Mathematical physics with partial differential equations / James Kirkwood.

Format:

Book

Author/Creator:

Kirkwood, James R.

Contributor:

Christine Hikawa Fund.

Language:

English

Subjects (All):

Mathematical physics.

Differential equations, Partial.

Physical Description:

xii, 418 pages ; 24 cm

Place of Publication:

Amsterdam ; Boston : Academic Press, [2013]

Summary:

Mathematical Physics with Partial Differential Equations presents material from vector calculus and each of the three basic partial differential equations (PDEs) of mathematical physics: the heat equation, the wave equation, and Laplace's equation. The vector calculus sections focus on differential operators in different coordinate systems and the major integration theorems of the field. The PDE sections are concerned not only with the equations themselves but also the most widely-accepted techniques used to solve partial differential equations, including separation of variables, Green's functions, the Fourier transform, and the Laplace transform. The text serves as a valuable resource for advanced undergraduate and graduate students in mathematics, physics, and engineering, yet its rigor supports those continuing further in mathematical physics. It also covers detailed derivations, including computations of well-known topics such as the kernel for the heat equation.

Key Features

Combines classical physical concepts in a mathematical framework

Numerous applications useful in research

Detailed mathematical derivations and solutions -reinforcing the material through repetition of both the equations and the techniques

Several examples solved by multiple methods-highlighting the strengths and weaknesses of various techniques and providing additional practice Book jacket.

Contents:

