Synopses & Reviews
A broad introduction to PDEs with an emphasis on specialized topics and applications occurring in a variety of fieldsFeaturing a thoroughly revised presentation of topics, Beginning Partial Differential Equations, Third Edition provides a challenging, yet accessible, combination of techniques, applications, and introductory theory on the subject of partial differential equations. The new edition offers nonstandard coverage on material including Burgers’ equation, the telegraph equation, damped wave motion, and the use of characteristics to solve nonhomogeneous problems.
The Third Edition is organized around four themes: methods of solution for initial-boundary value problems; applications of partial differential equations; existence and properties of solutions; and the use of software to experiment with graphics and carry out computations. With a primary focus on wave and diffusion processes, Beginning Partial Differential Equations, Third Edition also includes:
- Proofs of theorems incorporated within the topical presentation, such as the existence of a solution for the Dirichlet problem
- The incorporation of Maple^{™} to perform computations and experiments
- Unusual applications, such as Poe’s pendulum
- Advanced topical coverage of special functions, such as Bessel, Legendre polynomials, and spherical harmonics
- Fourier and Laplace transform techniques to solve important problems
Beginning Partial Differential Equations, Third Edition is an ideal textbook for upper-undergraduate and first-year graduate-level courses in analysis and applied mathematics, science, and engineering.
Review
O'Neil's introductory textbook pivots on four themes: methods ofsolving initial-boundary value problems, properties and existence of solutions, applications of partial differential equations, and usingsoftware to carry out computations and graphics. He focuses on equations of diffusion processes and wave motion, and on Dirichletand Neumann problems. In addition to the standard material, he discusses traveling-wave solutions of Burger's equation, damped wavedmotion, heat and wave equations with forcing terms, a general treatment of eigenfunction expansions, a complete solution of the telegraph equation using the Fourier transform, and other topics.Annotation ©2014 Ringgold, Inc., Portland, OR (protoview.com)
Synopsis
Featuring a challenging, yet accessible, introduction to partial differential equations, Beginning Partial Differential Equations provides a solid introduction to partial differential equations, particularly methods of solution based on characteristics, separation of variables, as well as Fourier series, integrals, and transforms. Thoroughly updated with novel applications, such as Poe's pendulum and Kepler's problem in astronomy, this third edition is updated to include the latest version of Maples, which is integrated throughout the text. New topical coverage includes novel applications, such as Poe's pendulum and Kepler's problem in astronomy.
About the Author
PETER V. O’NEIL, PHD, is Professor Emeritus in the Department of Mathematics at the University of Alabama at Birmingham. He has over forty years of experience in teaching and writing and is the recipient of the Lester R. Ford Award from the Mathematical Association of America. Dr. O’Neil is also a member of the American Mathematical Society, the Mathematical Association of America, the Society for Industrial and Applied Mathematics, and the American Association for the Advancement of Science.
Table of Contents
1 First Ideas 11.1 Two Partial Differential Equations 1
1.2 Fourier Series 10
1.3 Two Eigenvalue Problems 28
1.4 A Proof of the Fourier Convergence Theorem 30
2. Solutions of the Heat Equation 392.1 Solutions on an Interval (0, L) 39
2.2 A Nonhomogeneous Problem 64
2.3 The Heat Equation in Two space Variables 71
2.4 The Weak Maximum Principle 75
3. Solutions of the Wave Equation 813.1 Solutions on Bounded Intervals 81
3.2 The Cauchy Problem 109
3.3 The Wave Equation in Higher Dimensions 137
4. Dirichlet and Neumann Problems 1474.1 Laplace’s Equation and Harmonic Functions 147
4.2 The Dirichlet Problem for a Rectangle 153
4.3 The Dirichlet Problem for a Disk 158
4.4 Properties of Harmonic Functions 165
4.5 The Neumann Problem 187
4.6 Poisson’s Equation 197
4.7 Existence Theorem for a Dirichlet Problem 200
5. Fourier Integral Methods of Solution 2135.1 The Fourier Integral of a Function 213
5.2 The Heat Equation on a Real Line 220
5.3 The Debate over the Age of the Earth 230
5.4 Burger’s Equation 233
5.5 The Cauchy Problem for a Wave Equation 239
5.6 Laplace’s Equation on Unbounded Domains 244
6. Solutions Using Eigenfunction Expansions 2536.1 A Theory of Eigenfunction Expansions 253
6.2 Bessel Functions 266
6.3 Applications of Bessel Functions 279
6.4 Legendre Polynomials and Applications 288
7. Integral Transform Methods of Solution 3077.1 The Fourier Transform 307
7.2 Heat and Wave Equations 318
7.3 The Telegraph Equation 332
7.4 The Laplace Transform 334
8 First-Order Equations 3418.1 Linear First-Order Equations 342
8.2 The Significance of Characteristics 349
8.3 The Quasi-Linear Equation 354
9 End Materials 3619.1 Notation 361
9.2 Use of MAPLE 363
9.3 Answers to Selected Problems 370
Index 434