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Beginning Partial Differential Equations / Edition 2

Beginning Partial Differential Equations / Edition 2

by Peter V. O'Neil


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Product Details

ISBN-13: 9780470133903
Publisher: Wiley
Publication date: 04/04/2008
Series: Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts Series , #85
Edition description: Older Edition
Pages: 496
Product dimensions: 6.52(w) x 9.47(h) x 1.33(d)

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.

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Table of Contents

First-Order Equations     1
Notation and Terminology     1
The Linear First-Order Equation     4
The Significance of Characteristics     12
The Quasi-Linear Equation     16
Linear Second-Order Equations     23
Classification     23
The Hyperbolic Canonical Form     25
The Parabolic Canonical Form     30
The Elliptic Canonical Form     33
Some Equations of Mathematical Physics     38
The Second-Order Cauchy Problem     46
Characteristics and the Cauchy Problem     49
Characteristics as Carriers of Discontinuities     56
Elements of Fourier Analysis     59
Why Fourier Series?     59
The Fourier Series of a Function     60
Convergence of Fourier Series     63
Sine and Cosine Expansions     81
The Fourier Integral     89
The Fourier Transform     95
Convolution     101
Fourier Sine and Cosine Transforms     106
The Wave Equation     109
d'Alembert Solution of the Cauchy Problem     109
d'Alembert's Solution as a Sum of Waves     117
The Characteristic Triangle     126
The Wave Equation on a Half-Line     131
A Half-Line with Moving End     134
A Nonhomogeneous Problem on the Real Line     137
A General Problem on a Closed Interval     141
Fourier Series Solutions on a Closed Interval     150
A Nonhomogeneous Problem on a Closed Interval     159
The Cauchy Problem by Fourier Integral     168
A Wave Equation in Two Space Dimensions     173
The Kirchhoff-Poisson Solution     177
Hadamard's Method of Descent     182
The Heat Equation     185
The Cauchy Problem and Initial Conditions     185
The Weak Maximum Principle     188
Solutions on Bounded Intervals     192
The Heat Equation on the Real Line     210
The Heat Equation on the Half-Line     218
The Debate Over the Age of the Earth     224
The Nonhomogeneous Heat Equation     227
The Heat Equation in Two Space Variables     234
Dirichlet and Neumann Problems     239
The Setting of the Problems     239
Some Harmonic Functions     247
Representation Theorems     251
Two Properties of Harmonic Functions     257
Is the Dirichlet Problem Well Posed?     263
Dirichlet Problem for a Rectangle     266
Dirichlet Problem for a Disk     269
Poisson's Integral Representation for a Disk     272
Dirichlet Problem for the Upper Half-Plane     276
Dirichlet Problem for the Right Quarter-Plane     279
Dirichlet Problem for a Rectangular Box     282
The Neumann Problem     285
Neumann Problem for a Rectangle     288
Neumann Problem for a Disk     290
Neumann Problem for the Upper Half-Plane     294
Green's Function for a Dirichlet Problem     296
Conformal Mapping Techniques     303
Conformal Mappings     303
Bilinear Transformations     308
Construction of Conformal Mappings between Domains     313
An Integral Solution of the Dirichlet Problem for a Disk     320
Solution of Dirichlet Problems by Conformal Mapping     323
Existence Theorems     327
A Classical Existence Theorem     327
A Hilbert Space Approach     336
Distributions and an Existence Theorem     344
Additional Topics     351
Solutions by Eigenfunction Expansions     351
Numerical Approximations of Solutions      370
Burger's Equation     377
The Telegraph Equation     383
Poisson's Equation     390
End Materials     395
Historical Notes     395
Glossary     398
Answers to Selected Problems     399
Index     473

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