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Differential Equations: A Modern Approach with Wavelets

Differential Equations: A Modern Approach with Wavelets

Autorzy
Wydawnictwo Taylor & Francis Ltd
Data wydania 21/01/2020
Liczba stron 467
Forma publikacji książka w twardej oprawie
Poziom zaawansowania Dla szkół wyższych i kształcenia podyplomowego
Język angielski
ISBN 9780367444099
Kategorie Rachunek różniczkowy i formuły, Nauki środowiskowe, inżynieria i technologia
609.00 PLN (z VAT)
$136.99 / €130.57 / £113.35 /
Produkt na zamówienie
Dostawa 5-6 tygodni
Ilość
Do schowka

Opis książki

This new book from one of the most published authors in all of mathematics is an attempt to offer a new, more modern take on the Differential Equations course. The world is changing. Because of the theory of wavelets, Fourier analysis is ever more important and central. And applications are a driving force behind much of mathematics.This text text presents a more balanced picture. The text covers differential equations (both ordinary and partial), Fourier analysis and applications in equal measure and with equal weight. The Riemann integral is used throughout. We do not assume that the student knows any functional analysis. We likewise do not assume that the student has had a course in undergraduate real analysis. To make the book timely and exciting, a substantial chapter on basic properties of wavelets, with applications to signal processing and image processing is included. This should give students and instructors alike a taste of what is happening in the subject today.

Differential Equations: A Modern Approach with Wavelets

Spis treści

Preface for the Instructor xv

Preface for the Student xvii

1 What Is a Differential Equation? 1

1.1 Introductory Remarks 1

1.2 A Taste of Ordinary Differential Equations 5

1.3 The Nature of Solutions 7

1.4 Separable Equations 15

1.5 First-Order, Linear Equations 18

1.6 Exact Equations 24

1.7 Orthogonal Trajectories and Curves 30

1.8 Homogeneous Equations 36

1.9 Integrating Factors 41

1.10 Reduction of Order 46

1.10.1 Dependent Variable Missing 46

1.10.2 Independent Variable Missing 48

1.11 The Hanging Chain and Pursuit Curves 52

1.11.1 The Hanging Chain 52

1.11.2 Pursuit Curves 57

1.12 Electrical Circuits 62

1.13 The Design of a Dialysis Machine 67

Problems for Review and Discovery 72

2 Second-Order Linear Equations 77

2.1 Second-Order Linear Equations with Constant Coefficients 77

2.2 The Method of Undetermined Coefficients 85

ix

2.3 The Method of Variation of Parameters 90

2.4 The Use of a Known Solution to Find Another 95

2.5 Vibrations and Oscillations 100

2.5.1 Undamped Simple Harmonic Motion 100

2.5.2 Damped Vibrations 102

2.5.3 Forced Vibrations 106

2.5.4 A Few Remarks About Electricity 109

2.6 Newton's Law of Gravitation and Kepler's Laws 112

2.6.1 Kepler's Second Law 116

2.6.2 Kepler's First Law 118

2.6.3 Kepler's Third Law 121

2.7 Higher-Order Equations 128

Historical Note 135

2.8 Bessel Functions and the Vibrating Membrane 136

Problems for Review and Discovery 142

3 Power Series Solutions and Special Functions 145

3.1 Introduction and Review of Power Series 145

3.1.1 Review of Power Series 146

3.2 Series Solutions of First-Order Equations 157

3.3 Ordinary Points 163

3.4 Regular Singular Points 173

3.5 More on Regular Singular Points 180

Historical Note 190

Historical Note 192

3.6 Steady State Temperature in a Ball 193

Problems for Review and Discovery 196

4 Sturm-Liouville Problems and Boundary Value Problems 199

4.1 What is a Sturm-Liouville Problem? 199

4.2 Analyzing a Sturm-Liouville Problem 206

4.3 Applications of the Sturm-Liouville Theory 213

4.4 Singular Sturm-Liouville 220

4.5 Some Ideas from Quantum Mechanics 227

Problems for Review and Discovery 231

5 Numerical Methods 235

5.1 Introductory Remarks 236

5.2 The Method of Euler 238

5.3 The Error Term 242

5.4 An Improved Euler Method 246

5.5 The Runge-Kutta Method 252

5.6 A Constant Perturbation Method 256

Problems for Review and Discovery 260

6 Fourier Series: Basic Concepts 265

6.1 Fourier Coefficients 265

6.2 Some Remarks about Convergence 275

6.3 Even and Odd Functions: Cosine and Sine Series 282

6.4 Fourier Series on Arbitrary Intervals 289

6.5 Orthogonal Functions 293

Historical Note 299

6.6 Introduction to the Fourier Transform 300

6.6.1 Convolution and Fourier Inversion 309

6.6.2 The Inverse Fourier Transform 309

Problems for Review and Discovery 312

7 Laplace Transforms 317

7.0 Introduction 317

7.1 Applications to Differential Equations 321

7.2 Derivatives and Integrals 327

7.3 Convolutions 334

7.3.1 Abel's Mechanics Problem 337

7.4 The Unit Step and Impulse Functions 342

Historical Note 352

7.5 Flow on an Impulsively Started Flat Plate 353

Problems for Review and Discovery 356

8 Distributions 363

8.1 Schwartz Distributions 363

Problems for Review and Discovery 371

9 Wavelets 373

9.1 Localization in the Time and Space Variables 373

9.2 Building a Custom Fourier Analysis 376

9.3 The Haar Basis 379

9.4 Some Illustrative Examples 384

9.5 Construction of a Wavelet Basis 394

9.5.1 A Combinatorial Construction of the Daubechies Wavelets398

9.5.2 The Daubechies Wavelets from the Point of View of Fourier Analysis 399

9.5.3 Wavelets as an Unconditional Basis 402

9.5.4 Wavelets and Almost Diagonalizability 403

9.6 The Wavelet Transform 406

9.7 More on the Wavelet Transform 423

9.8 Decomposition and its Obverse 429

9.9 Some Applications 435

9.10 Cumulative Energy and Entropy 445

Problems for Review and Discovery 451

10 Partial Differential Equations and Boundary Value Problems455

10.1 Introduction and Historical Remarks 455

10.2 Eigenvalues and the Vibrating String 460

10.2.1 Boundary Value Problems 460

10.2.2 Derivation of the Wave Equation 461

10.2.3 Solution of the Wave Equation 463

10.3 The Heat Equation 469

10.4 The Dirichlet Problem for a Disc 478

10.4.1 The Poisson Integral 481

Historical Note 488

Historical Note 489

Problems for Review and Discovery 491

Table of Notation 495

Glossary 501

Solutions to Selected Exercises 527

Bibliography 527

Index 530

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