Steven Krantz

Differential Equations

A Modern Approach with Wavelets

• Gebundenes Buch

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. 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.

Produktdetails

• Textbooks in Mathematics
• Verlag: CRC Press / Taylor & Francis
• Seitenzahl: 467
• Englisch
• Abmessung: 29mm x 156mm x 234mm
• Gewicht: 816g
• ISBN-13: 9780367444099
• Artikelnr.: 58564959

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 InstructorxvPreface for the Studentxvii1What Is a Differential Equation?11.1Introductory Remarks11.2A Taste of Ordinary Differential Equations51.3The Nature of Solutions71.4Separable Equations151.5First-Order, Linear Equations181.6Exact Equations241.7Orthogonal Trajectories and Curves301.8Homogeneous Equations361.9Integrating Factors411.10Reduction of Order461.10.1Dependent Variable Missing461.10.2Independent Variable Missing481.11The Hanging Chain and Pursuit Curves521.11.1The Hanging Chain521.11.2Pursuit Curves571.12Electrical Circuits621.13The Design of a Dialysis Machine67Problems for Review and Discovery722Second-Order Linear Equations772.1Second-Order Linear Equations with Constant Coefficients772.2The Method of Undetermined Coefficients85ix2.3The Method of Variation of Parameters902.4The Use of a Known Solution to Find Another952.5Vibrations and Oscillations1002.5.1Undamped Simple Harmonic Motion1002.5.2Damped Vibrations1022.5.3Forced Vibrations1062.5.4A Few Remarks About Electricity1092.6Newton's Law of Gravitation and Kepler's Laws1122.6.1Kepler's Second Law1162.6.2Kepler's First Law1182.6.3Kepler's Third Law1212.7Higher-Order Equations128Historical Note1352.8Bessel Functions and the Vibrating Membrane136Problems for Review and Discovery1423Power Series Solutions and Special Functions1453.1Introduction and Review of Power Series1453.1.1Review of Power Series1463.2Series Solutions of First-Order Equations1573.3Ordinary Points1633.4Regular Singular Points1733.5More on Regular Singular Points180Historical Note190Historical Note1923.6Steady State Temperature in a Ball193Problems for Review and Discovery1964Sturm-Liouville Problems and Boundary Value Problems1994.1What is a Sturm-Liouville Problem?1994.2Analyzing a Sturm-Liouville Problem2064.3Applications of the Sturm-Liouville Theory2134.4Singular Sturm-Liouville2204.5Some Ideas from Quantum Mechanics227Problems for Review and Discovery2315Numerical Methods2355.1Introductory Remarks2365.2The Method of Euler2385.3The Error Term2425.4An Improved Euler Method2465.5The Runge-Kutta Method2525.6A Constant Perturbation Method256Problems for Review and Discovery2606Fourier Series: Basic Concepts2656.1Fourier Coefficients2656.2Some Remarks about Convergence2756.3Even and Odd Functions: Cosine and Sine Series2826.4Fourier Series on Arbitrary Intervals2896.5Orthogonal Functions293Historical Note2996.6Introduction to the Fourier Transform3006.6.1Convolution and Fourier Inversion3096.6.2The Inverse Fourier Transform309Problems for Review and Discovery3127Laplace Transforms3177.0Introduction3177.1Applications to Differential Equations3217.2Derivatives and Integrals3277.3Convolutions3347.3.1Abel's Mechanics Problem3377.4The Unit Step and Impulse Functions342Historical Note3527.5Flow on an Impulsively Started Flat Plate353Problems for Review and Discovery3568Distributions3638.1Schwartz Distributions363Problems for Review and Discovery3719Wavelets3739.1Localization in the Time and Space Variables3739.2Building a Custom Fourier Analysis3769.3The Haar Basis3799.4Some Illustrative Examples3849.5Construction of a Wavelet Basis3949.5.1A Combinatorial Construction of the Daubechies Wavelets3989.5.2The Daubechies Wavelets from the Point of View of Fourier Analysis3999.5.3Wavelets as an Unconditional Basis4029.5.4Wavelets and Almost Diagonalizability4039.6The Wavelet Transform4069.7More on the Wavelet Transform4239.8Decomposition and its Obverse4299.9Some Applications4359.10Cumulative Energy and Entropy445Problems for Review and Discovery45110Partial Differential Equations and Boundary Value Problems45510.1Introduction and Historical Remarks45510.2Eigenvalues and the Vibrating String46010.2.1Boundary Value Problems46010.2.2Derivation of the Wave Equation46110.2.3Solution of the Wave Equation46310.3The Heat Equation46910.4The Dirichlet Problem for a Disc47810.4.1The Poisson Integral481Historical Note488Historical Note489Problems for Review and Discovery491Table of Notation495Glossary501Solutions to Selected Exercises527Bibliography527Index530