Tài liệu Boundary Value Problems, Sixth Edition: and Partial Differential Equations pptx

BOUNDARY

VALUE PROBLEMS

FIFTH EDITION

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BOUNDARY

VALUE PROBLEMS

AND PARTIAL DIFFERENTIAL EQUATIONS

DAVID L. POWERS

Clarkson University

FIFTH EDITION

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Elsevier Academic Press

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05 06 07 08 09 10 9 8 7 6 5 4 3 2 1

Contents

Preface ix

CHAPTER 0 Ordinary Differential Equations 1

0.1 Homogeneous Linear Equations 1

0.2 Nonhomogeneous Linear Equations 14

0.3 Boundary Value Problems 26

0.4 Singular Boundary Value Problems 38

0.5 Green’s Functions 43

Chapter Review 51

Miscellaneous Exercises 51

CHAPTER 1 Fourier Series and Integrals 59

1.1 Periodic Functions and Fourier Series 59

1.2 Arbitrary Period and Half-Range Expansions 64

1.3 Convergence of Fourier Series 73

1.4 Uniform Convergence 79

1.5 Operations on Fourier Series 85

1.6 Mean Error and Convergence in Mean 90

1.7 Proof of Convergence 95

1.8 Numerical Determination of Fourier Coefficients 100

1.9 Fourier Integral 106

1.10 Complex Methods 113

1.11 Applications of Fourier Series and Integrals 117

1.12 Comments and References 124

Chapter Review 125

Miscellaneous Exercises 125

v

vi Contents

CHAPTER 2 The Heat Equation 135

2.1 Derivation and Boundary Conditions 135

2.2 Steady-State Temperatures 143

2.3 Example: Fixed End Temperatures 149

2.4 Example: Insulated Bar 157

2.5 Example: Different Boundary Conditions 163

2.6 Example: Convection 170

2.7 Sturm–Liouville Problems 175

2.8 Expansion in Series of Eigenfunctions 181

2.9 Generalities on the Heat Conduction Problem 184

2.10 Semi-Infinite Rod 188

2.11 Infinite Rod 193

2.12 The Error Function 199

2.13 Comments and References 204

Chapter Review 206

Miscellaneous Exercises 206

CHAPTER 3 The Wave Equation 215

3.1 The Vibrating String 215

3.2 Solution of the Vibrating String Problem 218

3.3 d’Alembert’s Solution 227

3.4 One-Dimensional Wave Equation: Generalities 233

3.5 Estimation of Eigenvalues 236

3.6 Wave Equation in Unbounded Regions 239

3.7 Comments and References 246

Chapter Review 247

Miscellaneous Exercises 247

CHAPTER 4 The Potential Equation 255

4.1 Potential Equation 255

4.2 Potential in a Rectangle 259

4.3 Further Examples for a Rectangle 264

4.4 Potential in Unbounded Regions 270

4.5 Potential in a Disk 275

4.6 Classification and Limitations 280

4.7 Comments and References 283

Chapter Review 285

Miscellaneous Exercises 285

CHAPTER 5 Higher Dimensions and Other Coordinates 295

5.1 Two-Dimensional Wave Equation: Derivation 295

5.2 Three-Dimensional Heat Equation 298

5.3 Two-Dimensional Heat Equation: Solution 303

Contents vii

5.4 Problems in Polar Coordinates 308

5.5 Bessel’s Equation 311

5.6 Temperature in a Cylinder 316

5.7 Vibrations of a Circular Membrane 321

5.8 Some Applications of Bessel Functions 329

5.9 Spherical Coordinates; Legendre Polynomials 335

5.10 Some Applications of Legendre Polynomials 345

5.11 Comments and References 353

Chapter Review 354

Miscellaneous Exercises 354

CHAPTER 6 Laplace Transform 363

6.1 Definition and Elementary Properties 363

6.2 Partial Fractions and Convolutions 369

6.3 Partial Differential Equations 376

6.4 More Difficult Examples 383

6.5 Comments and References 389

Miscellaneous Exercises 389

CHAPTER 7 Numerical Methods 397

7.1 Boundary Value Problems 397

7.2 Heat Problems 403

7.3 Wave Equation 408

7.4 Potential Equation 414

7.5 Two-Dimensional Problems 420

7.6 Comments and References 428

Miscellaneous Exercises 428

Bibliography 433

Appendix: Mathematical References 435

Answers to Odd-Numbered Exercises 441

Index 495

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Preface

This text is designed for a one-semester or two-quarter course in partial dif￾ferential equations given to third- and fourth-year students of engineering and

