Intended for a college senior or first-year graduate-level course in partial differential equations, this text offers students in mathematics, engineering, and the applied sciences a solid foundation for advanced studies in mathematics. Classical topics presented in a modern context include coverage of integral equations and basic scattering theory. This complete and accessible treatment includes a variety of examples of inverse problems arising from improperly posed applications. Exercises at the ends of chapters, many with answers, offer a clear progression in developing an understanding of this essential area of mathematics. 1988 edition.
Author(s): David Colton
Publisher: Dover
Year: 2012
Language: English
Pages: 661
Tags: Partial Differential Equations
Title Page......Page 2
Copyright Page......Page 4
Preface......Page 5
ACKNOWLEDGMENTS......Page 8
Table of Contents......Page 9
Chapter 1 - Introduction......Page 10
1.1 PHYSICAL EXAMPLES......Page 12
1.2 FIRST ORDER LINEAR EQUATIONS......Page 21
1.3 CLASSIFICATION OF SECOND ORDER EQUATIONS AND CANONICAL FORMS......Page 31
1.4 FOURIER SERIES AND INTEGRALS......Page 49
* 1.5 ANALYTIC FUNCTIONS......Page 70
1.6 A BRIEF HISTORY OF THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS......Page 111
2.1 THE WAVE EQUATION IN TWO INDEPENDENT VARIABLES......Page 130
2.2 THE CAUCHY PROBLEM FOR HYPERBOLIC EQUATIONS IN TWO INDEPENDENT VARIABLES......Page 137
2.3 THE CAUCHY PROBLEM FOR THE WAVE EQUATION IN MORE THAN TWO INDEPENDENT VARIABLES......Page 146
2.4 THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE WAVE EQUATION IN TWO INDEPENDENT VARIABLES......Page 162
2.5 FOURIER’S METHOD FOR THE WAVE EQUATION IN THREE INDEPENDENT VARIABLES......Page 175
2.6 THE EQUATIONS OF GAS DYNAMICS......Page 190
3.1 THE WEAK MAXIMUM PRINCIPLE FOR PARABOLIC EQUATIONS......Page 223
3.2 THE INITIAL-BOUNDARY VALUE PROBLEM FOR THE HEAT EQUATION IN A RECTANGLE......Page 233
3.3 CAUCHY’S PROBLEM FOR THE HEAT EQUATION......Page 242
3.4 REGULARITY OF SOLUTIONS TO THE HEAT EQUATION......Page 257
* 3.5 THE STRONG MAXIMUM PRINCIPLE FOR THE HEAT EQUATION......Page 264
* 3.6 THE STEFAN PROBLEM AND ANALYTIC CONTINUATION......Page 271
3.7 HERMITE POLYNOMIALS AND THE NUMERICAL SOLUTION OF THE HEAT EQUATION IN A RECTANGLE......Page 281
3.8 NONLINEAR PROBLEMS IN HEAT CONDUCTION......Page 287
Chapter 4 - Laplace’s Equation......Page 319
4.1 GREEN’S FORMULAS......Page 320
4.2 BASIC PROPERTIES OF HARMONIC FUNCTIONS......Page 323
4.3 BOUNDARY VALUE PROBLEMS FOR LAPLACE’S EQUATION......Page 328
4.4 SEPARATION OF VARIABLES IN POLAR AND SPHERICAL COORDINATES......Page 332
4.5 GREEN’S FUNCTION AND POISSON’S FORMULA......Page 349
4.6 FINITE DIFFERENCE METHODS FOR LAPLACE’S EQUATION......Page 362
4.7 POISSON’S EQUATION......Page 366
4.8 TIME HARMONIC WAVE PROPAGATION IN A NONHOMOGENEOUS MEDIUM......Page 378
Chapter 5 - Potential Theory and Fredholm Integral Equations......Page 407
5.1 POTENTIAL THEORY......Page 408
5.2 The Fredholm Alternative......Page 428
5.3 APPLICATIONS TO THE DIRICHLET AND NEUMANN PROBLEMS......Page 450
5.4 HILBERT-SCHMIDT THEORY AND EIGENVALUE PROBLEMS......Page 465
5.5 THE NUMERICAL SOLUTION OF FREDHOLM INTEGRAL EQUATIONS OF THE SECOND KIND......Page 493
*Chapter 6 - Scattering Theory......Page 507
6.1 THE GAMMA FUNCTION......Page 508
6.2 BESSEL FUNCTIONS......Page 516
6.3 THE SCATTERING OF ACOUSTIC WAVES......Page 547
6.4 MAXWELL’S EQUATIONS......Page 557
6.5 SCATTERING BY A CYLINDER OF ARBITRARY CROSS SECTION......Page 563
6.6 THE INVERSE SCATTERING PROBLEM......Page 583
References......Page 627
Index......Page 638
