A Very Applied First Course in Partial Differential Equations

Preface xvii

1 Introduction 1

2 The One-Dimensional Heat Equation 5
2.1 INTRODUCTION 5
2.2 DERIVATION OF HEAT CONDUCTION IN A ONE-DIMENSIONAL ROD 5
2.2.1 Derivation of the Mathematical Model 7
2.2.2 Initial Temperature 14
2.3 BOUNDARY CONDITIONS FOR A ONE-DIMENSIONAL ROD 17
2.3.1 Boundary Conditions of the First Kind 17
2.3.2 Boundary Conditions of the Second Kind 18
2.3.3 Boundary Conditions of the Third Kind 19
2.4 THE MAXIMUM PRINCIPLE AND UNIQUENESS 23
2.5 STEADY-STATE TEMPERATURE DISTRIBUTION 30

3 The One-Dimensional Wave Equation 41
3.1 INTRODUCTION 41
3.2 DERIVATION OF THE ONE-DIMENSIONAL WAVE EQUATION 41
3.3 BOUNDARY CONDITIONS 49
3.3.1 Boundary Conditions of the First Kind 50
3.3.2 Boundary Conditions of the Second Kind 50
3.3.3 Boundary Conditions of the Third Kind 50
3.4 CONSERVATION OF ENERGY FOR A VIBRATING STRING 54
3.5 METHOD OF CHARACTERISTICS 57
3.6 D'ALEMBERT'S SOLUTION TO THE ONE-DIMENSIONAL WAVE EQUATION 64

4 The Essentials of Fourier Series 73
4.1 INTRODUCTION 73
4.2 ELEMENTS OF LINEAR ALGEBRA 74
4.2.1 Vector Space 75
4.2.2 Linear Dependence, Linear Independence, and Basis 76
4.2.3 Orthogonality and Inner Product 78
4.2.4 Eigenvalues and Eigenvectors 83
4.2.5 Significance 85
4.3 A NEW SPACE: THE FUNCTION SPACE OF PIECEWISE SMOOTH FUNCTIONS 90
4.3.1 Inner Product, Orthogonality, and Basis in a Function Space 92
4.3.2 Definition of Trigonometric Fourier series 100
4.3.3 Fourier series Representation of Piecewise Smooth Functions 103
4.4 EVEN AND ODD FUNCTIONS AND FOURIER SERIES 119

5 Separation of Variables: The Homogeneous Problem 131
5.1 INTRODUCTION 131
5.2 OPERATORS: LINEAR AND HOMOGENEOUS EQUATIONS 132
5.2.1 Linear Operators 132
5.2.2 Linear Equations 136
5.3 SEPARATION OF VARIABLES: HEAT EQUATION 139
5.3.1 Spatial Problem Solution 141
5.3.2 Time Problem Solution 144
5.3.3 The Complete Solution 145
5.4 SEPARATION OF VARIABLES: WAVE EQUATION 153
5.4.1 Spatial Problem Solution 155
5.4.2 Time Problem Solution 156
5.4.3 The Complete Solution 156
5.5 THE MULTIDIMENSIONAL SPATIAL PROBLEM 164
5.5.1 Spatial Problem for $X(x)$ 167
5.5.2 Spatial Problem for $Y (y)$ 168
5.5.3 Time Problem 168
5.5.4 The Complete Solution 168
5.6 LAPLACE'S EQUATION 180
5.6.1 An Electrostatics Derivation of Laplace's Equation 180
5.6.2 Uniqueness of Solution 181
5.6.3 Laplace's Equation in Cartesian Coordinate System 182

6 The Calculus of Fourier Series 191
6.1 INTRODUCTION 191
6.2 FOURIER SERIES REPRESENTATION OF A FUNCTION: FOURIER SERIES AS A FUNCTION 191
6.3 DIFFERENTIATION OF FOURIER SERIES 195
6.4 INTEGRATION OF FOURIER SERIES 206
6.5 FOURIER SERIES AND THE GIBBS PHENOMENON 213

7 Separation of Variables: The Nonhomogeneous Problem 223
7 .1 INTRODUCTION 223
7.2 NONHOMOGENEOUS PDES WITH HOMOGENEOUS BCS 224
7.3 HOMOGENEOUS PDE WITH NONHOMOGENEOUS BCS 236
7.3.1 Homogeneous PDE-Nonhomogeneous Constant BCs 236
7.3.2 Homogeneous PDE-Nonhomogeneous Variable BCs 242
7.4 NONHOMOGENEOUS PDE AND BCS 258
7.5 SUMMARY 277

