotting Results
The Plot Command
The Show Command
Plotting Several Curves on the Same Graph
The ListPlot Function
Parametric Plots
3D Plots
Animations
Add-On Packages
Exercises for Sec. 9.6
Help for Mathematica Users
Computer Algebra
Manipulating Expressions
Replacement
Defining Functions
Applying Functions
Delayed Evaluation of Functions
Putting Conditions on Function Definitions
Exercises for Sec. 9.8
Calculus
Derivatives
Power Series
Integration
Exercises for Sec. 9.9
Analytic Solution of Algebraic Equations
Solve and NSolve
Exercises for Sec. 9.10
Numerical Analysis
Numerical Solution of Algebraic Equations
Numerical Integration
Interpolation
Fitting
Exercises for Sec. 9.11
Summary of Basic Mathematica Commands
Elementary Functions
Using Previous Results; Substitution and Defining Variables
Lists, Tables, Vectors and Matrices
Graphics
Symbolic Mathematics
References
Finite-Differenced Derivatives
Index

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