Cover
Front Matter
A Very Applied First Course in Partial Differential Equations
ยฉ 2002 Prentice-Hal
ISBN: 0-13-030417-4
QA377.K38 2002 515'.353-dc21
LCCN 2001040032
Dedication
Contents
List of Figures
Preface
Course Outline
Acknowledgments
Chapter 1 Introduction
EXERCISES 1
Chapter 2 The One-Dimensional Heat Equation
2.1 INTRODUCTION
2.2 DERIVATION OF HEAT CONDUCTION IN A ONE-DIMENSIONAL ROD
2.2.1 Derivation of the Mathematical Model
2.2.2 Initial Temperature
EXERCISES 2.2
2.3 BOUNDARY CONDITIONS FOR A ONE-DIMENSIONAL ROD
2.3.1 Boundary Conditions of the First Kind
2.3.2 Boundary Conditions of the Second Kind
2.3.3 Boundary Conditions of the Third Kind
EXERCISES 2.3
2.4 THE MAXIMUM PRINCIPLE AND UNIQUENESS
EXERCISES 2.4
2.5 STEADY-STATE TEMPERATURE DISTRIBUTION
EXERCISES 2.5
Chapter 3 The One-Dimensional Wave Equation
3.1 INTRODUCTION
3.2 DERIVATION OF THE ONE-DIMENSIONAL WAVE EQUATION
EXERCISES 3.2
3.3 BOUNDARY CONDITIONS
3.3.1 Boundary Conditions of the First Kind
3.3.2 Boundary Conditions of the Second Kind
3.3.3 Boundary Conditions of the Third Kind
EXERCISES 3.3
3.4 CONSERVATION OF ENERGY FOR A VIBRATING STRING
EXERCISES 3.4
3.5 FIRST-ORDER PDES: METHOD OF CHARACTERISTICS
EXERCISES 3.5
3.6 D'ALEMBERT'S SOLUTION TO THE ONE-DIMENSIONAL WAVE EQUATION
EXERCISES 3.6
Chapter 4 The Essentials of Fourier Series
4.1 INTRODUCTION
4.2 ELEMENTS OF LINEAR ALGEBRA
4.2.1 Vector Space
4.2.2 Linear Dependence, Linear Independence, and Basis
4.2.3 Orthogonality and Inner Product
4.2.4 Eigenvalues and Eigenvectors
4.2.5 Significance
EXERCISES 4.2
4.3 A NEW SPACE: THE FUNCTION SPACE OF PIECEWISE SMOOTH FUNCTIONS
4.3.1 Inner Product, Orthogonality, and Basis in a Function Space
4.3.2 Definition of Trigonometric Fourier series
4.3.3 Fourier series Representation of P iecewise Smooth Functions
EXERCISES 4.3
4.4 EVEN AND ODD FUNCTIONS AND FOURIER SERIES
EXERCISES 4.4
Chapter 5 Separation of Variables: The Homogeneous Problem
5.1 INTRODUCTION
5.2 OPERATORS: LINEAR AND HOMOGENEOUS EQUATIONS
5.2.1 Linear Operators
5.2.2 Linear Equations
EXERCISES 5.2
5.3 SEPARATION OF VARIABLES: THE HEAT EQUATION IN A ONE-DIMENSIONAL ROD
5.3.1 Spatial Problem Solution
5.3.2 Time Problem Solution
5.3.3 The Complete Solution
EXERCISES 5.3
5.4 SEPARATION OF VARIABLES: THE WAVE EQUATION IN A ONE-DIMENSIONAL STRING
5.4.1 Spatial Problem Solution
5.4.2 Time Problem Solution
5.4.3 The Complete Solution
EXERCISES 5.4
5.5 THE MULTIDIMENSIONAL SPATIAL PROBLEM
5.5.1 Spatial Problem for X (x)
5.5.2 Spatial Problem for Y(y)
5.5.3 Time Problem
5.5.4 The Complete Solution
EXERCISES 5.5
5.6 LAPLACE'S EQUATION
5.6.1 An Electrostatics Derivation of Laplace's Equatio
5.6.2 Uniqueness of Solution
5.6.3 Laplace's Equation in Cartesian Coordinate System
EXERCISES 5.6
Chapter 6 The Calculus of Fourier Series
6.1 INTRODUCTION
6.2 FOURIER SERIES REPRESENTATION OF A FUNCTION: FOURIER SERIES AS A FUNCTION
6.3 DIFFERENTIATION OF FOURIER SERIES
EXERCISES 6.3
6.4 INTEGRATION OF FOURIER SERIES
EXERCISES 6.4
6.5 FOURIER SERIES AND THE GIBBS PHENOMENON
EXERCISES 6.5
Chapter 7 Separation of Variables: The Nonhomogeneous Problem
7.1 INTRODUCTION
7.2 NONHOMOGENEOUS PDES WITH HOMOGENEOUS BCS
