Drabek P. Elements of partial differential equations(Berlin; Boston, 2014). - ОГЛАВЛЕНИЕ / CONTENTS

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ОбложкаDrábek P. Elements of partial differential equations / P.Drábek, G.Holubová - 2nd, revised and extended edition. - Berlin; Boston: De Gruyter, 2014. - xiii, 277 p.: il. - Bibliogr.: p.271-272. - Ind.: p.273-277. - ISBN 978-3-11-031665-0
 

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Оглавление / Contents
 
Preface ......................................................... v

1    Motivation, Derivation of Basic Mathematical Models ........ 1
1.1  Conservation Laws .......................................... 1
     1.1.1  Evolution Conservation Law .......................... 3
     1.1.2  Stationary Conservation Law ......................... 5
     1.1.3  Conservation Law in One Dimension ................... 5
1.2  Constitutive Laws .......................................... 6
1.3  Basic Models ............................................... 7
     1.3.1  Convection and Transport Equation ................... 7
     1.3.2  Diffusion in One Dimension .......................... 9
     1.3.3  Heat Equation in One Dimension ..................... 10
     1.3.4  Heat Equation in Three Dimensions .................. 10
     1.3.5  String Vibrations and Wave Equation in One
            Dimension .......................................... 11
     1.3.6  Wave Equation in Two Dimensions - Vibrating
            Membrane ........................................... 15
     1.3.7  Laplace and Poisson Equations - Steady States ...... 16
1.4  Exercises ................................................. 18

2    Classification, Types of Equations, Boundary and Initial
     Conditions ................................................ 21
2.1  Basic Types of Equations .................................. 21
2.2  Classical, General, Generic and Particular Solutions ...... 23
2.3  Boundary and Initial Conditions ........................... 26
2.4  Well-Posed and Ill-Posed Problems ......................... 28
2.5  Classification of Linear Equations of the Second Order .... 29
2.6  Exercises ................................................. 32

3    Linear Partial Differential Equations of the First
     Order ..................................................... 37
3.1  Equations with Constant Coefficients ...................... 37
     3.1.1  Geometric Interpretation - Method of
            Characteristics .................................... 38
     3.1.2  Coordinate Method .................................. 42
     3.1.3  Method of Characteristic Coordinates ............... 43
3.2  Equations with Non-Constant Coefficients .................. 45
     3.2.1  Method of Characteristics .......................... 45
     3.2.2  Method of Characteristic Coordinates ............... 48
3.3  Problems with Side Conditions ............................. 50
3.4  Solution in Parametric Form ............................... 55
3.5  Exercises ................................................. 60

4    Wave Equation in One Spatial Variable - Cauchy Problem
     in M ...................................................... 65
4.1  General Solution of the Wave Equation ..................... 65
     4.1.1  Transformation to System of Two First Order
            Equations .......................................... 65
     4.1.2  Method of Characteristics .......................... 66
4.2  Cauchy Problem on the Real Line ........................... 67
4.3  Principle of Causality .................................... 73
4.4  Wave Equation with Sources ................................ 74
     4.4.1  Use of Green's Theorem ............................. 76
     4.4.2  Operator Method .................................... 77
4.5  Exercises ................................................. 79

5    Diffusion Equation in One Spatial Variable - Cauchy
     Problem in fig.1 .............................................. 83
5.1  Cauchy Problem on the Real Line ........................... 83
5.2  Diffusion Equation with Sources ........................... 91
5.3  Exercises ................................................. 94

6    Laplace and Poisson Equations in Two Dimensions ........... 97
6.1  Invariance of the Laplace Operator ........................ 97
6.2  Transformation of the Laplace Operator into Polar
     Coordinates ............................................... 98
6.3  Solutions of Laplace and Poisson Equations in ............. 99
     6.3.1  Laplace Equation ................................... 99
     6.3.2  Poisson Equation .................................. 100
6.4  Exercises ................................................ 101

