A Very Applied First Course in Partial Differential Equations

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Front Matter

     A Very Applied First Course in Partial Differential Equations

     © 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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