Elements of Partial Differential Equations

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ISBN: 9783110191240 | 3110191245
Cover: Paperback
Copyright: 2/28/2007

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This book presents a first introduction to PDEs on an elementary level, enabling the reader to understand what partial differential equations are, where they come from and how they can be solved. The intention is that the reader understands the basic principles which are valid for particular types of PDEsand learns some classical methods to solve them, thus the authors restrict their considerations to fundamental types of equations and basic methods. Only basic facts from calculus and linear ordinary differential equations of first and second order are needed as a prerequisite. An elementary introduction to the basic principles of partial differential equations. With many illustrations.

Preface	p. v
Mathematical Models, Conservation and Constitutive Laws	p. 1
Basic Notions	p. 1
Evolution Conservation Law	p. 3
Stationary Conservation Law	p. 4
Conservation Law in One Dimension	p. 4
Constitutive Laws	p. 6
Exercises	p. 8
Classification, Types of Equations, Boundary and Initial Conditions	p. 9
Basic Types of Equations, Boundary and Initial Conditions	p. 9
Classification of Linear Equations of the Second Order	p. 14
Exercises	p. 17
Linear Partial Differential Equations of the First Order	p. 21
Convection and Transport Equation	p. 21
Equations with Constant Coefficients	p. 22
Equations with Non-Constant Coefficients	p. 28
Exercises	p. 32
Wave Equation in One Spatial Variable-Cauchy Problem in R	p. 37
String Vibrations and Wave Equation in One Dimension	p. 37
Cauchy Problem on the Real Line	p. 40
Wave Equation with Sources	p. 49
Exercises	p. 53
Diffusion Equation in One Spatial Variable-Cauchy Problem in R	p. 57
Diffusion and Heat Equations in One Dimension	p. 57
Cauchy Problem on the Real Line	p. 58
Diffusion Equation with Sources	p. 65
Exercises	p. 68
Laplace and Poisson Equations in Two Dimensions	p. 71
Steady States and Laplace and Poisson Equations	p. 71
Invariance of the Laplace Operator, Its Transformation into Polar Coordinates	p. 73
Solution of Laplace and Poisson Equations in R[superscript 2]	p. 74
Exercises	p. 76
Solutions of Initial Boundary Value Problems for Evolution Equations	p. 78
Initial Boundary Value Problems on Half-Line	p. 78
Initial Boundary Value Problem on Finite Interval, Fourier Method	p. 84
Fourier Method for Nonhomogeneous Problems	p. 99
Transformation to Simpler Problems	p. 103
Exercises	p. 104
Solutions of Boundary Value Problems for Stationary Equations	p. 113
Laplace Equation on Rectangle	p. 113
Laplace Equation on Disc	p. 115
Poisson Formula	p. 117
Exercises	p. 119
Methods of Integral Transforms	p. 123
Laplace Transform	p. 123
Fourier Transform	p. 128
Exercises	p. 134
General Principles	p. 139
Principle of Causality (Wave Equation)	p. 139
Energy Conservation Law (Wave Equation)	p. 141
Ill-Posed Problem (Diffusion Equation for Negative t)	p. 144
Maximum Principle (Heat Equation)	p. 145
Energy Method (Diffusion Equation)	p. 147
Maximum Principle (Laplace Equation)	p. 148
Consequences of Poisson Formula (Laplace Equation)	p. 150
Comparison of Wave, Diffusion and Laplace Equations	p. 152
Exercises	p. 152
Laplace and Poisson equations in Higher Dimensions	p. 157
Invariance of the Laplace Operator	p. 157
Green's First Identity	p. 159
Properties of Harmonic Functions	p. 161
Green's Second Identity and Representation Formula	p. 164
Boundary Value Problems and Green's Function	p. 166
Dirichlet Problem on Half-Space and on Ball	p. 168
Exercises	p. 174
Diffusion Equation in Higher Dimensions	p. 178
Heat Equation in Three Dimensions	p. 178
Cauchy Problem in R[superscript 3]	p. 179
Diffusion on Bounded Domains, Fourier Method	p. 182
Exercises	p. 192
Wave Equation in Higher Dimensions	p. 195
Membrane Vibrations and Wave Equation in Two Dimensions	p. 195
Cauchy Problem in R[superscript 3]-Kirchhoff's Formula	p. 196
Cauchy problem in R[superscript 2]	p. 199
Wave with sources in R[superscript 3]	p. 202
Characteristics, Singularities, Energy and Principle of Causality	p. 204
Wave on Bounded Domains, Fourier Method	p. 208
Exercises	p. 224
Appendix	p. 228
Sturm-Liouville problem	p. 228
Bessel Functions	p. 229
Some Typical Problems Considered in This Book	p. 235
Notation	p. 237
Bibliography	p. 239
Index	p. 241
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Elements of Partial Differential Equations

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