7    Solutions of Initial Boundary Value Problems for
     Evolution Equations ...................................... 103
7.1  Initial Boundary Value Problems on Half-Line ............. 103
     7.1.1  Diffusion and Heat Flow on Half-Line .............. 103
     7.1.2  Wave on the Half-Line ............................. 105
     7.1.3  Problems with Nonhomogeneous Boundary Condition ... 109
7.2  Initial Boundary Value Problem on Finite Interval,
     Fourier Method ........................................... 109
     7.2.1  Dirichlet Boundary Conditions, Wave Equation ...... 111
     7.2.2  Dirichlet Boundary Conditions, Diffusion
            Equation .......................................... 116
     7.2.3  Neumann Boundary Conditions ....................... 118
     7.2.4  Robin Boundary Conditions ......................... 120
     7.2.5  Principle of the Fourier Method ................... 124
7.3  Fourier Method for Nonhomogeneous Problems ............... 125
     7.3.1  Nonhomogeneous Equation ........................... 125
     7.3.2  Nonhomogeneous Boundary Conditions and Their
            Transformation .................................... 127
7.4  Transformation to Simpler Problems ....................... 129
     7.4.1  Lateral Heat Transfer in Bar ...................... 129
     7.4.2  Problem with Convective Term ...................... 130
7.5  Exercises ................................................ 131

8    Solutions of Boundary Value Problems for Stationary
     Equations ................................................ 140
8.1  Laplace Equation on Rectangle ............................ 141
8.2  Laplace Equation on Disc ................................. 143
8.3  Poisson Formula .......................................... 145
8.4  Exercises ................................................ 146

9    Methods of Integral Transforms ........................... 150
9.1  Laplace Transform ........................................ 150
9.2  Fourier Transform ........................................ 156
9.3  Exercises ................................................ 162

10   General Principles ....................................... 166
10.1 Principle of CausaUty (Wave Equation) .................... 166
10.2 Energy Conservation Law (Wave Equation) .................. 169
10.3 Ill-Posed Problem (Diffusion Equation for Negative t) .... 171
10.4 Maximum Principle (Heat Equation) ........................ 173
10.5 Energy Method (Diffusion Equation) ....................... 176
10.6 Maximum Principle (Laplace Equation) ..................... 177
10.7 Consequences of Poisson Formula (Laplace Equation) ....... 179
10.8 Comparison of Wave, Diffusion and Laplace Equations ...... 182
10.9 Exercises ................................................ 182

11   Laplace and Poisson equations in Higher Dimensions ....... 187
11.1 Invariance of the Laplace Operator and its
     Transformation into Spherical Coordinates ................ 187
11.2 Green's First Identity ................................... 190
11.3 Properties of Harmonic Functions ......................... 190
     11.3.1 Mean Value Property and Strong Maximum Principle .. 190
     11.3.2 Dirichlet Principle ............................... 192
     11.3.3 Uniqueness of Solution of Dirichlet Problem ....... 193
     11.3.4 Necessary Condition for the Solvability of
            Neumann Problem ................................... 194
11.4 Green's Second Identity and Representation Formula ....... 195
11.5 Boundary Value Problems and Green's Function ............. 197
11.6 Dirichlet Problem on Half-Space and on Ball .............. 199
     11.6.1 Dirichlet Problem on Half-Space ................... 199
     11.6.2 Dirichlet Problem on a Ball ....................... 202
11.7 Exercises ................................................ 206

12   Diffusion Equation in Higher Dimensions .................. 209
12.1 Cauchy Problem in fig.13 ..................................... 209
     12.1.1 Homogeneous Problem ............................... 209
     12.1.2 Nonhomogeneous Problem ............................ 211
12.2 Diffusion on Bounded Domains, Fourier Method ............. 212
     12.2.1 Fourier Method .................................... 213
     12.2.2 Nonhomogeneous Problems ........................... 220
12.3 General Principles for Diffusion Equation ................ 222
12.4 Exercises ................................................ 223

13   Wave Equation in Higher Dimensions ....................... 225
13.1 Cauchy Problem in fig.13 - Kirchhoff's Formula ............... 225
13.2 Cauchy Problem in fig.13 ..................................... 228
13.3 Wave with Sources in fig.13 .................................. 231
13.4 Characteristics, Singularities, Energy and Principle
     of Causality ............................................. 233
     13.4.1 Characteristics ................................... 233
     13.4.2 Energy ............................................ 234
     13.4.3 Principle of Causality ............................ 235
13.5 Wave on Bounded Domains, Fourier Method .................. 238
13.6 Exercises ................................................ 254

A  Sturm-Liouville Problem .................................... 259
В  Bessel Functions ........................................... 261

Some Typical Problems Considered in this Book ................. 267
Notation ...................................................... 269
Bibliography .................................................. 271

Index ......................................................... 273


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