#
Differential Equations

A Modern Approach with Wavelets

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## Book Description

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.

## Table of Contents

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

## Author(s)

### Biography

**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.