8 The Sturm-Liouville Eigenvalue Problem 279
8 .1 INTRODUCTION 279
8.2 DEFINITION OF THE STURM-LIOUVILLE EIGENVALUE PROBLEM 281
8.3 RAYLEIGH QUOTIENT 293
8.4 THE GENERAL PDE EXAMPLE 298
8.5 PROBLEMS INVOLVING HOMOGENEOUS BCS OF THE THIRD KIND 302

9 Solution of Linear Homogeneous Variable-Coefficient ODE 317
9 .1 INTRODUCTION 317
9.2 SOME FACTS ABOUT THE GENERAL SECOND-ORDER ODE 318
9.3 EULER'S EQUATION 320
9.4 BRIEF REVIEW OF POWER SERIES 325
9.5 THE POWER SERIES SOLUTION METHOD 327
9.6 LEGENDER'S EQUATION AND LEGENDRE POLYNOMIALS 334
9.7 METHOD OF FROBENIUS AND BESSEL'S EQUATION 341

10 Classical PDE Problems 351
10.1 INTRODUCTION 351
10.2 LAPLACE'S EQUATION 351
10.2.1 Laplace's equation in the Polar Coordinate System 352
10.2.2 Laplace's equation in the Spherical Coordinate System 356
10.3 TRANSVERSE VIBRATIONS OF A THIN BEAM 364
10.3.1 Derivation of the Beam Equation 364
10.3.2 Transverse Vibrations of a Simply Supported Thin Beam 367
10.4 HEAT CONDUCTION IN A CIRCULAR PLATE 373
10.5 SCHRODINGER'S EQUATION 380
10.6 THE TELEGRAPHER'S EQUATION 389
10.6.1 Development of the Telegrapher's Equation 389
10.6.2 Application of the Telegrapher's Equation to a Neuron 391
10.7 INTERESTING PROBLEMS IN DIFFUSION 398

11 Fourier Integrals and Transform Methods 403
11.1 INTRODUCTION 403
11.2 THE FOURIER INTEGRAL 404
11.2.1 Development of the Fourier Integral 405
11.2.2 The Fourier Sine and Cosine Integrals 414
11.3 THE LAPLACE TRANSFORM 421
11.3.1 Laplace transform Solution Method of ODEs 422
11.3.2 The Error Function 424
11.3.3 Laplace Transform Solution Method of PDEs 425
11.4 THE FOURIER TRANSFORM 439
11.4.1 Fourier Cosine and Sine Transforms 442
11.4.2 Fourier Transform Theorems 444
11.5 FOURIER TRANSFORM SOLUTION METHOD OF PDES 450

Appendices 462
A Summary of the Spatial Problem 463

B Proofs of Related Theorems 465
B.l THEOREMS FROM CHAPTER 2 465
B.l.l Leibniz's Formula 465
B.l.2 Maximum-Minimum Theorem 466
B.2 THEOREMS FROM CHAPTER 4 469
B.2.1 Eigenvectors of Distinct Eigenvalues Are Linearly Independent 469
B.2.2 Eigenvectors of Distinct Eigenvalues of an $n$ by $n$ Matrix Form a Basis for $\mathbb{R}^n$ 470
B.3 THEOREM FROM CHAPTER 5. .

C Basics from Ordinary Differential Equations 473
C.l SOME SOLUTION METHODS FOR FIRST-ORDER ODES 473
C.l.l First-Order ODE Where $k(t)$ Is a Constant . 473
C.l.2 First-Order ODE Where $k(t)$ Is a Function 475
C.2 SOME SOLUTION METHODS OF SECOND-ORDER ODES 479
C.2.1 Second-Order Linear Homogeneous ODEs 479
C.2.2 Second-Order Linear Nonhomogeneous ODEs 485

D Mathematical Notation 489

E Summary of Thermal Diffusivity of Common Materials 491

F Tables of Fourier and Laplace Transforms 493
F.l TABLES OF FOURIER, FOURIER COSINE, AND FOURIER SINE TRANSFORMS 494
F.2 TABLE OF LAPLACE TRANSFORMS 498

Bibliography 501

Index 504