An introduction to partial differential equations with MATLAB / Matthew P. Coleman.

Format:

Book

Author/Creator:

Coleman, Matthew P.

Contributor:

Thea Marie Garfield Fund.

Series:

Chapman & Hall/CRC applied mathematics and nonlinear science series

Language:

English

Subjects (All):

Differential equations, Partial--Computer-assisted instruction.

Differential equations, Partial.

MATLAB.

Computer-assisted instruction.

Physical Description:

xiv, 669 pages : illustrations ; 25 cm.

Edition:

Second edition.

Place of Publication:

Boca Raton, FL : CRC Press, [2013]

Summary:

"Preface Many problems in the physical world can be modeled by partial differential equations, from applications as diverse as the flow of heat, the vibration of a ball, the propagation of sound waves, the diffusion of ink in a glass of water, electric and magnetic fields, the spread of algae along the ocean's surface, the fluctuation in the price of a stock option, and the quantum mechanical behavior of a hydrogen atom. However, as with any area of applied mathematics, the field of PDEs is interesting not only because of its applications, but because it has taken on a mathematical life of its own. The author has written this book with both ideas in mind, in the hope that the student will appreciate the usefulness of the subject and, at the same time, get a glimpse into the beauty of some of the underlying mathematics. This text is suitable for a two-semester introduction to partial differential equations and Fourier series for students who have had basic courses in multivariable calculus (through Stokes's and the Divergence Theorems) and ordinary differential equations. Over the years, the author has taught much of the material to undergraduate mathematics, physics and engineering students at Penn State and Fairfield Universities, as well as to engineering graduate students at Penn State and mathematics and engineering graduate students at Fairfield. It is assumed that the student has not had a course in real analysis. Thus, we treat pointwise convergence of Fourier series and do not talk about mean-square convergence until Chapter 8 (and, there, in terms of the Riemann, and not the Lebesgue, integral). Further, we feel that it is not appropriate to introduce so subtle an idea as uniform convergence in this setting, so we discuss it only in the Appendices"-- Provided by publisher.

Contents:

Prelude to Chapter 1 1

1 Introduction 3

1.1 What are Partial Differential Equations? 3

1.2 PDEs We Can Already Solve 6

1.3 Initial and Boundary Conditions 10

1.4 Linear PDEs-Definitions 12

1.5 Linear PDEs-The Principle of Superposition 16

1.6 Separation of Variables for Linear, Homogeneous PDEs 19

1.7 Eigenvalue Problems 25

Prelude to Chapter 2 41

2 The Big Three PDEs 43

2.1 Second-Order, Linear, Homogeneous PDEs with Constant Coefficients 43

2.2 The Heat Equation and Diffusion 44

2.3 The Wave Equation and the Vibrating String 54

2.4 Initial and Boundary Conditions for the Heat and Wave Equations 59

2.5 Laplace's Equation-The Potential Equation 66

2.6 Using Separation of Variables to Solve the Big Three PDEs 71

Prelude to Chapter 3 77

3 Fourier Series 79

3.1 Introduction 79

3.2 Properties of Sine and Cosine 80

3.3 The Fourier Series 89

3.4 The Fourier Series, Continued 95

3.5 The Fourier Series-Proof of Pointwise Convergence 104

3.6 Fourier Sine and Cosine Series 117

3.7 Completeness 124

Prelude to Chapter 4 127

4 Solving the Big Three PDEs on Finite Domains 129

4.1 Solving the Homogeneous Heat Equation for a Finite Rod 129

4.2 Solving the Homogeneous Wave Equation for a Finite String 138

4.3 Solving the Homogeneous Laplace's Equation on a Rectangular Domain 147

4.4 Nonhomogeneous Problems 153

Prelude to Chapter 5 161

5 Characteristics 163

5.1 First-Order PDEs with Constant Coefficients 163

5.2 First-Order PDEs with Variable Coefficients 174

5.3 The Infinite String 180

5.4 Characteristics for Semi-Infinite and Finite String Problems 192

5.5 General Second-Order Linear PDEs and Characteristics 201

Prelude to Chapter 6 211

6 Integral Transforms 213

6.1 The Laplace Transform for PDEs 213

6.2 Fourier Sine and Cosine Transforms 220

6.3 The Fourier Transform 230

6.4 The Infinite and Semi-Infinite Heat Equations 242

6.5 Distributions, the Dirac Delta Function and Generalized Fourier Transforms 254

6.6 Proof of the Fourier Integral Formula 266

Prelude to Chapter 7 275

7 Special Functions and Orthogonal Polynomials 277

7.1 The Special Functions and Their Differential Equations 277

7.2 Ordinary Points and Power Series Solutions; Chebyshev, Her-mite and Legendre Polynomials 285

7.3 The Method of Frobenius; Laguerre Polynomials 292

7.4 Interlude: The Gamma Function 300

7.5 Bessel Functions 305

7.6 Recap: A List of Properties of Bessel Functions and Orthogonal Polynomials 317

Prelude to Chapter 8 327

8 Sturm-Liouville Theory and Generalized Fourier Series 329

8.1 Sturm-Liouville Problems 329

8.2 Regular and Periodic Stunri-Liouville Problems 337

8.3 Singular Sturm-Liouville Problems; Self-Adjoint Problems 345

8.4 The Mean-Square or L² Norm and Convergence in the Mean 354

8.5 Generalized Fourier Series; Parseval's Equality and Completeness 361

Prelude to Chapter 9 373

9 PDEs in Higher Dimensions 375

9.1 PDEs in Higher Dimensions: Examples and Derivations 375

9.2 The Heat and Wave Equations on a Rectangle; Multiple Fourier Series 386

9.3 Laplace's Equation in Polar Coordinates: Poisson's Integral Formula 402

9.4 The Wave and Heat Equations in Polar Coordinates 414

9.5 Problems in Spherical Coordinates 425

9.6 The Infinite Wave Equation and Multiple Fourier Transforms 439

9.7 Postlude: Eigenvalues and Eigenfunctions of the Laplace Operator; Green's Identities for the Laplacian 456

Prelude to Chapter 10 463

10 Nonhomogeneous Problems and Green's Functions 465

10.1 Green's Functions for ODEs 465

10.2 Green's Function and the Dirac Delta Function 484

10.3 Green's Functions for Elliptic PDEs (I): Poisson's Equation in Two Dimensions 500

10.4 Green's Functions for Elliptic PDEs (II): Poisson's Equation in Three Dimensions; the Helmholtz Equation 516

10.5 Green's Functions for Equations of Evolution 525

Prelude to Chapter 11 537

11 Numerical Methods 539

11.1 Finite Difference Approximations for ODEs 539

11.2 Finite Difference Approximations for PDEs 551

11.3 Spectral Methods and the Finite Element Method 565.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Thea Marie Garfield Fund.

ISBN:

1439898464

9781439898468

1439898472

9781439898475

1439898499

9781439898499

OCLC:

822959644

Publisher Number:

99954928386