Differential Equations: A Modern Approach with Wavelets (Textbooks in Mathematics)

482 Pages·2020·13.9 MB·English

by  Steven Krantz

Differential Equations
Textbooks in Mathematics
Series editors:

Al Boggess and Ken Rosen

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AN ELEMENTARY TRANSITION TO ABSTRACT MATHEMATICS
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AN INTRODUCTION TO MATHEMATICAL PROOFS
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MATHEMATICAL MODELING WITH EXCEL
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THE SHAPE OF SPACE, THIRD EDITION
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CHROMATIC GRAPH THEORY, SECOND EDITION
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PARTIAL DIFFERENTIAL EQUATIONS: ANALYTICAL METHODS AND
APPLICATIONS
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ADVANCED PROBLEM SOLVING USING MAPLE: APPLIED MATHEMATICS,
OPERATION RESEARCH, BUSINESS ANALYTICS, AND DECISION ANALYSIS
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DIFFERENTIAL EQUATIONS: A MODERN APPROACH WITH WAVELETS
Steven G. Krantz
https://www.crcpress.com/Textbooks-in-Mathematics/book-series/CANDHTEXBOOMTH
Differential Equations

A Modern Approach with Wavelets
Steven G. Krantz
CRC Press
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Contents
Preface for the Instructor xi

Preface for the Student xiii

1 What Is a Differential Equation? 1

1.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 1

1.2 A Taste of Ordinary Differential Equations . . . . . . . . . . 4

1.3 The Nature of Solutions . . . . . . . . . . . . . . . . . . . . . 6

1.4 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . 13

1.5 First-Order, Linear Equations . . . . . . . . . . . . . . . . . 16

1.6 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.7 Orthogonal Trajectories and Families of Curves . . . . . . . 26

1.8 Homogeneous Equations . . . . . . . . . . . . . . . . . . . . 31

1.9 Integrating Factors . . . . . . . . . . . . . . . . . . . . . . . 35

1.10 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . 39

1.10.1 Dependent Variable Missing . . . . . . . . . . . . . . . 40

1.10.2 Independent Variable Missing . . . . . . . . . . . . . . 42

1.11 Hanging Chain . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1.11.1 The Hanging Chain . . . . . . . . . . . . . . . . . . . 45

1.11.2 Pursuit Curves . . . . . . . . . . . . . . . . . . . . . . 50

1.12 Electrical Circuits . . . . . . . . . . . . . . . . . . . . . . . . 54

1.13 The Design of a Dialysis Machine . . . . . . . . . . . . . . . 58

Problems for Review and Discovery . . . . . . . . . . . . . . . . . 62

2 Second-Order Linear Equations 67

2.1 Second-Order Linear Equations with Constant

Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.2 The Method of Undetermined Coefficients . . . . . . . . . . 73

2.3 The Method of Variation of Parameters . . . . . . . . . . . . 78

2.4 The Use of a Known Solution to Find Another . . . . . . . . 82

2.5 Vibrations and Oscillations . . . . . . . . . . . . . . . . . . . 86

2.5.1 Undamped Simple Harmonic Motion . . . . . . . . . . 86

2.5.2 Damped Vibrations . . . . . . . . . . . . . . . . . . . 88

2.5.3 Forced Vibrations . . . . . . . . . . . . . . . . . . . . 92

2.5.4 A Few Remarks about Electricity . . . . . . . . . . . . 94

2.6 Newton’s Law of Gravitation and Kepler’s Laws . . . . . . . 97

vii
viii CONTENTS
2.6.1 Kepler’s Second Law . . . . . . . . . . . . . . . . . . . 101

2.6.2 Kepler’s First Law . . . . . . . . . . . . . . . . . . . . 103

2.6.3 Kepler’s Third Law. . . . . . . . . . . . . . . . . . . . 106

2.7 Higher-Order Coupled Harmonic Oscillators . . . . . . . . . 111

Historical Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

2.8 Bessel Functions and the Vibrating Membrane . . . . . . . . 118

Problems for Review and Discovery . . . . . . . . . . . . . . . . . 122

3 Power Series Solutions and Special Functions 125

3.1 Introduction and Review of Power Series . . . . . . . . . . . 125

3.1.1 Review of Power Series . . . . . . . . . . . . . . . . . 126

3.2 Series Solutions of First-Order Differential Equations . . . . 136

3.3 Second-Order Linear Equations: Ordinary Points . . . . . . . 141

3.4 Regular Singular Points . . . . . . . . . . . . . . . . . . . . . 149

3.5 More on Regular Singular Points . . . . . . . . . . . . . . . . 155

Historical Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Historical Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

