Drábek / Holubová Elements of Partial Differential Equations

2. revised and extended Auflage 2014
ISBN: 978-3-11-031667-4
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 290 Seiten

Reihe: De Gruyter Textbook

This textbook is an elementary introduction to the basic principles of partial differential equations. With many illustrations it introduces 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 PDEs, and to acquire 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.The book is addressed to students who intend to specialize in mathematics as well as to students of physics, engineering, and economics.

Zielgruppe

Students of Mathematics, Physics, and Engineering; Academic Libra

Autoren/Hrsg.

Drábek, Pavel

Holubová, Gabriela

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure

Inhaltsverzeichnis

1;Preface;5
2;Contents;9
3;1 Motivation, Derivation of Basic Mathematical Models;15
3.1;1.1 Conservation Laws;15
3.1.1;1.1.1 Evolution Conservation Law;17
3.1.2;1.1.2 Stationary Conservation Law;19
3.1.3;1.1.3 Conservation Law in One Dimension;19
3.2;1.2 Constitutive Laws;20
3.3;1.3 Basic Models;21
3.3.1;1.3.1 Convection and Transport Equation;21
3.3.2;1.3.2 Diffusion in One Dimension;23
3.3.3;1.3.3 Heat Equation in One Dimension;24
3.3.4;1.3.4 Heat Equation in Three Dimensions;24
3.3.5;1.3.5 String Vibrations and Wave Equation in One Dimension;25
3.3.6;1.3.6 Wave Equation in Two Dimensions – Vibrating Membrane;29
3.3.7;1.3.7 Laplace and Poisson Equations – Steady States;30
3.4;1.4 Exercises;32
4;2 Classification, Types of Equations, Boundary and Initial Conditions;35
4.1;2.1 Basic Types of Equations;35
4.2;2.2 Classical, General, Generic and Particular Solutions;37
4.3;2.3 Boundary and Initial Conditions;40
4.4;2.4 Well-Posed and Ill-Posed Problems;42
4.5;2.5 Classification of Linear Equations of the Second Order;43
4.6;2.6 Exercises;46
5;3 Linear Partial Differential Equations of the First Order;51
5.1;3.1 Equations with Constant Coefficients;51
5.1.1;3.1.1 Geometric Interpretation – Method of Characteristics;52
5.1.2;3.1.2 Coordinate Method;56
5.1.3;3.1.3 Method of Characteristic Coordinates;57
5.2;3.2 Equations with Non-Constant Coefficients;59
5.2.1;3.2.1 Method of Characteristics;59
5.2.2;3.2.2 Method of Characteristic Coordinates;62
5.3;3.3 Problems with Side Conditions;64
5.4;3.4 Solution in Parametric Form;69
5.5;3.5 Exercises;74
6;4 Wave Equation in One Spatial Variable – Cauchy Problem in R;79
6.1;4.1 General Solution of the Wave Equation;79
6.1.1;4.1.1 Transformation to System of Two First Order Equations;79
6.1.2;4.1.2 Method of Characteristics;80
6.2;4.2 Cauchy Problem on the Real Line;81
6.3;4.3 Principle of Causality;87
6.4;4.4 Wave Equation with Sources;88
6.4.1;4.4.1 Use of Green’s Theorem;90
6.4.2;4.4.2 Operator Method;91
6.5;4.5 Exercises;93
7;5 Diffusion Equation in One Spatial Variable – Cauchy Problem in R;97
7.1;5.1 Cauchy Problem on the Real Line;97
7.2;5.2 Diffusion Equation with Sources;105
7.3;5.3 Exercises;108
8;6 Laplace and Poisson Equations in Two Dimensions;111
8.1;6.1 Invariance of the Laplace Operator;111
8.2;6.2 Transformation of the Laplace Operator into Polar Coordinates;112
8.3;6.3 Solutions of Laplace and Poisson Equations in R2;113
8.3.1;6.3.1 Laplace Equation;113
8.3.2;6.3.2 Poisson Equation;114
8.4;6.4 Exercises;115
9;7 Solutions of Initial Boundary Value Problems for Evolution Equations;117
9.1;7.1 Initial Boundary Value Problems on Half-Line;117
