Fourier Analysis and Boundary Value Problems (eBook)

Enrique A. Gonzalez-Velasco (Autor)

eBook Download: EPUB

1996 | 1. Auflage
551 Seiten
Elsevier Science (Verlag)
978-0-08-053193-9 (ISBN)

Infos

Fourier Analysis and Boundary Value Problems provides a thorough examination of both the theory and applications of partial differential equations and the Fourier and Laplace methods for their solutions. Boundary value problems, including the heat and wave equations, are integrated throughout the book. Written from a historical perspective with extensive biographical coverage of pioneers in the field, the book emphasizes the important role played by partial differential equations in engineering and physics. In addition, the author demonstrates how efforts to deal with these problems have lead to wonderfully significant developments in mathematics.
A clear and complete text with more than 500 exercises, Fourier Analysis and Boundary Value Problems is a good introduction and a valuable resource for those in the field.

Key Features
* Topics are covered from a historical perspective with biographical information on key contributors to the field
* The text contains more than 500 exercises
* Includes practical applications of the equations to problems in both engineering and physics

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Fourier Analysis and Boundary Value Problems provides a thorough examination of both the theory and applications of partial differential equations and the Fourier and Laplace methods for their solutions. Boundary value problems, including the heat and wave equations, are integrated throughout the book. Written from a historical perspective with extensive biographical coverage of pioneers in the field, the book emphasizes the important role played by partial differential equations in engineering and physics. In addition, the author demonstrates how efforts to deal with these problems have lead to wonderfully significant developments in mathematics. A clear and complete text with more than 500 exercises, Fourier Analysis and Boundary Value Problems is a good introduction and a valuable resource for those in the field. - Topics are covered from a historical perspective with biographical information on key contributors to the field- The text contains more than 500 exercises- Includes practical applications of the equations to problems in both engineering and physics

Inhalt

Front Cover 1
Fourier Analysis and Boundary Value Problems 4
Copyright Page 5
Table of Contents 6
Preface 10
CHAPTER 1. A HEATED DISCUSSION 14
1.1 Historical Prologue 14
1.2 The Heat Equation 17
1.3 Boundary Value Problems 19
1.4 The Method of Separation of Variables 22
1.5 Linearity and Superposition of Solutions 24
1.6 Historical Epilogue 27
Exercises 29
CHAPTER 2. FOURIER SERIES 36
2.1 Introduction 36
2.2 Fourier Series 38
2.3 The Riemann-Lebesgue Theorem 43
2.4 The Convergence of Fourier Series 49
2.5 Fourier Series on Arbitrary Intervals 58
2.6 The Gibbs Phenomenon 62
2.7 Fejér Sums 66
2.8 Integration of Fourier Series 71
2.9 Historical Epilogue 75
Exercises 82
CHAPTER 3. RETURN TO THE HEATED BAR 97
3.1 Existence of a Solution 97
3.2 Uniqueness and Stability of the Solution 103
3.3 Nonzero Temperature at the Endpoints 106
3.4 Bar Insulated at the Endpoints 108
3.5 Mixed Endpoint Conditions 110
3.6 Heat Convection at One Endpoint 112
3.7 Time-Independent Problems 114
3.8 The Steady-State Solution 117
3.9 The Transient Solution 121
3.10 The Complete Solution 124
3.11 Time-Dependent Problems 127
Exercises 133
CHAPTER 4. GENERALIZED FOURIER SERIES 146
4.1 Sturm-Liouville Problems 146
4.2 The Eigenvalues and Eigenfunctions 152
4.3 The Existence of the Eigenvalues 154
4.4 Generalized Fourier Series 164
4.5 Approximations 167
4.6 Historical Epilogue 170
Exercises 172
CHAPTER 5. THE WAVE EQUATION 178
5.1 Introduction 178
5.2 The Vibrating String 180
5.3 D'Alembert's Solution 183
5.4 A Struck String 191
5.5 Bernoulli's Solution 194
5.6 Time-Independent Problems 200
5.7 Time-Dependent Problems 204
5.8 Historical Epilogue 208
Exercises 211
CHAPTER 6. ORTHOGONAL SYSTEMS 220
6.1 Fourier Series and Parseval's Identity 220
6.2 An Approximation Problem 226
6.3 The Uniform Convergence of Fourier Series 228
6.4 Convergence in the Mean 230
6.5 Applications to the Vibrating String 236
6.6 The Riesz-Fischer Theorem 237
Exercises 243
CHAPTER 7. FOURIER TRANSFORMS 250
7.1 The Laplace Equation 250
7.2 Fourier Transforms 255
7.3 Properties of the Fourier Transform 260
7.4 Convolution 261
7.5 Solution of the Dirichlet Problem for the Half-Plane 265
7.6 The Fourier Transform Method 269
Exercises 272
CHAPTER 8. LAPLACE TRANSFORMS 279
8.1 The Laplace Transform and the Inversion Theorem 279
8.2 Properties of the Laplace Transform 284
8.3 Convolution 291
8.4 The Telegraph Equation 293
8.5 The Method of Residues 299
8.6 Historical Epilogue 306
Exercises 309
CHAPTER 9. BOUNDARY VALUE PROBLEMS IN HIGHER DIMENSIONS 315
9.1 Electrostatic Potential in a Charged Box 315
9.2 Double Fourier Series 323
9.3 The Dirichlet Problem in a Box 332
9.4 Return to the Charged Box 336
9.5 The Multiple Fourier Transform Method 337
9.6 The Double Laplace Transform Method 347
Exercises 353
CHAPTER 10. BOUNDARY VALUE PROBLEMS WITH CIRCULAR SYMMETRY 364
10.1 Vibrations of a Circular Membrane 364
10.2 The Gamma Function 368
10.3 Bessel Functions of the First Kind 370
10.4 Recursion Formulas for Bessel Functions 374
10.5 Bessel Functions of the Second Kind 376
10.6 The Zeros of Bessel Functions 378
10.7 Orthogonal Systems of Bessel Functions 383
10.8 Fourier-Bessel Series and Dini-Bessel Series 386
10.9 Return to the Vibrating Membrane 391
10.10 Modified Bessel Functions 396
10.11 The Skin Effect 401
Exercises 407
CHAPTER 11. BOUNDARY VALUE PROBLEMS WITH SPHERICAL SYMMETRY 423
11.1 The Potbellied Stove 423
11.2 Solutions of the Legendre Equation 426
11.3 The Norms of the Legendre Polynomials 430
11.4 Fourier-Legendre Series 431
11.5 Return to the Potbellied Stove 438
11.6 The Dirichlet Problem for the Sphere 440
11.7 The Associated Legendre Functions 441
11.8 Solution of the Dirichlet Problem for the Sphere 445
11.9 Poisson's Integral Formula for the Sphere 447
11.10 The Cooling of a Sphere 454
Exercises 456
CHAPTER 12. DISTRIBUTIONS AND GREEN'S FUNCTIONS 464
12.1 Historical Prologue 464
12.2 Distributions 469
12.3 Basic Properties of Distributions 473
12.4 Differentiation of Distributions 476
12.5 Sequences and Series of Distributions 481
12.6 Convolution 484
12.7 The Poisson Equation on the Sphere 493
12.8 Distributions Depending on a Parameter 498
12.9 The Cauchy Problem for Time-Dependent Equations 501
12.10 Conclusion 506
Exercises 508
APPENDIX A. UNIFORM CONVERGENCE 519
Excercise 528
APPENDIX B. IMPROPER INTEGRALS 531
Exercises 546
APPENDIX C. TABLES OF FOURIER AND LAPLACE TRANSFORMS 548
APPENDIX D. HISTORICAL BIBLIOGRAPHY 552
Index 556

