Elements of Partial Differential Equations (eBook)

Pavel Drábek, Gabriela Holubová (Autoren)

eBook Download: PDF | EPUB

2014 | 2. Auflage
290 Seiten
Walter de Gruyter GmbH & Co.KG (Verlag)
978-3-11-031667-4 (ISBN)

Infos

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 PDEs and 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.

The book is addressed to students who intend to specialize in mathematics as well as to students of physics, engineering, and economics.

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

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 19.8.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• ISBN-10: 3-11-031667-6 / 3110316676
• ISBN-13: 978-3-11-031667-4 / 9783110316674