Numerical and Analytical Methods for Scientists and Engineers Using Mathematica (HAR/CDR)

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Numerical and Analytical Methods for Scientists and Engineers Using Mathematica (HAR/CDR)

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  • Wiley-Interscience(2003/04発売)
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  • ポイント 1,740pt
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  • 製本 Hardcover:ハードカバー版/ページ数 633 p.
  • 言語 ENG
  • 商品コード 9780471266105
  • DDC分類 530.150285

Full Description

* The electronic component of the book is based on the widely used and highly praised Mathematica software package.* Each chapter of the bookis a Mathematica notebook with links to web-based material.* The methods are applied to a range of problems taken from physics and engineering.* The book covers elementary and advaned numerical methods used in modern scientific computing.

Table of Contents

Preface                                            xiii
Ordinary Differential Equations in the 1 (86)
Physical Sciences
Introduction 1 (4)
Definitions 1 (4)
Exercises for Sec. 1.1 5 (1)
Graphical Solution of Initial-Value 5 (12)
Problems
Direction Fields; Existence and 5 (4)
Uniqueness of Solutions
Direction Fields for Second-Order ODEs: 9 (5)
Phase-Space Portraits
Exercises for Sec. 1.2 14 (3)
Analytic Solution of Initial-Value 17 (6)
Problems via DSolve
DSolve 17 (3)
Exercises for Sec. 1.3 20 (3)
Numerical Solution of Initial-Value 23 (39)
Problems
NDSolve 23 (4)
Error in Chaotic Systems 27 (4)
Euler's Method 31 (7)
The Predictor-Corrector Method of Order 38 (3)
2
Euler's Method for Systems of ODEs 41 (2)
The Numerical N-Body Problem: An 43 (7)
Introduction to Molecular Dynamics
Exercises for Sec. 1.4 50 (12)
Boundary-Value Problems 62 (8)
Introduction 62 (2)
Numerical Solution of Boundary-Value 64 (3)
Problems: The Shooting Method
Exercises for Sec. 1.5 67 (3)
Linear ODEs 70 (17)
The Principle of Superposition 70 (1)
The General Solution to the Homogeneous 71 (3)
Equation
Linear Differential Operators and 74 (4)
Linear Algebra
Inhomogeneous Linear ODEs 78 (6)
Exercises for Sec. 1.6 84 (2)
References 86 (1)
Fourier Series and Transforms 87 (104)
Fourier Representation of Periodic 87 (24)
Functions
Introduction 87 (3)
Fourier Coefficients and Orthogonality 90 (2)
Relations
Triangle Wave 92 (3)
Square Wave 95 (2)
Uniform and Nonuniform Convergence 97 (2)
Gibbs Phenomenon for the Square Wave 99 (3)
Exponential Notation for Fourier Series 102 (3)
Response of a Damped Oscillator to 105 (1)
Periodic Forcing
Fourier Analysis, Sound, and Hearing 106 (3)
Exercises for Sec. 2.1 109 (2)
Fourier Representation of Functions 111 (11)
Defined on a Finite Interval
Periodic Extension of a Function 111 (2)
Even Periodic Extension 113 (3)
Odd Periodic Extension 116 (2)
Solution of Boundary-Value Problems 118 (3)
Using Fourier Series
Exercises for Sec. 2.2 121 (1)
Fourier Transforms 122 (47)
Fourier Representation of Functions on 122 (7)
the Real Line
Fourier sine and cosine Transforms 129 (2)
Some Properties of Fourier Transforms 131 (4)
The Dirac δ-Function 135 (9)
Fast Fourier Transforms 144 (14)
Response of a Damped Oscillator to 158 (6)
General Forcing, Green's Function for
the Oscillator
Exercises for Sec. 2.3 164 (5)
Green's Functions 169 (22)
Introduction 169 (2)
Constructing the Green's Function from 171 (3)
Homogeneous Solutions
Discretized Green's Function I: 174 (4)
Initial-Value Problems by Matrix
Inversion
Green's Function for Boundary-Value 178 (3)
Problems
Discretized Green's Functions II: 181 (6)
Boundary-Value Problems by Matrix
Inversion
Exercises for Sec. 2.4 187 (3)
References 190 (1)
Introduction to Linear Partial Differential 191 (70)
Equations