science. It can also be used as the basis for an introductory course for graduate

students. Mathematical prerequisites have been kept to a minimum — calculus

and differential equations. Vector calculus is used for only one derivation, and

necessary linear algebra is limited to determinants of order two. A reader needs

enough background in physics to follow the derivations of the heat and wave

equations.

The principal objective of the book is solving boundary value problems

involving partial differential equations. Separation of variables receives the

greatest attention because it is widely used in applications and because it pro￾vides a uniform method for solving important cases of the heat, wave, and

potential equations. One technique is not enough, of course. D’Alembert’s so￾lution of the wave equation is developed in parallel with the series solution,

and the distributed-source solution is constructed for the heat equation. In

addition, there are chapters on Laplace transform techniques and on numeri￾cal methods.

The second objective is to tie together the mathematics developed and the

student’s physical intuition. This is accomplished by deriving the mathemati￾cal model in a number of cases, by using physical reasoning in the mathemat￾ical development, by interpreting mathematical results in physical terms, and

by studying the heat, wave, and potential equations separately.

In the service of both objectives, there are many fully worked examples and

now about 900 exercises, including miscellaneous exercises at the end of each

chapter. The level of difficulty ranges from drill and verification of details

to development of new material. Answers to odd-numbered exercises are in

ix

x Preface

the back of the book. An Instructor’s Manual is available both online and in

print (ISBN: 0-12-369435-3), with the answers to the even-numbered prob￾lems. A Student Solutions Manual is available both online and in print (ISBN:

0-12-088586-7), that contains detailed solutions of odd-numbered problems.

There are many ways of choosing and arranging topics from the book to

provide an interesting and meaningful course. The following sections form

the core, requiring at least 14 hours of lecture: Sections 1.1–1.3, 2.1–2.5, 3.1–

3.3, 4.1–4.3, and 4.5. These cover the basics of Fourier series and the solutions

of heat, wave, and potential equations in finite regions. My choice for the next

most important block of material is the Fourier integral and the solution of

problems on unbounded regions: Sections 1.9, 2.10–2.12, 3.6, and 4.4. These

require at least six more lectures.

The tastes of the instructor and the needs of the audience will govern the

choice of further material. A rather theoretical flavor results from including:

Sections 1.4–1.7 on convergence of Fourier series; Sections 2.7–2.9 on Sturm–

Liouville problems, and the sequel, Section 3.4; and the more difficult parts of

Chapter 5, Sections 5.5–5.10 on Bessel functions and Legendre polynomials.

On the other hand, inclusion of numerical methods in Sections 1.8 and 3.5

and Chapter 7 gives a very applied flavor.

Chapter 0 reviews solution techniques and theory of ordinary differential

equations and boundary value problems. Equilibrium forms of the heat and

wave equations are derived also. This material belongs in an elementary differ￾ential equations course and is strictly optional. However, many students have

either forgotten it or never seen it.

For this fifth edition, I have revised in response to students’ changing needs

and abilities. Many sections have been rewritten to improve clarity, provide

extra detail, and make solution processes more explicit. In the optional Chap￾ter 0, free and forced vibrations are major examples for solution of differential

equations with constant coefficients. In Chapter 1, I have returned to deriving

the Fourier integral as a “limit” of Fourier series. New exercises are included

for applications of Fourier series and integrals. Solving potential problems on a

rectangle seems to cause more difficulty than expected. A new section 4.3 gives

more guidance and examples as well as some information about the Poisson

equation. New exercises have been added and old ones revised throughout.

