Beginning partial differential equations / Peter V. O'Neil.

Format:

Book

Author/Creator:

O'Neil, Peter V.

Series:

Pure and applied mathematics (John Wiley & Sons : Unnumbered)

Pure and applied mathematics

Language:

English

Subjects (All):

Differential equations, Partial.

Physical Description:

ix, 477 pages : illustrations ; 25 cm.

Edition:

Second edition.

Place of Publication:

Hoboken, N.J. : Wiley-Interscience, [2008]

Summary:

Beginning Partial Differential Equations, Second Edition provides a comprehensive introduction to partial differential equations (PDEs) with a special focus on the significance of characteristics, solutions by Fourier series, integrals and transforms, properties and physical interpretations of solutions, and a transition to the modern function space approach to PDEs. With its breadth of coverage, this new edition continues to present a broad introduction to the field, while also addressing more specialized topics and applications.

Maintaining the hallmarks of the previous edition, the book begins with first-order linear and quasi-linear PDEs and the role of characteristics in the existence and uniqueness of solutions. Canonical forms are discussed for the linear second-order equation, along with the Cauchy problem, existence and uniqueness of solutions, and characteristics as carriers of discontinuities in solutions. Fourier series, integrals, and transforms are followed by their rigorous application to wave and difficusion equations as well as to Dirichlet and Neumann problems. In addition, solutions are viewed through physical interpretations of PDEs. The book concludes with a transition to more advanced topics, including the proof of an existence theorem for the Dirichlet problem and an introduction to distributions.

Additional features of the Second Edition include solutions by both general eigenfunction expansions and numerical methods. Explicit solutions of Burger's equation, the telegraph equation (with an asymptotic analysis of the solution), and Poisson's equation are provided. A historical sketch of the field of PDEs and an extensive section with solutions to selected problems are also included.

Beginning Partial Differential Equations, Second Edition is an excellent book for advanced undergraduate- and beginning graduate-level courses in mathematics, science, and engineering.

Contents:

1 First-Order Equations 1

1.1 Notation and Terminology 1

1.2 The Linear First-Order Equation 4

1.3 The Significance of Characteristics 12

1.4 The Quasi-Linear Equation 16

2 Linear Second-Order Equations 23

2.1 Classification 23

2.2 The Hyperbolic Canonical Form 25

2.3 The Parabolic Canonical Form 30

2.4 The Elliptic Canonical Form 33

2.5 Some Equations of Mathematical Physics 38

2.6 The Second-Order Cauchy Problem 46

2.7 Characteristics and the Cauchy Problem 49

2.8 Characteristics as Carriers of Discontinuities 56

3 Elements of Fourier Analysis 59

3.1 Why Fourier Series? 59

3.2 The Fourier Series of a Function 60

3.3 Convergence of Fourier Series 63

3.4 Sine and Cosine Expansions 81

3.5 The Fourier Integral 89

3.6 The Fourier Transform 95

3.7 Convolution 101

3.8 Fourier Sine and Cosine Transforms 106

4 The Wave Equation 109

4.1 d'Alembert Solution of the Cauchy Problem 109

4.2 d'Alembert's Solution as a Sum of Waves 117

4.3 The Characteristic Triangle 126

4.4 The Wave Equation on a Half-Line 131

4.5 A Half-Line with Moving End 134

4.6 A Nonhomogeneous Problem on the Real Line 137

4.7 A General Problem on a Closed Interval 141

4.8 Fourier Series Solutions on a Closed Interval 150

4.9 A Nonhomogeneous Problem on a Closed Interval 159

4.10 The Cauchy Problem by Fourier Integral 168

4.11 A Wave Equation in Two Space Dimensions 173

4.12 The Kirchhoff-Poisson Solution 177

4.13 Hadamard's Method of Descent 182

5 The Heat Equation 185

5.1 The Cauchy Problem and Initial Conditions 185

5.2 The Weak Maximum Principle 188

5.3 Solutions on Bounded Intervals 192

5.4 The Heat Equation on the Real Line 210

5.5 The Heat Equation on the Half-Line 218

5.6 The Debate Over the Age of the Earth 224

5.7 The Nonhomogeneous Heat Equation 227

5.8 The Heat Equation in Two Space Variables 234

6 Dirichlet and Neumann Problems 239

6.1 The Setting of the Problems 239

6.2 Some Harmonic Functions 247

6.3 Representation Theorems 251

6.4 Two Properties of Harmonic Functions 257

6.5 Is the Dirichlet Problem Well Posed? 263

6.6 Dirichlet Problem for a Rectangle 266

6.7 Dirichlet Problem for a Disk 269

6.8 Poisson's Integral Representation for a Disk 272

6.9 Dirichlet Problem for the Upper Half-Plane 276

6.10 Dirichlet Problem for the Right Quarter-Plane 279

6.11 Dirichlet Problem for a Rectangular Box 282

6.12 The Neumann Problem 285

6.13 Neumann Problem for a Rectangle 288

6.14 Neumann Problem for a Disk 290

6.15 Neumann Problem for the Upper Half-Plane 294

6.16 Green's Function for a Dirichlet Problem 296

6.17 Conformal Mapping Techniques 303

6.17.1 Conformal Mappings 303

6.17.2 Bilinear Transformations 308

6.17.3 Construction of Conformal Mappings between Domains 313

6.17.4 An Integral Solution of the Dirichlet Problem for a Disk 320

6.17.5 Solution of Dirichlet Problems by Conformal Mapping 323

7 Existence Theorems 327

7.1 A Classical Existence Theorem 327

7.2 A Hilbert Space Approach 336

7.3 Distributions and an Existence Theorem 344

8.1 Solutions by Eigenfunction Expansions 351

8.2 Numerical Approximations of Solutions 370

8.3 Burger's Equation 377

8.4 The Telegraph Equation 383

8.5 Poisson's Equation 390

9 End Materials 395

9.1 Historical Notes 395.

Notes:

Includes index.

ISBN:

9780470133903

0470133902

OCLC:

173368446

Online:

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