**Pure and Applied Undergraduate Texts**

Volume: 15;
1998;
526 pp;
Hardcover

MSC: Primary 35;

Print ISBN: 978-0-8218-6889-8

Product Code: AMSTEXT/15

List Price: $83.00

AMS Member Price: $66.40

MAA member Price: $74.70

**Electronic ISBN: 978-1-4704-1128-2
Product Code: AMSTEXT/15.E**

List Price: $83.00

AMS Member Price: $66.40

MAA member Price: $74.70

#### Supplemental Materials

# Partial Differential Equations and Boundary-Value Problems with Applications: Third Edition

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*Mark A. Pinsky*

Building on the basic techniques of separation of variables and
Fourier series, the book presents the solution of boundary-value
problems for basic partial differential equations: the heat equation,
wave equation, and Laplace equation, considered in various standard
coordinate systems—rectangular, cylindrical, and spherical. Each of
the equations is derived in the three-dimensional context; the
solutions are organized according to the geometry of the coordinate
system, which makes the mathematics especially transparent. Bessel and
Legendre functions are studied and used whenever appropriate
throughout the text. The notions of steady-state solution of closely
related stationary solutions are developed for the heat equation;
applications to the study of heat flow in the earth are presented. The
problem of the vibrating string is studied in detail both in the
Fourier transform setting and from the viewpoint of the explicit
representation (d'Alembert formula). Additional chapters include the
numerical analysis of solutions and the method of Green's functions
for solutions of partial differential equations. The exposition also
includes asymptotic methods (Laplace transform and stationary phase).

With more than 200 working examples and 700 exercises (more than
450 with answers), the book is suitable for an undergraduate course in
partial differential equations.

#### Readership

Undergraduate students interested in PDE and applied PDE.

#### Reviews & Endorsements

With more than 200 working examples and 700 exercises (more than 450 with answers) this book is suitable for an undergraduate course in PDEs.

-- Zentralblatt MATH

I have been one of the cheerleaders for Mark's book PDE and BVP over the years. . . . [M]ost texts for undergraduates are either too advanced or lacking mathematical rigor. Mark's book captures just the right balance. I found [it] easy to use and the problems were doable by my students. His latest edition added some rather important topics that were not covered earlier and emphasized points where the grind it out Fourier methods did not apply.

-- Marshall Slemrod, University of Wisconsin-Madison, Madison, WI

I have used Partial Differential Equations and Boundary-Value Problems with Applications by Mark Pinsky to teach a one semester undergraduate course on Partial Differential Equations since we first offered the course in 1990. Major strengths [of the book]: The book is very well and concisely written. There is an excellent collection of problems. There is a good appendix with a review of ODE. There is a good appendix on a 'review of infinite series.' There are numerous interesting examples. There is a chapter on asymptomatic analysis. There is a chapter on numerical analysis. . . . Most students have liked the book and I have found it very convenient to teach out of.

-- Nancy Stanton, University of Notre Dame, South Bend, IN

I have taught from an earlier edition of this very nice book. Both the students and I have been happy with it. It is an important and useful topic in math [both pure and applied], and it is especially relevant and central to the service courses offered by most math departments...Pinsky's book is the best text for teaching [the] classical tools...When students need to look up one of the classical formulas in the theory of boundary value problems, I often refer to Pinsky's book which has always been on target.

-- Palle E. T. Jorgensen, University of Iowa

#### Table of Contents

# Table of Contents

## Partial Differential Equations and Boundary-Value Problems with Applications: Third Edition

Table of Contents pages: 1 2

- COVER Cover11 free
- TITLE iii4 free
- COPYRIGHT iv5 free
- PREFACE v6 free
- CONTENTS ix10 free
- Chapter 0. PRELIMINARIES 116 free
- 0.1. Partial Differential Equations 116
- 0.2. Separation of Variables 1025
- 0.3. Orthogonal Functions 2136
- 0.3.1. Inner product space of functions 2136
- 0.3.2. Projection of a function onto an orthogonal set 2439
- 0.3.3. Orthonormal sets of functions 2843
- 0.3.4. Parseval's equality, completeness, and mean square convergence 2944
- 0.3.5. Weighted inner product 3045
- 0.3.6. Gram-Schmidt orthogonalization 3146
- 0.3.7. Complex inner product 3247

- Chapter 1. FOURIER SERIES 3550
- 1.1. Definitions and Examples 3550
- 1.2. Convergence of Fourier Series 4661
- 1.3. Uniform Convergence and the Gibbs Phenomenon 5873
- 1.3.1. Example of Gibbs overshoot 5873
- 1.3.2. Implementation with Mathematica 6176
- 1.3.3. Uniform and nonuniform convergence 6479
- 1.3.4. Two criteria for uniform convergence 6479
- 1.3.5. Differentiation of Fourier series 6580
- 1.3.6. Integration of Fourier series 6681
- 1.3.7. A continuous function with a divergent Fourier series 6782