Textauszug

1

A Heated Discussion

§1.1 Historical Prologue

Napoléon Bonaparte's expedition to Egypt took place in the summer of 1798, the expeditionary forces arriving on July 1 and capturing Alexandria the following day.1 On the previous March 27—7 Germinal Year VI in the chronology of the French Republic—a young professor at the newly founded École Polytechnique, Jean Joseph Fourier (1768-1830), was summoned by the Minister of the Interior in no uncertain terms:

Citizen, the Executive Directory having in the present circumstances a particular need of your talents and of your zeal has just disposed of you for the sake of public service. You should prepare yourself and be ready to depart at the first order.2

It was in this manner, perhaps not entirely reconcilable with the idea of Liberté, that Fourier joined the Commission of Arts and Sciences of Bonaparte's expedition and sailed for Egypt on May 19. While in temporary quarters in the town of Rosetta, near Alexandria, where he held an administrative position, the military forces marched on Cairo. They entered on July 24 after successfully defeating the Mameluks in the Battle of the Pyramids. By August 20 Bonaparte had decreed the foundation of the Institut d'Egypte in Cairo, modeled on the Institut de France3 of whose second classe (mechanical arts) he was a proud member, to serve as an advisory body to the administration, to engage on studies on Egypt, and, what is more important, to devote itself to the advancement of science in Egypt.

The first meeting of the Institut d'Egypte, with Fourier already appointed as its permanent secretary, was held on August 25. From this moment on until his departure in 1801 on the English brig Good Design, Fourier devoted his time not only to his administrative duties but also to scientific research, presenting numerous papers on several subjects. In the autumn of 1799 he was appointed leader of one of two scientific expeditions to study the monuments and inscriptions in Upper Egypt and was put in charge of cataloguing and describing all its discoveries.