3.6 Steady-State Temperature in a Ball . . . . . . . . . . . . . . 165

Problems for Review and Discovery . . . . . . . . . . . . . . . . . 168

4 Sturm–Liouville Problems and Boundary Value

Problems 171

4.1 What Is a Sturm–Liouville Problem? . . . . . . . . . . . . . 171

4.2 Analyzing a Sturm–Liouville Problem . . . . . . . . . . . . . 177

4.3 Applications of the Sturm–Liouville Theory . . . . . . . . . . 182

4.4 Singular Sturm–Liouville . . . . . . . . . . . . . . . . . . . . 188

4.5 Some Ideas from Quantum Mechanics . . . . . . . . . . . . . 195

Problems for Review and Discovery . . . . . . . . . . . . . . . . . 198

5 Numerical Methods 201

5.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . 202

5.2 The Method of Euler . . . . . . . . . . . . . . . . . . . . . . 203

5.3 The Error Term . . . . . . . . . . . . . . . . . . . . . . . . . 207

5.4 An Improved Euler Method . . . . . . . . . . . . . . . . . . . 211

5.5 The Runge–Kutta Method . . . . . . . . . . . . . . . . . . . 215

5.6 A Constant Perturbation Method for Linear,

Second-Order Equations . . . . . . . . . . . . . . . . . . . . 219

Problems for Review and Discovery . . . . . . . . . . . . . . . . . 222

6 Fourier Series: Basic Concepts 227

6.1 Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . 227

6.2 Some Remarks about Convergence . . . . . . . . . . . . . . . 237

6.3 Even and Odd Functions: Cosine and Sine Series . . . . . . . 242

6.4 Fourier Series on Arbitrary Intervals . . . . . . . . . . . . . . 248

6.5 Orthogonal Functions . . . . . . . . . . . . . . . . . . . . . . 252

Historical Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
CONTENTS ix

6.6 Introduction to the Fourier Transform . . . . . . . . . . . . . 258

6.6.1 Convolution and Fourier Inversion . . . . . . . . . . . 266

6.6.2 The Inverse Fourier Transform . . . . . . . . . . . . . 266

Problems for Review and Discovery . . . . . . . . . . . . . . . . . 268

7 Laplace Transforms 273

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

7.2 Applications to Differential Equations . . . . . . . . . . . . . 276

7.3 Derivatives and Integrals of Laplace Transforms . . . . . . . 281

7.4 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 288

7.4.1 Abel’s Mechanics Problem . . . . . . . . . . . . . . . . 290

7.5 The Unit Step and Impulse Functions . . . . . . . . . . . . . 295

Historical Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

7.6 Flow Initiated by an Impulsively Started Flat Plate . . . . . 304

Problems for Review and Discovery . . . . . . . . . . . . . . . . . 307

8 Distributions 313

8.1 Schwartz Distributions . . . . . . . . . . . . . . . . . . . . . 313

8.1.1 The Topology of the Space . . . . . . . . . . . . . . 314

S
8.1.2 Algebraic Properties of Distributions . . . . . . . . . . 316

8.1.3 The Fourier Transform . . . . . . . . . . . . . . . . . . 317

8.1.4 Other Spaces of Distributions . . . . . . . . . . . . . . 317

8.1.5 More on the Topology of and . . . . . . . . . . . 320

′
D D
Problems for Review and Discovery . . . . . . . . . . . . . . . . . 321

9 Wavelets 323

9.1 Localization in Both Variables . . . . . . . . . . . . . . . . . 323

9.2 Building a Custom Fourier Analysis . . . . . . . . . . . . . . 326

9.3 The Haar Basis . . . . . . . . . . . . . . . . . . . . . . . . . 328

9.4 Some Illustrative Examples . . . . . . . . . . . . . . . . . . . 333

9.5 Construction of a Wavelet Basis . . . . . . . . . . . . . . . . 343

9.5.1 A Combinatorial Construction of the Daubechies

Wavelets. . . . . . . . . . . . . . . . . . . . . . . . . . 346

9.5.2 The Daubechies Wavelets from the Point of View

of Fourier Analysis . . . . . . . . . . . . . . . . . . . . 348

9.5.3 Wavelets as an Unconditional Basis . . . . . . . . . . 350

9.5.4 Wavelets and Almost Diagonalizability . . . . . . . . . 352

9.6 The Wavelet Transform . . . . . . . . . . . . . . . . . . . . . 355

9.7 More on the Wavelet Transform . . . . . . . . . . . . . . . . 369

9.7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 374

9.8 Decomposition and Its Obverse . . . . . . . . . . . . . . . . . 375

9.9 Some Applications . . . . . . . . . . . . . . . . . . . . . . . . 381

9.10 Cumulative Energy and Entropy . . . . . . . . . . . . . . . . 390

Problems for Review and Discovery . . . . . . . . . . . . . . . . . 394