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  .............................................. 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 3 ..................................... 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 3 - Kirchhoff's Formula ............... 225
13.2 Cauchy Problem in 3 ..................................... 228
13.3 Wave with Sources in 3 .................................. 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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