Numerical and Analytical Methods for Scientists and Engineers Using Mathematica

by Dubin, Daniel

• ISBN13: 9780471266105
• ISBN10: 0471266108
• Edition: 1st
• Format: Hardcover
• Copyright: 2003-05-05
• Publisher: Wiley-Interscience
• More Book Details
• Purchase Benefits

Summary

* 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.

Author Biography

DANIEL DUBIN holds a Ph.D. in Astrophysics from Princeton University and is a Fellow of the American Physical Society. He is currently a Professor of Physics at the University of California at San Diego.

Table of Contents

Preface	xiii
Ordinary Differential Equations in the Physical Sciences	1	(86)
Introduction	1	(4)
Definitions	1	(4)
Exercises for Sec. 1.1	5	(1)
Graphical Solution of Initial-Value Problems	5	(12)
Direction Fields; Existence and Uniqueness of Solutions	5	(4)
Direction Fields for Second-Order ODEs: Phase-Space Portraits	9	(5)
Exercises for Sec. 1.2	14	(3)
Analytic Solution of Initial-Value Problems via DSolve	17	(6)
DSolve	17	(3)
Exercises for Sec. 1.3	20	(3)
Numerical Solution of Initial-Value Problems	23	(39)
NDSolve	23	(4)
Error in Chaotic Systems	27	(4)
Euler's Method	31	(7)
The Predictor-Corrector Method of Order 2	38	(3)
Euler's Method for Systems of ODEs	41	(2)
The Numerical N-Body Problem: An Introduction to Molecular Dynamics	43	(7)
Exercises for Sec. 1.4	50	(12)
Boundary-Value Problems	62	(8)
Introduction	62	(2)
Numerical Solution of Boundary-Value Problems: The Shooting Method	64	(3)
Exercises for Sec. 1.5	67	(3)
Linear ODEs	70	(17)
The Principle of Superposition	70	(1)
The General Solution to the Homogeneous Equation	71	(3)
Linear Differential Operators and Linear Algebra	74	(4)
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 Functions	87	(24)
Introduction	87	(3)
Fourier Coefficients and Orthogonality Relations	90	(2)
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 Periodic Forcing	105	(1)
Fourier Analysis, Sound, and Hearing	106	(3)
Exercises for Sec. 2.1	109	(2)
Fourier Representation of Functions Defined on a Finite Interval	111	(11)
Periodic Extension of a Function	111	(2)
Even Periodic Extension	113	(3)
Odd Periodic Extension	116	(2)
Solution of Boundary-Value Problems Using Fourier Series	118	(3)
Exercises for Sec. 2.2	121	(1)
Fourier Transforms	122	(47)
Fourier Representation of Functions on the Real Line	122	(7)
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 General Forcing, Green's Function for the Oscillator	158	(6)
Exercises for Sec. 2.3	164	(5)
Green's Functions	169	(22)
Introduction	169	(2)
Constructing the Green's Function from Homogeneous Solutions	171	(3)
Discretized Green's Function I: Initial-Value Problems by Matrix Inversion	174	(4)
Green's Function for Boundary-Value Problems	178	(3)
Discretized Green's Functions II: Boundary-Value Problems by Matrix Inversion	181	(6)
Exercises for Sec. 2.4	187	(3)
References	190	(1)
Introduction to Linear Partial Differential Equations	191	(70)
Separation of Variables and Fourier Series Methods in Solutions of the Wave and Heat Equations	191	(40)
Derivation of the Wave Equation	191	(4)
Solution of the Wave Equation Using Separation of Variables	195	(11)
Derivation of the Heat Equation	206	(4)
Solution of the Heat Equation Using Separation of Variables	210	(14)
Exercises for Sec. 3.1	224	(7)
Laplace's Equation in Some Separable Geometries	231	(30)
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 General Solution of the 1D Wave and Heat Equations	277	(23)
Standard Form for the PDE	278	(2)
Generalized Fourier Series Expansion for the Solution	280	(14)
Exercises for Sec. 4.2	294	(6)
Poisson's Equation in Two and Three Dimensions	300	(33)
Introduction. Uniqueness and Standard Form	300	(1)
Green's Function	301	(1)
Expansion of g and φ in Eigenmodes of the Laplacian Operator	302	(2)
Eigenmodes of 2 in Separable Geometries	304	(20)
Exercises for Sec. 4.3	324	(9)
The Wave and Heat Equations in Two and Three Dimensions	333	(22)
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 Domains	355	(80)
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 Differential Equations	435	(76)
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 First-Order PDEs	511	(25)
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 Solitons	537	(9)
Time-Dependent Solutions: The Galerkin Method	546	(8)
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