After several military encounters the French surrendered to invading British forces on August 30, 1801. While forced to depart from Egypt, they were allowed to keep their scientific papers and collections of antiques with the exception of a precious find: the Rosetta stone.1

Upon his return to France in November of 1801, Fourier resumed his post at the École Polytechnique but only briefly. In February of 1802 Bonaparte himself appointed him Préfet of the Department of Isère in the French Alps. It was here, in the city of Grenoble, that Fourier returned to his physical and mathematical research, with which we shall presently occupy ourselves.2

But Fourier's stay in Egypt had left a permanent mark on his health which was to influence, perhaps, the direction of his research. He claimed to have contracted chronic rheumatic pains during the siege of Alexandria and that the sudden change of climate from that of Egypt to that of the Alps was too distressful for him. The facts are that he seemed to need large amounts of heat, that he lived in overheated rooms, that he covered himself with an excessive amount of clothing even in the heat of summer, and that his preoccupation with heat extended to the subject of heat propagation in solid bodies, heat loss by radiation, and heat conservation. It was then on the subject of heat that he concentrated his main research efforts, for which he had ample time after he settled down to the routine of his administrative duties. These efforts, of which there is documentary evidence as early as 1804, were first made public when Fourier read his work Mémoire sur la propagation de la chaleur before the first classe of the Institut de France on December 21, 1807.3

From our present point of view, approximately two centuries after the fact, this memoir stands as one of the most daring, original, complete, and influential works of the nineteenth century on mathematical physics. The methods that Fourier used to deal with heat problems were those of a true pioneer because he had to work with concepts that were not yet properly formulated. He worked with discontinuous functions when others dealt with continuous ones, used integral as an area when integral as a prederivative was popular, and talked about the convergence of a series of functions before there was a definition of convergence. But the methods that Fourier used to deal with heat problems were to prove fruitful in many other physical disciplines such as electricity, acoustics and hydrodynamics. It was the success of Fourier's work in applications that made necessary a redefinition of the concept of function, the introduction of a definition of convergence, a reexamination of the concept of integral, and the ideas of uniform continuity and uniform convergence. It also provided motivation for the discovery of the theory of sets, was in the background of ideas leading to measure theory, and contained the germ of the theory of distributions.

However, back in 1807 his memoir was not well received. A committee consisting of Lacroix, Lagrange, Laplace, and Monge was to judge the memoir and publish a report on it, but never did so. Instead, criticisms were made personally to Fourier, either in 1808 or in 1809, on occasion of his visits to Paris to supervise the printing of his Préface historique—this title was personally chosen by the former First Consul who had since crowned himself Emperor Napoléon—to the Description de l'Egypte, a book on the Egyptian discoveries of the 1799 expedition. The criticisms came mainly from Lagrange and Laplace and referred to two major points: Fourier's derivation of the equations of heat propagation and his use of some series of trigonometric functions, known today as Fourier series. Fourier replied to their objections but, by this time, Jean-Baptiste Biot published some new criticisms in the Mercure de France, a fact that Fourier resented and moved him to write, in 1810, angry letters of protest and pointed attacks against Biot and Laplace, although Laplace had already become supportive of Fourier's work by 1809. In one of these letters—to unknown correspondents—he suggested that, as a means to settle the question, a public competition be set up and a prize be awarded by the Institut to the best work on the propagation of heat. If not because of this suggestion, it is at least possible that the Institut considered the question of a prize essay on the theory of heat in view of Fourier's vigorous defense of his own work. The fact is that in 1810 this was the subject chosen for a prize essay for the year 1811, and Laplace was probably instrumental in converting Fourier's suggestion into reality. A committee consisting of Hatiy, Lagrange, Laplace, Legendre, and Malus was to judge on the only two entries. On January 6, 1812, the prize was awarded to Fourier's Théorie du mouvement de la chaleur dans les corps solides, an expanded version of his 1807 memoir. However, the committee's report expressed some reservations:

This essay contains the correct differential equations of the transmission of heat, both in the interior of solid bodies or on their surface: and the novelty of the subject, added to its importance, has induced the Class to reward this Work, but observing meanwhile that the manner in which the Author arrives at his equations is not exempt from difficulties, and that his analysis, to integrate them, still leaves something to be desired in the realms of both generality and even rigor.1

Fourier protested but to no avail, and his new work, like his previous memoir, was not published by the Institut at this time. He was to ultimately prevail and enjoy a well deserved fame, but the time has come when we should interrupt the telling of this story and present one of the problems, the earliest, considered by Fourier: that of a thin heated bar. This will show the originality of his methods and also the nature of those insidious analytical difficulties, as they were referred to by some of Fourier's opponents.

§1.2 The Heat Equation

Consider the problem of finding an equation describing the temperature distribution in a thin bar of some conducting material, which we suppose is located along the x-axis. We shall work under the following hypotheses:

1. the bar is insulated along its lateral surface so that there is no exchange of heat with the surrounding medium through this surface,

2. the bar has a uniform cross section, whose area is denoted by A, and constant density ρ,

3. at any given time t all the points of abscissa x have the same temperature, denoted by u(x, t), and

4. the temperature varies so smoothly in time and along the bar that the function u has continuous first and second partial derivatives with respect to both variables.

In order to derive the desired equation we shall apply the law of conservation of energy to a small piece of the bar, the slice situated between the abscissas x and x + h as...

Produkt-Details

• Erscheint lt. Verlag: 28.11.1996
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• ISBN-10: 0-08-053193-8 / 0080531938
• ISBN-13: 978-0-08-053193-9 / 9780080531939