Differential Equations : An Introduction to Modern Methods and Applications (3TH)

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Differential Equations : An Introduction to Modern Methods and Applications (3TH)

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  • 製本 Hardcover:ハードカバー版/ページ数 673 p.
  • 言語 ENG,ENG
  • 商品コード 9781118531778
  • DDC分類 515

Table of Contents

Chapter 1 Introduction                         1  (36)
1.1 Mathematical Models and Solutions 2 (10)
1.2 Qualitative Methods: Phase Lines and 12 (16)
Direction Fields
1.3 Definitions, Classification, and 28 (9)
Terminology
Chapter 2 First Order Differential Equations 37 (79)
2.1 Separable Equations 38 (7)
2.2 Linear Equations: Method of 45 (10)
Integrating Factors
2.3 Modeling with First Order Equations 55 (15)
2.4 Differences Between Linear and 70 (10)
Nonlinear Equations
2.5 Autonomous Equations and Population 80 (13)
Dynamics
2.6 Exact Equations and Integrating 93 (8)
Factors
2.7 Substitution Methods 101(15)
Projects
2.P.1 Harvesting a Renewable Resource 110(1)
2.P.2 A Mathematical Model of a 111(2)
Groundwater Contaminant Source
2.P.3 Monte Carlo Option Pricing: 113(3)
Pricing Financial Options by Flipping a
Coin
Chapter 3 Systems of Two First Order 116(87)
Equations
3.1 Systems of Two Linear Algebraic 117(12)
Equations
3.2 Systems of Two First Order Linear 129(16)
Differential Equations
3.3 Homogeneous Linear Systems with 145(22)
Constant Coefficients
3.4 Complex Eigenvalues 167(11)
3.5 Repeated Eigenvalues 178(11)
3.6 A Brief Introduction to Nonlinear 189(14)
Systems
Projects
3.P.1 Estimating Rate Constants for an 199(2)
Open Two-Compartment Model
3.P.2 A Blood-Brain Pharmacokinetic 201(2)
Model
Chapter 4 Second Order Linear Equations 203(91)
4.1 Definitions and Examples 203(13)
4.2 Theory of Second Order Linear 216(12)
Homogeneous Equations
4.3 Linear Homogeneous Equations with 228(13)
Constant Coefficients
4.4 Mechanical and Electrical Vibrations 241(11)
4.5 Nonhomogeneous Equations; Method of 252(9)
Undetermined Coefficients
4.6 Forced Vibrations, Frequency 261(13)
Response, and Resonance
4.7 Variation of Parameters 274(20)
Projects
4.P.1 A Vibration Insulation Problem 285(1)
4.P.2 Linearization of a Nonlinear 286(2)
Mechanical System
4.P.3 A Spring-Mass Event Problem 288(1)
4.P.4 Euler-Lagrange Equations 289(5)
Chapter 5 The Laplace Transform 294(83)
5.1 Definition of the Laplace Transform 295(9)
5.2 Properties of the Laplace Transform 304(7)
5.3 The Inverse Laplace Transform 311(9)
5.4 Solving Differential Equations with 320(8)
Laplace Transforms
5.5 Discontinuous Functions and Periodic 328(9)
Functions
5.6 Differential Equations with 337(7)
Discontinuous Forcing Functions
5.7 Impulse Functions 344(7)
5.8 Convolution Integrals and Their 351(10)
Applications
5.9 Linear Systems and Feedback Control 361(16)
Projects
5.P.1 An Electric Circuit Problem 371(1)
5.P.2 The Watt Governor, Feedback 372(5)
Control, and Stability
Chapter 6 Systems of First Order Linear 377(79)
Equations
6.1 Definitions and Examples 378(11)
6.2 Basic Theory of First Order Linear 389(10)
Systems
6.3 Homogeneous Linear Systems with 399(11)
Constant Coefficients