- 1.4. Parseval's Theorem and Mean Square Error 7186
- 1.5. Complex Form of Fourier Series 7893
- 1.6. Sturm-Liouville Eigenvalue Problems 8499
- 1.6.1. Examples of Sturm-Liouville eigenvalue problems 85100
- 1.6.2. Some general properties of S-L eigenvalue problems 86101
- 1.6.3. Example of transcendental eigenvalues 87102
- 1.6.4. Further properties: completeness and positivity 89104
- 1.6.5. General Sturm-Liouville problems 92107
- 1.6.6. Complex-valued eigenfunctions and eigenvalues 95110

- Chapter 2. BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 99114
- 2.1. The Heat Equation 99114
- 2.2. Homogeneous Boundary Conditions on a Slab 110125
- 2.3. Nonhomogeneous Boundary Conditions 121136
- 2.4. The Vibrating String 134149
- 2.4.1. Derivation of the equation 134149
- 2.4.2. Linearized model 137152
- 2.4.3. Motion of the plucked string 138153
- 2.4.4. Acoustic interpretation 140155
- 2.4.5. Explicit (d'Alembert) representation 141156
- 2.4.6. Motion of the struck string 145160
- 2.4.7. d'Alembert's general solution 146161
- 2.4.8. Vibrating string with external forcing 148163

- 2.5. Applications of Multiple Fourier Series 152167
- 2.5.1. The heat equation (homogeneous boundary conditions) 153168
- 2.5.2. Laplace's equation 155170
- 2.5.3. The heat equation (nonhomogeneous boundary conditions) 157172
- 2.5.4. The wave equation (nodal lines) 159174
- 2.5.5. Multiplicities of the eigenvalues 162177
- 2.5.6. Implementation with Mathematica 164179
- 2.5.7. Application to Poisson's equation 165180

- Chapter 3. BOUNDARY-VALUE PROBLEMS IN CYLINDRICAL COORDINATES 171186
- 3.1. Laplace's Equation and Applications 171186
- 3.1.1. Laplacian in cylindrical coordinates 171186
- 3.1.2. Separated solutions of Laplace's equation in p, p 173188
- 3.1.3. Application to boundary-value problems 174189
- 3.1.4. Regularity 177192
- 3.1.5. Uniqueness of solutions 177192
- 3.1.6. Exterior problems 178193
- 3.1.7. Wedge domains 178193
- 3.1.8. Neumann problems 179194
- 3.1.9. Explicit representation by Poisson's formula 179194

- 3.2. Bessel Functions 183198
- 3.2.1. Bessel's equation 183198
- 3.2.2. The power series solution of Bessel's equation 184199
- 3.2.3. Integral representation of Bessel functions 188203
- 3.2.4. The second solution of Bessel's equation 191206
- 3.2.5. Zeros of the Bessel function J[sub(0)]. 192207
- 3.2.6. Asymptotic behavior and zeros of Bessel functions 193208
- 3.2.7. Fourier-Bessel series 197212
- 3.2.8. Implementation with Mathematica 202217

- 3.3. The Vibrating Drumhead 209224
- 3.4. Heat Flow in the Infinite Cylinder 216231
- 3.5. Heat Flow in the Finite Cylinder 227242

- Chapter 4. BOUNDARY-VALUE PROBLEMS IN SPHERICAL COORDINATES 235250
- Chapter 5. FOURIER TRANSFORMS AND APPLICATIONS 277292
- 5.1. Basic Properties of the Fourier Transform 277292
- 5.1.1. Passage from Fourier series to Fourier integrals 277292
- 5.1.2. Definition and properties of the Fourier transform 279294
- 5.1.3. Fourier sine and cosine transforms 285300
- 5.1.4. Generalized h-transform 287302
- 5.1.5. Fourier transforms in several variables 288303
- 5.1.6. The uncertainty principle 289304
- 5.1.7. Proof of convergence 291306

- 5.2. Solution of the Heat Equation for an Infinite Rod 294309
- 5.2.1. First method: Fourier series and passage to the limit 294309
- 5.2.2. Second method: Direct solution by Fourier transform 295310
- 5.2.3. Verification of the solution 296311
- 5.2.4. Explicit representation by the Gauss-Weierstrass kernel 297312
- 5.2.5. Some explicit formulas 300315
- 5.2.6. Solutions on a half-line: The method of images 303318
- 5.2.7. The Black-Scholes model 310325
- 5.2.8. Hermite polynomials 314329

- 5.3. Solutions of the Wave Equation and Laplace's Equation 318333
- 5.3.1. One-dimensional wave equation and d'Alembert's formula 318333
- 5.3.2. General solution of the wave equation 321336
- 5.3.3. Three-dimensional wave equation and Huygens' principle 323338
- 5.3.4. Extended validity of the explicit representation 327342
- 5.3.5. Application to one-and two-dimensional wave equations 329344
- 5.3.6. Laplace's equation in a half-space: Poisson's formula 332347

- 5.4. Solution of the Telegraph Equation 335350

- Chapter 6. ASYMPTOTIC ANALYSIS 345360

Table of Contents pages: 1 2