9.1.1;7.1.1 Diffusion and Heat Flow on Half-Line;117
9.1.2;7.1.2 Wave on the Half-Line;119
9.1.3;7.1.3 Problems with Nonhomogeneous Boundary Condition;123
9.2;7.2 Initial Boundary Value Problem on Finite Interval, Fourier Method;123
9.2.1;7.2.1 Dirichlet Boundary Conditions, Wave Equation;125
9.2.2;7.2.2 Dirichlet Boundary Conditions, Diffusion Equation;130
9.2.3;7.2.3 Neumann Boundary Conditions;132
9.2.4;7.2.4 Robin Boundary Conditions;134
9.2.5;7.2.5 Principle of the Fourier Method;138
9.3;7.3 Fourier Method for Nonhomogeneous Problems;139
9.3.1;7.3.1 Nonhomogeneous Equation;139
9.3.2;7.3.2 Nonhomogeneous Boundary Conditions and Their Transformation;141
9.4;7.4 Transformation to Simpler Problems;143
9.4.1;7.4.1 Lateral Heat Transfer in Bar;143
9.4.2;7.4.2 Problem with Convective Term;144
9.5;7.5 Exercises;145
10;8 Solutions of Boundary Value Problems for Stationary Equations;154
10.1;8.1 Laplace Equation on Rectangle;155
10.2;8.2 Laplace Equation on Disc;157
10.3;8.3 Poisson Formula;159
10.4;8.4 Exercises;160
11;9 Methods of Integral Transforms;164
11.1;9.1 Laplace Transform;164
11.2;9.2 Fourier Transform;170
11.3;9.3 Exercises;176
12;10 General Principles;180
12.1;10.1 Principle of Causality (Wave Equation);180
12.2;10.2 Energy Conservation Law (Wave Equation);183
12.3;10.3 Ill-Posed Problem (Diffusion Equation for Negative t);185
12.4;10.4 Maximum Principle (Heat Equation);187
12.5;10.5 Energy Method (Diffusion Equation);190
12.6;10.6 Maximum Principle (Laplace Equation);191
12.7;10.7 Consequences of Poisson Formula (Laplace Equation);193
12.8;10.8 Comparison of Wave, Diffusion and Laplace Equations;196
12.9;10.9 Exercises;196
13;11 Laplace and Poisson equations in Higher Dimensions;201
13.1;11.1 Invariance of the Laplace Operator and its Transformation into Spherical Coordinates;201
13.2;11.2 Green’s First Identity;204
13.3;11.3 Properties of Harmonic Functions;204
13.3.1;11.3.1 Mean Value Property and Strong Maximum Principle;204
13.3.2;11.3.2 Dirichlet Principle;206
13.3.3;11.3.3 Uniqueness of Solution of Dirichlet Problem;207
13.3.4;11.3.4 Necessary Condition for the Solvability of Neumann Problem;208
13.4;11.4 Green’s Second Identity and Representation Formula;209
13.5;11.5 Boundary Value Problems and Green’s Function;211
13.6;11.6 Dirichlet Problem on Half-Space and on Ball;213
13.6.1;11.6.1 Dirichlet Problem on Half-Space;213
13.6.2;11.6.2 Dirichlet Problem on a Ball;216
13.7;11.7 Exercises;220
14;12 Diffusion Equation in Higher Dimensions;223
14.1;12.1 Cauchy Problem in R3;223
14.1.1;12.1.1 Homogeneous Problem;223
14.1.2;12.1.2 Nonhomogeneous Problem;225
14.2;12.2 Diffusion on Bounded Domains, Fourier Method;226
14.2.1;12.2.1 Fourier Method;227
14.2.2;12.2.2 Nonhomogeneous Problems;234
14.3;12.3 General Principles for Diffusion Equation;236
14.4;12.4 Exercises;237
15;13 Wave Equation in Higher Dimensions;239
15.1;13.1 Cauchy Problem in R3 – Kirchhoff’s Formula;239
15.2;13.2 Cauchy Problem in R2;242
15.3;13.3 Wave with Sources in R3;245
15.4;13.4 Characteristics, Singularities, Energy and Principle of Causality;247
15.4.1;13.4.1 Characteristics;247
15.4.2;13.4.2 Energy;248
15.4.3;13.4.3 Principle of Causality;249
15.5;13.5 Wave on Bounded Domains, Fourier Method;252
15.6;13.6 Exercises;268
16;A Sturm-Liouville Problem;273
17;B Bessel Functions;275
18;Some Typical Problems Considered in this Book;281
19;Notation;283
20;Bibliography;285
21;Index;287

Über Autor(innen)

Pavel Drábek and Gabriela Holubová, University of West Bohemia, Czech Republic.