Separation of Variables and Fourier 191 (40)
Series Methods in Solutions of the Wave
and Heat Equations
Derivation of the Wave Equation 191 (4)
Solution of the Wave Equation Using 195 (11)
Separation of Variables
Derivation of the Heat Equation 206 (4)
Solution of the Heat Equation Using 210 (14)
Separation of Variables
Exercises for Sec. 3.1 224 (7)
Laplace's Equation in Some Separable 231 (30)
Geometries
Existence and Uniqueness of the Solution 232 (1)
Rectangular Geometry 233 (5)
2D Cylindrical Geometry 238 (2)
Spherical Geometry 240 (7)
3D Cylindrical Geometry 247 (9)
Exercises for Sec. 3.2 256 (4)
References 260 (1)
Eigenmode Analysis 261 (94)
Generalized Fourier Series 261 (16)
Inner Products and Orthogonal Functions 261 (5)
Series of Orthogonal Functions 266 (2)
Eigenmodes of Hermitian Operators 268 (4)
Eigenmodes of Non-Hermitian Operators 272 (1)
Exercises for Sec. 4.1 273 (4)
Beyond Separation of Variables: The 277 (23)
General Solution of the 1D Wave and Heat
Equations
Standard Form for the PDE 278 (2)
Generalized Fourier Series Expansion 280 (14)
for the Solution
Exercises for Sec. 4.2 294 (6)
Poisson's Equation in Two and Three 300 (33)
Dimensions
Introduction. Uniqueness and Standard 300 (1)
Form
Green's Function 301 (1)
Expansion of g and φ in Eigenmodes 302 (2)
of the Laplacian Operator
Eigenmodes of 2 in Separable Geometries 304 (20)
Exercises for Sec. 4.3 324 (9)
The Wave and Heat Equations in Two and 333 (22)
Three Dimensions
Oscillations of a Circular Drumhead 334 (7)
Large-Scale Ocean Modes 341 (3)
The Rate of Cooling of the Earth 344 (2)
Exercises for Sec. 4.4 346 (8)
References 354 (1)
Partial Differential Equations in Infinite 355 (80)
Domains
Fourier Transform Methods 356 (40)
The Wave Equation in One Dimension 356 (3)
Dispersion; Phase and Group Velocities 359 (7)
Waves in Two and Three Dimensions 366 (20)
Exercises for Sec. 5.1 386 (10)
The WKB Method 396 (36)
WKB Analysis without Dispersion 396 (19)
WKB with Dispersion: Geometrical Optics 415 (9)
Exercises for Sec. 5.2 424 (8)
Wave Action (Electronic Version Only)
The Eikonal Equation
Conservation of Wave Action
Exercises for Sec. 5.3
References 432 (3)
Numerical Solution of Linear Partial 435 (76)
Differential Equations
The Galerkin Method 435 (29)
Introduction 435 (1)
Boundary-Value Problems 435 (16)
Time-Dependent Problems 451 (10)
Exercises for Sec. 6.1 461 (3)
Grid Methods 464 (46)
Time-Dependent Problems 464 (22)
Boundary-Value Problems 486 (18)
Exercises for Sec. 6.2 504 (6)
Numerical Eigenmode Methods (Electronic
Version Only)
Introduction
Grid-Method Eigenmodes
Galerkin-Method Eigenmodes
WKB Eigenmodes
Exercises for Sec. 6.3
References 510 (1)
Nonlinear Partial Differential Equations 511 (56)
The Method of Characteristics for 511 (25)
First-Order PDEs
Characteristics 511 (2)
Linear Cases 513 (16)
Nonlinear Waves 529 (5)
Exercises for Sec. 7.1 534 (2)
The KdV Equation 536 (30)
Shallow-Water Waves with Dispersion 536 (1)
Steady Solutions: Cnoidal Waves and 537 (9)
Solitons
Time-Dependent Solutions: The Galerkin 546 (8)
Method
Shock Waves: Burgers' Equation 554 (6)
Exercises for Sec. 7.2 560 (6)
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 566 (1)
Introduction to Random Processes 567 (50)
Random Walks 567 (25)
Introduction 567 (1)
The Statistics of Random Walks 568 (18)
Exercises for Sec. 8.1 586 (6)
Thermal Equilibrium 592 (23)
Random Walks with Arbitrary Steps 592 (6)
Simulations 598 (7)
Thermal Equilibrium 605 (4)
Exercises for Sec. 8.2 609 (6)
The Rosenbluth-Teller-Metropolis Monte
Carlo Method (Electronic Version Only)
Theory
Simulations
Exercises for Sec. 8.3
References 615 (2)
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
Appendix Finite-Differenced Derivatives 617 (4)
Index 621