In particular I have included exercises based on engineering research publica￾tions. These provide genuine problems with real data.

A new feature of this edition is a CD with auxiliary materials: animations

of convergence of Fourier series; animations of solutions of the heat and wave

equations as well as ordinary initial value problems; color graphics of solu￾tions of potential problems; additional exercises in a workbook style; review

questions for each chapter; text material on using a spreadsheet for numerical

methods. All files are readable with just a browser and Adobe Reader, available

without cost.

Preface xi

I wish to acknowledge the skillful work of Cindy Smith, who was the LaTeX

compositor and corrected many of my mistakes, the help of Academic Press

editors and consultants, and the guidance of reviewers for this edition:

Darryl Yong, Harvey Mudd College

Ken Luther, Valparaiso University

Alexander Kirillov, SUNY at Stony Brook

James V. Herod, Georgia Tech University

Hilary Davies, University of Alaska Anchorage

Catherine Crawford, Elmhurst College

Ahmed Mohammed, Ball State University

I also wish to acknowledge the guidance of reviewers for the previous edi￾tion:

Linda Allen, Texas Tech University

Ilya Bakelman, Texas A&M University

Herman Gollwitzer, Drexel University

James Herod, Georgia Institute of Technology

Robert Hunt, Humboldt State University

Mohammad Khavanin, University of North Dakota

Jeff Morgan, Texas A&M University

Jim Mueller, California Polytechnic State University

Ron Perline, Drexel University

William Royalty, University of Idaho

Lawrence Schovanec, Texas Tech University

Al Shenk, University of California at San Diego

Michael Smiley, Iowa State University

Monty Strauss, Texas Tech University

Kathie Yerion, Gonzaga University

David L. Powers

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Ordinary Differential

Equations CHAPTER

0

0.1 Homogeneous Linear Equations

The subject of most of this book is partial differential equations: their physical

meaning, problems in which they appear, and their solutions. Our principal

solution technique will involve separating a partial differential equation into

ordinary differential equations. Therefore, we begin by reviewing some facts

about ordinary differential equations and their solutions.

We are interested mainly in linear differential equations of first and second

orders, as shown here:

du

dt = k(t)u + f(t), (1)

d2u

dt2 + k(t)

du

dt + p(t)u = f(t). (2)

In either equation, if f(t) is 0, the equation is homogeneous. (Another test: If

the constant function u(t) ≡ 0 is a solution, the equation is homogeneous.) In

the rest of this section, we review homogeneous linear equations.

A. First-Order Equations

The most general first-order linear homogeneous equation has the form

du

dt = k(t)u. (3)

1

2 Chapter 0 Ordinary Differential Equations

This equation can be solved by isolating u on one side and then integrating:

1

u

du

dt = k(t),

ln |u| =

k(t) dt + C,

u(t) = ±e

C e

k(t) dt = ce

k(t) dt. (4)

It is easy to check directly that the last expression is a solution of the differential

equation for any value of c. That is, c is an arbitrary constant and can be used

to satisfy an initial condition if one has been specified.

Example.

Solve the homogeneous differential equation

du

dt = −tu.

The procedure outlined here gives the general solution

u(t) = ce−t

2/2

for any c. If an initial condition such as u(0) = 5 is specified, then c must be

chosen to satisfy it (c = 5).

The most common case of this differential equation has k(t) = k constant.

The differential equation and its general solution are

du

dt = ku, u(t) = cekt. (5)

If k is negative, then u(t) approaches 0 as t increases. If k is positive, then u(t)

increases rapidly in magnitude with t. This kind of exponential growth often

signals disaster in physical situations, as it cannot be sustained indefinitely.

B. Second-Order Equations

It is not possible to give a solution method for the general second-order linear

homogeneous equation,

d2u

dt2 + k(t)

du

dt + p(t)u = 0. (6)

Nevertheless, we can solve some important cases that we detail in what follows.

The most important point in the general theory is the following.