1 Preliminaries 1

1-1 Self-Adjoint Operators 1

Fourier Coefficients 5

Exercises 11

1-2 Curvilinear Coordinates 14

Scaling Factors 17

Volume Integrals 18

The Gradient 22

The Laplacian 23

Spherical Coordinates 25

Other Curvilinear Systems 25

Applications 31

An Alternate Approach (Optional) 33

Exercises 33

1-3 Approximate Identities and the Dirac-δ Function 34

Approximate Identities 35

The Dirac-δ Function in Physics 37

Some Calculus for the Dirac-δ Function 40

The Dirac-δ Function in Curvilinear Coordinates 42

Exercises 44

1-4 The Issue of Convergence 45

Series of Real Numbers 45

Convergence versus Absolute Convergence 47

Series of Functions 48

Power Series 54

Taylor Series 56

Exercises 60

1-5 Some Important Integration Formulas 64

Other Facts We Will Use Later 68

Another Important Integral 69

Exercises 70

2 Vector Calculus 73

2-1 Vector Integration 73

Path Integrals 74

Line Integrals 77

Surfaces 80

Parameterized Surfaces 82

Integrals of Scalar Functions Over Surfaces 83

Surface Integrals of Vector Functions 85

Exercises 91

2-2 Divergence and Curl 93

Cartesian Coordinate Case 94

Cylindrical Coordinate Case 97

Spherical Coordinate Case 100

The Curl 104

The Curl in Cartesian Coordinates 104

The Curl in Cylindrical Coordinates 109

The Curl in Spherical Coordinates 114

Exercises 122

2-3 Green's Theorem, the Divergence Theorem, and Stokes' Theorem 122

The Divergence (Gauss') Theorem 127

Stokes' Theorem 135

An Application of Stokes' Theorem 140

An Application of the Divergence Theorem 141

Conservative Fields 142

Exercises 148

3 Green's Functions 155

Introduction 155

3-1 Construction of Green's Function Using the Dirac-δ Function 156

Exercises 164

3-2 Construction of Green's Function Using Variation of Parameters 164

Exercises 168

3-3 Construction of Green's Function from Eigenfunctions 168

Exercises 171

3-4 More General Boundary Conditions 171

Exercises 173

3-5 The Fredholm Alternative (or, What If 0 Is an Eigenvalue?) 173

Exercises 180

3-6 Green's Function for the Laplacian in Higher Dimensions 180

Exercises 186

4 Fourier Series 187

Introduction 187

4-1 Basic Definitions 188

Exercises 191

4-2 Methods of Convergence of Fourier Series 193

Fourier Series on Arbitrary Intervals 203

Exercises 204

4-3 The Exponential Form of Fourier Series 206

Exercises 207

4-4 Fourier Sine and Cosine Series 208

Exercises 210

4-5 Double Fourier Series 210

Exercise 212

5 Three Important Equations 213

Introduction 213

5-1 Laplace's Equation 215

Exercises 216

5-2 Derivation of the Heat Equation in One Dimension 216

Exercise 218

5-3 Derivation of the Wave Equation in One Dimension 218

Exercises 222

5-4 An Explicit Solution of the Wave Equation 222

Exercises 227

5-5 Converting Second-Order PDEs to Standard Form 228

Exercise 232

6 Sturm-Liouville Theory 233

Introduction 233

Exercises 234

6-1 The Self-Adjoint Property of a Sturm-Liouville Equation 234

Exercises 236

6-2 Completeness of Eigenfunctions for Sturm-Liouville Equations 237

Exercises 245

6-3 Uniform Convergence of Fourier Series 245

7 Separation of Variables in Cartesian Coordinates 251

Introduction 251

7-1 Solving Laplace's Equation on a Rectangle 251

Exercises 256

7-2 Laplace's Equation on a Cube 258

Exercises 261

7-3 Solving the Wave Equation in One Dimension by Separation of Variables 262

Exercises 267

7-4 Solving the Wave Equation in Two Dimensions in Cartesian Coordinates by Separation of Variables 269

Exercises 271

7-5 Solving the Heat Equation in One Dimension Using Separation of Variables 271

The Initial Condition Is the Dirac-δ Function 274

Exercises 276

7-6 Steady State of the Heat Equation 277

Exercises 281

7-7 Checking the Validity of the Solution 283

8 Solving Partial Differential Equations in Cylindrical Coordinates Using Separation of Variables 287

Introduction 287

An Example Where Bessel Functions Arise 287

Exercises 292

8-1 The Solution to Bessel's Equation in Cylindrical Coordinates 292

Exercises 294

8-2 Solving Laplace's Equation in Cylindrical Coordinates Using Separation of Variables 295

Exercises 299

8-3 The Wave Equation on a Disk (Drum Head Problem) 299

Exercises 303

8-4 The Heat Equation on a Disk 303

Exercises 306

9 Solving Partial Differential Equations in Spherical Coordinates Using Separation of Variables 307

9-1 An Example Where Legendre Equations Arise 307

9-2 The Solution to Bessel's Equation in Spherical Coordinates 310

9-3 Legendre's Equation and Its Solutions 315

Exercises 318

9-4 Associated Legendre Functions 319

Exercise 322

9-5 Laplace's Equation in Spherical Coordinates 322

Exercise 325

10 The Fourier Transform 327

Introduction 327

10-1 The Fourier Transform as a Decomposition 328

10-2 The Fourier Transform from the Fourier Series 329

10-3 Some Properties of the Fourier Transform 331

Exercises 334

10-4 Solving Partial Differential Equations Using the Fourier Transform 335

Exercises 341

10-5 The Spectrum of the Negative Laplacian in One Dimension 343

10-6 The Fourier Transform in Three Dimensions 346

Exercise 350

11 The Laplace Transform 351

Introduction 351

Exercises 352

11-1 Properties of the Laplace Transform 352

Exercises 356

11-2 Solving Differential Equations Using the Laplace Transform 356

Exercises 360

11-3 Solving the Heat Equation Using the Laplace Transform 361

Exercises 366

11-4 The Wave Equation and the Laplace Transform 368

Exercises 373

12 Solving PDEs with Green's Functions 375

12-1 Solving the Heat Equation Using Green's Function 375

Green's Function for the Nonhomogeneous Heat Equation 377

Exercises 379

12-2 The Method of Images 379

Method of Images for a Semi-infinite Interval 379

Method of Images for a Bounded Interval 383

Exercises 389

12-3 Green's Function for the Wave Equation 390

Exercises 397

12-4 Green's Function and Poisson's Equation 398

Exercises 401.

Notes:

Includes bibliographical references and index.

Machine generated contents note: Chapter 1 Prelimininaries- Introduction Chapter 2 Vector Calculus Chapter 3 Green's Functions Chapter 4 Fourier Series Chapter 5 Three Important Equations Chapter 6 Sturm-Liouville Theory Chapter 7 Bessel Equations and Bessel Functions Chapter 8 Legendre Equations and Legendre Polynomials Chapter 9 The Fourier Transform Chapter 10 The Laplace Transform Chapter 11 The Heat Equation Chapter 12 The Wave Equation Chapter 13 Laplace's Equation.

Local Notes:

Acquired for the Penn Libraries with assistance from the Christine Hikawa Fund.

ISBN:

0123869110

9780123869111

OCLC:

693812426

Publisher Number:

99947894736