6.4 Nondefective Matrices with Complex 410(10)
Eigenvalues
6.5 Fundamental Matrices and the 420(11)
Exponential of a Matrix
6.6 Nonhomogeneous Linear Systems 431(7)
6.7 Defective Matrices 438(18)
Projects
6.P.1 Earthquakes and Tall Buildings 446(3)
6.P.2 Controlling a Spring-Mass System 449(7)
to Equilibrium
Chapter 7 Nonlinear Differential Equations 456(63)
and Stability
7.1 Autonomous Systems and Stability 456(10)
7.2 Almost Linear Systems 466(10)
7.3 Competing Species 476(12)
7.4 Predator-Prey Equations 488(8)
7.5 Periodic Solutions and Limit Cycles 496(10)
7.6 Chaos and Strange Attractors: The 506(13)
Lorenz Equations
Projects
7.P.1 Modeling of Epidemics 514(2)
7.P.2 Harvesting in a Competitive 516(2)
Environment
7.P.3 The Rossler System 518(1)
Chapter 8 Numerical Methods 519(36)
8.1 Numerical Approximations: Euler's 519(11)
Method
8.2 Accuracy of Numerical Methods 530(7)
8.3 Improved Euler and Runge--Kutta 537(9)
Methods
8.4 Numerical Methods for Systems of 546(9)
First Order Equations
Projects
8.P.1 Designing a Drip Dispenser for a 550(1)
Hydrology Experiment
8.p.2 Monte Carlo Option Pricing: 551(4)
Pricing Financial Options by Flipping a
Coin
Chapter 9 Series Solutions of Second Order
Equations (online only)
9.1 Review of Power Series
9.2 Series Solutions Near an Ordinary
Point, Part I
9.3 Series Solutions Near an Ordinary
Point, Part II
9.4 Regular Singular Points
9.5 Series Solutions Near a Regular
Singular Point, Part I
9.6 Series Solutions Near a Regular
Singular Point, Part II
9.7 Bessel's Equation Projects
9.P.1 Diffraction Through a Circular
Aperature
9.P.2 Hermite Polynomials and the
Quantum Mechanical Harmonic Oscillator
9.P.3 Perturbation Methods
Chapter 10 Orthogonal Functions, Fourier
Series, and Boundary Value Problems (online
only)
10.1 Orthogonal Families in the Space
PC[a, b]
10.2 Fourier Series
10.3 Elementary Two-Point Boundary Value
Problems
10.4 General Sturm--Liouville Boundary
Value Problems
10.5 Generalized Fourier Series and
Eigenfunction Expansions
10.6 Singular Boundary Value Problems
10.7 Convergence Issues
Chapter 11 Elementary Partial Differential
Equations (online only)
11.1 Terminology
11.2 Heat Conduction in a
Rod---Homogeneous Case
11.3 Heat Conduction in a
Rod---Nonhomogeneous Case
11.4 Wave Equation---Vibrations of an
Elastic String
11.5 Wave Equation---Vibrations of a
Circular Membrane
11.6 Laplace Equation Projects
11.p.1 Estimating the Diffusion
Coefficient in the Heat Equation
11.p.2 The Transmission Line Problem
11.p.3 Solving Poisson's Equation by
Finite Differences
11.p.4 Dynamic Behavior of a Hanging
Cable
11.p.5 Advection Dispersion: A Model
for Solute Transport in Saturated
Porous Media
11.p.6 Fisher's Equation for Population
Growth and Dispersion
Appendices (available on companion web
site)
11.a Derivation of the Heat Equation
11.b Derivation of the Wave Equation
APPENDIX A Matrices and Linear Algebra 555(46)
A.1 Matrices 555(9)
A.2 Systems of Linear Algebraic 564(17)
Equations, Linear Independence, and Rank
A.3 Determinants and Inverses 581(9)
A.4 The Eigenvalue Problem 590(11)
APPENDIX B Complex Variables (online only)
Review of Integration (online only)
Answers 601(63)
References 664(2)
Index 666