Boundary Value Problems

By: Chi Y. Lo

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Hardcover | 15 May 2000

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272 Pages

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23 x 16.1 x 2

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This text has been designed for use on a one-year graduate course on boundary value problems for students of mathematics, engineering, and the physical sciences. It deals mainly with the three fundamental equations of mathematical physics, namely the heat equation, the wave equation, and Laplace's equation. The goal of the book is to obtain a formal solution to a given problem either by the method of separation of variables or by the method of general solutions, and to verify that the formal solution possesses all the required properties. To provide the mathematical justification for this approach, the theory of Sturm-Liouville problems, the Fourier series, and the Fourier transform are fully developed. The book assumes a knowledge of advanced calculus and elementary differential equations.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Science Physics Science in General Maths for Scientists Engineering & Technology Technology in General Engineering in General Booktopia Publisher Services

World Scientific Publishing

Preface
Linear Partial Differential Equations	p. 1
Linear problems	p. 1
Classification	p. 5
Well-posed problems	p. 9
Method of general solutions	p. 12
Method of separation of variables	p. 14
The Wave Equation	p. 21
The vibrating string	p. 21
The initial value problem	p. 24
The nonhomogeneous wave equation	p. 29
Uniqueness of the initial value problem	p. 31
Initial-boundary value problems	p. 33
Initial-boundary value problems for semi-infinite string	p. 39
Green's Function and Sturm-Liouville Problems	p. 45
Solutions of second order linear equations	p. 45
Boundary value problems and Green's function	p. 50
Sturm-Liouville problems	p. 58
Convergence in the mean	p. 62
Integral operator with continuous, symmetric kernel	p. 66
Completeness of eigenfunctions of Sturm-Liouville problems	p. 75
Nonhomogeneous integral equation	p. 78
Further properties of eigenvalues and eigenfunctions	p. 81
Fourier Series and Fourier Transforms	p. 95
Trigonometric Fourier series	p. 95
Uniform convergence and completeness	p. 99
Other types of Fourier series	p. 102
Application to the wave equation	p. 105
Fourier integrals	p. 109
Fourier transforms	p. 113
Contour integration	p. 117
The Heat Equation	p. 127
Derivation of the heat equation	p. 127
Maximum principle	p. 130
The initial-boundary value problem	p. 133
Nonhomogeneous problems and finite Fourier transform	p. 135
The initial value problem	p. 138
The initial value problem for the nonhomogeneous equation	p. 142
Nonhomogeneous boundary conditions for initial-boundary value problems	p. 145
Laplace's Equation and Poisson's Equation	p. 151
Boundary value problems	p. 151
Green's identities and uniqueness theorems	p. 152
Maximum principle	p. 154
Laplace's equation in a rectangle	p. 155
Laplace's equation in disc	p. 158
Poisson's integral formula	p. 162
Green's function for Laplace's equation	p. 166
Poisson's equation in a disc	p. 173
Finite Fourier transform for Poisson's equation	p. 177
Dirichlet problem in the upper-half plane	p. 181
Problems in Higher Dimensions	p. 191
Classification	p. 191
Double Fourier series	p. 194
Laplace's equation in a cube	p. 200
The two-dimensional wave equation in a rectangular domain	p. 202
Bessel functions	p. 205
Singular Sturm-Liouville problem for Bessel's equation	p. 209
The two-dimensional wave equation in a circular domain	p. 212
Initial-boundary value problems for the heat equation	p. 217
Legendre's equation	p. 221
Properties of Legendre polynomials	p. 224
Legendre series and boundary value problems	p. 229
Laplace's equation in a sphere	p. 232
Poisson's integral formula in space	p. 235
Ascoli's Theorem	p. 243
App. B: Answers for Selected Problems	p. 245
Bibliography	p. 255
Index	p. 256
Table of Contents provided by Blackwell. All Rights Reserved.

Boundary Value Problems

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Hardcover

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