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Beginning Partial Differential Equations, Solutions Manual / Edition 2

Beginning Partial Differential Equations, Solutions Manual / Edition 2

by Peter V. O'Neil


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A rigorous, yet accessible, introduction to partial differential equations—updated in a valuable new edition

Beginning Partial Differential Equations, Second Edition provides a comprehensive introduction to partial differential equations (PDEs) with a special focus on the significance of characteristics, solutions by Fourier series, integrals and transforms, properties and physical interpretations of solutions, and a transition to the modern function space approach to PDEs. With its breadth of coverage, this new edition continues to present a broad introduction to the field, while also addressing more specialized topics and applications.

Maintaining the hallmarks of the previous edition, the book begins with first-order linear and quasi-linear PDEs and the role of characteristics in the existence and uniqueness of solutions. Canonical forms are discussed for the linear second-order equation, along with the Cauchy problem, existence and uniqueness of solutions, and characteristics as carriers of discontinuities in solutions. Fourier series, integrals, and transforms are followed by their rigorous application to wave and diffusion equations as well as to Dirichlet and Neumann problems. In addition, solutions are viewed through physical interpretations of PDEs. The book concludes with a transition to more advanced topics, including the proof of an existence theorem for the Dirichlet problem and an introduction to distributions.

Additional features of the Second Edition include solutions by both general eigenfunction expansions and numerical methods. Explicit solutions of Burger's equation, the telegraph equation (with an asymptotic analysis of the solution), and Poisson's equation are provided. A historical sketch of the field of PDEs and an extensive section with solutions to selected problems are also included.

Beginning Partial Differential Equations, Second Edition is an excellent book for advanced undergraduate- and beginning graduate-level courses in mathematics, science, and engineering.

Product Details

ISBN-13: 9780470133897
Publisher: Wiley
Publication date: 04/18/2008
Series: Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts Series
Edition description: Solution M
Pages: 180
Product dimensions: 6.24(w) x 9.31(h) x 0.50(d)

Table of Contents

First-Order Equations     1
Notation and Terminology     1
The Linear First Order Equation     3
The Significance of Characteristics     6
The Quasi-Linear Equation     11
Linear Second-Order Equations     17
Classification     17
The Hyperbolic Canonical Form     17
The Parabolic Canonical Form     20
The Elliptic Canonical Form     21
Problems for Sections 2.1 - 2.4     23
Some Equations of Mathematical Physics     31
The Second Order Cauchy Problem     31
Characteristics and the Cauchy Problem     33
Elements of Fourier Analysis     37
Why Fourier Series?     37
The Fourier Series of a Function     37
Convergence of Fourier Series     38
Sine and Cosine Expansions     50
The Fourier Integral     65
The Fourier Transform     66
Convolution     72
Fourier Sine and Cosine Transforms     74
The Wave Equation     77
The Cauchy Problem and d'Alembert's Solution     77
d'Alembert's Solution As a Sum of Waves     78
The Characteristic Triangle     78
The Wave Equation on a Half-Line     79
A Problem on a Half-Line With Moving End     82
A Nonhomogeneous Problem on the Real Line     84
A General Problem on a Closed Interval     87
Fourier Series Solutions on a Closed Interval     94
A Nonhomogeneous Problem on a Closed Interval     106
The Cauchy Problem by Fourier Integral     109
A Wave Equation in Two Space Dimensions     112
The Kirchhoff-Poisson Solution     112
Hadamard's Method of Descent     113
The Heat Equation     115
The Cauchy Problem and Initial Conditions     115
Solutions on Bounded Intervals     116
The Heat Equation on the Real Line     121
The Heat Equation on the Half-Line     124
The Nonhomogeneous Heat Equation     125
The Heat Equation In Several Space Variables     129
Dirichlet and Neumann Problems     133
The Setting of the Problems     133
Some Harmonic Functions     133
Representation Theorems     134
Two Properties of Harmonic Functions     136
Dirichlet Problem for a Rectangle     138
Dirichlet Problem for a Disk     141
Poisson's Integral Representation for a Disk     144
Dirichlet Problem for the Upper Half-Plane     146
Dirichlet Problem for the Right Quarter-Plane     147
Dirichlet Problem for a Rectangular Box     148
The Neumann Problem     148
Neumann Problem for a Rectangle     149
Neumann Problem for a Disk     151
Neumann Problem for the Upper Half-Plane     153
Green's Function for a Dirichlet Problem     153
Conformal Mapping Techniques     157
Conformal Mappings     157
Bilinear Transformations     157
Construction of Conformal Mappings Between Domains     159
Solution of Dirichlet Problems by Conformal Mapping     161
Existence Theorems     165
A Classical Existence Theorem     165
A Hilbert Space Approach     165
Distributions and an Existence Theorem     166
Additional Topics     167
Solutions by Eigenfunction Expansions     167
Numerical Approximations of Solutions     173
Burger's Equation     176
The Telegraph Equation     178
Poisson's Equation     179

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