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Summary
Summary
Solutions Manual to Accompany Beginning Partial Differential Equations, 3rd Edition
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.
Author Notes
Peter V. O'Neil, PhD, is Professor Emeritus in the Department of Mathematics at The University of Alabama at Birmingham. Dr. O'Neil has over forty years of academic experience and is the recipient of the Lester R. Ford Award from the Mathematical Association of America. He is a member of the American Mathematical Society, the Society for Industrial and Applied Mathematics, and the American Association for the Advancement of Science.
Table of Contents
| Preface | p. vii |
| 1 First Ideas | p. 1 |
| 1.1 Two Partial Differential Equations | p. 1 |
| 1.2 Fourier Series | p. 4 |
| 1.3 Two Eigenvalue Problems | p. 12 |
| 1.4 A Proof of the Convergence Theorem | p. 14 |
| 2 Solutions of the Heat Equation | p. 15 |
| 2.1 Solutions on an Interval [0,L] | p. 15 |
| 2.2 A Nonhomogeneous Problem | p. 19 |
| 3 Solutions of the Wave Equation | p. 25 |
| 3.1 Solutions on Bounded Intervals | p. 25 |
| 3.2 The Cauchy Problem | p. 32 |
| 3.2.1 d'Alembert's Solution | p. 32 |
| 3.2.2 The Cauchy Problem on a Half Line | p. 36 |
| 3.2.3 Characteristic Triangles and Quadrilaterals | p. 41 |
| 3.2.4 A Cauchy Problem with a Forcing Term | p. 41 |
| 3.2.5 String with Moving Ends | p. 42 |
| 3.3 The Wave Equation in Higher Dimensions | p. 46 |
| 3.3.1 Vibrations in a Membrane with Fixed Frame | p. 46 |
| 3.3.2 The Poisson Integral Solution | p. 47 |
| 3.3.3 Hadamard's Method of Descent | p. 47 |
| 4 Dirichlet and Neumann Problems | p. 49 |
| 4.1 Laplace's Equation and Harmonic Functions | p. 49 |
| 4.2 The Dirichlet Problem for a Rectangle | p. 50 |
| 4.3 The Dirichlet Problem for a Disk | p. 52 |
| 4.4 Properties of Harmonic Functions | p. 57 |
| 4.4.1 Topology of R n | p. 57 |
| 4.4.2 Representation Theorems | p. 58 |
| 4.4.3 The Mean Value Theorem and the Maximum Principle | p. 60 |
| 4.5 The Neumann Problem | p. 61 |
| 4.5.1 Uniqueness and Existence | p. 61 |
| 4.5.2 Neumann Problem for a Rectangle | p. 62 |
| 4.5.3 Neumann Problem for a Disk | p. 63 |
| 4.6 Poisson's Equation | p. 64 |
| 4.7 An Existence Theorem for the Dirichlet Problem | p. 65 |
| 5 Fourier Integral Methods of Solution | p. 67 |
| 5.1 The Fourier Integral of a Function | p. 67 |
| 5.2 The Heat Equation on the Real Line | p. 70 |
| 5.3 The Debate Over the Age of the Earth | p. 73 |
| 5.4 Burgers' Equation | p. 73 |
| 5.5 The Cauchy Problem for the Wave Equation | p. 74 |
| 5.6 Laplace's Equation on Unbounded Domains | p. 76 |
| 6 Solutions Using Eigenfunction Expansions | p. 79 |
| 6.1 A Theory of Eigenfunction Expansions | p. 79 |
| 6.2 Bessel Functions | p. 83 |
| 6.3 Applications of Bessel Functions | p. 87 |
| 6.3.1 Temperature Distribution in a Solid Cylinder | p. 87 |
| 6.3.2 Vibrations of a Circular Drum | p. 87 |
| 6.4 Legendre Polynomials and Applications | p. 90 |
| 7 Integral Transform Methods of Solution | p. 97 |
| 7.1 The Fourier Transform | p. 97 |
| 7.2 Heat and Wave Equations | p. 101 |
| 7.3 The Telegraph Equation | p. 104 |
| 7.4 The Laplace Transform | p. 106 |
| 8 First-Order Equations | p. 109 |
| 8.1 Linear First-Order Equations | p. 109 |
| 8.2 The Significance of Characteristics | p. 111 |
| 8.3 The Quasi-Linear Equation | p. 114 |
| Series List | p. 117 |
