Steven Krantz

Differential Equations (eBook, ePUB)

A Modern Approach with Wavelets

• Format: ePub

Produktbeschreibung

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. Because of the theory of wavelets, Fourier analysis is ever more important and central and applications are a driving force behind mathematics.

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Produktdetails

• Verlag: Taylor & Francis
• Seitenzahl: 481
• Erscheinungstermin: 3. Februar 2020
• Englisch
• ISBN-13: 9781000768466
• Artikelnr.: 58607173

Autorenporträt

Steven G. Krantz is a professor of mathematics at Washington University in St. Louis. He has previously taught at UCLA, Princeton University, and Pennsylvania State University. He has written more than 65 books and more than 175 scholarly papers and is the founding editor of the Journal of Geometric Analysis. An AMS Fellow, Dr. Krantz has been a recipient of the Chauvenet Prize, Beckenbach Book Award, and Kemper Prize. He received a Ph.D. from Princeton University.

Inhaltsangabe

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