EXERCISES 7.2
7.3 HOMOGENEOUS PDE WITH NONHOMOGENEOUS BCS
7.3.1 Homogeneous PDE Nonhomogeneous Constant BCs
7.3.2 Homogeneous PDE Nonhomogeneous Variable BCs
EXERCISES 7.3
7.4 NONHOMOGENEOUS PDE AND BCs
EXERCISES 7.4
7.5 SUMMARY
Chapter 8 The Sturm-Liouville Eigenvalue Problem
8.1 INTRODUCTION
8.2 DEFINITION OF THE STURM-LIOUVILLE EIGENVALUE PROBLEM
EXERCISES 8.2
8.3 RAYLEIGH QUOTIENT
EXERCISES 8.3
8.4 THE GENERAL PDE EXAMPLE
EXERCISES 8.4
8.5 PROBLEMS INVOLVING HOMOGENEOUS BCS OF THE THIRD KIND
EXERCISES 8.5
Chapter 9 Solution of Linear Homogeneous Variable-Coefficient ODE
9.1 INTRODUCTION
9.2 SOME FACTS ABOUT THE GENERAL SECOND-ORDER ODE
EXERCISES 9.2
9.3 EULER'S EQUATION
EXERCISES 9.3
9.4 BRIEF REVIEW OF POWER SERIES
EXERCISES 9.4
9.5 THE POWER SERIES SOLUTION METHOD
EXERCISES 9.5
9.6 LEGENDER'S EQUATION AND LEGENDRE POLYNOMIALS
EXERCISES 9.6
9.7 METHOD OF FROBENIUS AND BESSEL'S EQUATION
EXERCISES 9.7
Chapter 10 Classical PDE Problems
10.1 INTRODUCTION
10.2 LAPLACE'S EQUATION
10.2.1 Laplace's equation in the Polar Coordinate System
10.2.2 Laplace's equation in the Spherical Coordinate System
EXERCISES 10.2
10.3 TRANSVERSE VIBRATIONS OF A THIN BEAM
10.3.1 Derivation of the Beam Equation
10.3.2 Transverse Vibrations of a Simply Supported Thin Beam
EXERCISES 10.3
10.4 HEAT CONDUCTION IN A CIRCULAR PLATE
EXERCISES 10.4
10.5 SCHRODINGER'S EQUATION
EXERCISES 10.5
10.6 THE TELEGRAPHER'S EQUATION
10.6.1 Development of the Telegrapher's Equation
10.6.2 Application of the Telegrapher's Equation to a Neuron
EXERCISES 10.6
10.7 INTERESTING PROBLEMS IN DIFFUSION
EXERCISES 10.7
Chapter 11 Fourier Integrals and Transform Methods
11.1 INTRODUCTION
11.2 THE FOURIER INTEGRAL
11.2.1 Development of the Fourier Integral
11.2.2 The Fourier Sine and Cosine Integrals
EXERCISES 11.2
11.3 THE LAPLACE TRANSFORM
11.3.1 Laplace transform Solution Method of ODEs
11.3.2 The Error Function
11.3.3 Laplace Transform Solution Method of PDEs
EXERCISES 11.3
11.4 THE FOURIER TRANSFORM
11.4.1 Fourier Cosine and Sine Transforms
11.4.2 Fourier Transform Theorems
EXERCISES 11.4
11.5 FOURIER TRANSFORM SOLUTION METHOD OF PDES
EXERCISES 11.5
Back Matter
Appendix A Summary of the Spatial Problem
Appendix B Proofs of Related Theorems
B.1 THEOREMS FROM CHAPTER 2
B.1.1 Leibniz's Formula
B.1.2 Maximum-Minimum Theorem
B.2 THEOREMS FROM CHAPTER 4
B.2.1 Eigenvectors of Distinct Eigenvalues Are Linearly Independen
B.2.2 Eigenvectors of Distinct Eigenvalues of an n by n Matrix Form a Basis for Rn
B.3 THEOREM FROM CHAPTER 5
Appendix C Basics from Ordinary Differential Equations
C.1 SOME SOLUTION METHODS FOR FIRST-ORDER ODES
C.1.1 First-Order ODE Where k(t) Is a Constant
C.1.2 First-Order ODE Where k(t) Is a Function
C.2 SOME SOLUTION METHODS OF SECOND-ORDER ODES
C.2.1 Second-Order Linear Homogeneous ODES
C.2.2 Second-Order Linear Nonhomogeneous ODES
Appendix D Mathematical Notation
Appendix E Summary of Thermal Diffusivity of Common Materials
Appendix F Tables of Fourier and LaplaceTransforms
F.1 TABLES OF FOURIER, FOURIER COSINE, AND FOURIER SINE TRANSFORMS
F.2 TABLE OF LAPLACE TRANSFORMS
Bibliography
Index
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