 | Klyatskin V.I. Stochastic equations: theory and applications in acoustics, hydrodynamics, magnetohydrodynamics, and radiophysics. Vol.1: Basic concepts, exact results, and asymptotic approximations / transl. from Russ. by A.Vinogradov. - Cham: Springer, 2015. - xix, 418 p.: ill. - Bibliogr.: p.407-413. - Ind.: p.415-418. - (Understanding complex systems). - ISBN 978-3-319-07586-0
Шифр: (И/В17-K58/N 1) 02
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Preface ....................................................... VII
Introduction ................................................. XVII
Part I: Dynamical Description of Stochastic Systems
1 Examples, Basic Problems, Peculiar Features of
Solutions .................................................. 3
1.1 Ordinary Differential Equations: Initial-Value
Problems ................................................... 3
1.1.1 Particles under Random Velocity Field ............... 3
1.1.2 Particles under Random Forces ....................... 9
1.1.3 The Hopping Phenomenon ............................. 10
1.1.4 Systems with Blow-Up Singularities ................. 20
1.1.5 Oscillator with Randomly Varying Frequency
(Stochastic Parametric Resonance) .................. 21
1.2 Linear Ordinary Differential Equations: Boundary-Value
Problems .................................................. 22
1.2.1 Plane Waves in Layered Media: A Wave Incident
on a Medium Layer .................................. 22
1.2.2 Plane Waves in Layered Media: Source Inside the
Medium ............................................. 29
1.2.3 Plane Waves in Layered Media: Two-Layer Model ...... 32
1.3 First-Order Partial Differential Equations ................ 33
1.3.1 Linear First-Order Partial Differential Equations .. 34
1.3.2 Quasilinear Equations .............................. 44
1.3.3 Boundary-Value Problems for Nonlinear Ordinary
Differential Equations ............................. 49
1.3.4 Nonlinear First-Order Partial Differential
Equations .......................................... 50
1.4 Partial Differential Equations of Higher Orders ........... 52
1.4.1 Fundamental Solutions of Wave Problems in Free
Space and Layered Media ............................ 52
1.4.2 Stationary Problems for Maxwell's Equations ........ 59
1.4.3 The Helmholtz Equation (Boundary-Value Problem)
and the Parabolic Equation of Quasi-Optics (Waves
in Randomly Inhomogeneous Media) ................... 60
1.4.4 The Navier-Stokes Equation: Random Forces in
Hydrodynamic Theory of Turbulence .................. 67
1.4.5 Equations of Geophysical Hydrodynamics ............. 79
2 Solution Dependence on Problem Type, Medium Parameters,
and Initial Data .......................................... 81
2.1 Functional Representation of Problem Solution ............. 81
2.1.1 Variational (Functional) Derivatives ............... 81
2.1.2 Principle of Dynamic Causality ..................... 87
2.2 Solution Dependence on Problem's Parameters ............... 88
2.2.1 Solution Dependence on Initial Data ................ 88
2.2.2 Imbedding Method for Boundary-Value Problems ....... 90
3 Indicator Function and Liouville Equation ................. 95
3.1 Ordinary Differential Equations ........................... 95
3.2 First-Order Partial Differential Equations ................ 98
3.2.1 Linear Equations ................................... 99
3.2.2 Quasilinear Equations ............................. 104
3.2.3 General-Form Nonlinear Equations .................. 109
3.3 Higher-Order Partial Differential Equations .............. 109
3.3.1 Parabolic Equation of Quasi-Optics ................ 110
3.3.2 Random Forces in Hydrodynamic Theory of
Turbulence ........................................ 113
Part II: Random Quantities, Processes, and Fields
4 Random Quantities and Their Characteristics .............. 117
5 Random Processes and Their Characteristics ............... 125
5.1 General Remarks .......................................... 125
5.1.1 Random Process Typical Realization Curve .......... 127
5.1.2 Statistics of Random Process Cross Points with
a Line ............................................ 129
5.2 Gaussian Random Process .................................. 130
5.3 Logarithmically Normal Random Process .................... 132
5.4 Discontinuous Random Processes ........................... 133
5.4.1 Poisson (Impulse) Random Process .................. 134
5.4.2 Telegrapher's Random Process ...................... 137
5.4.3 Generalized Telegrapher's Random Process .......... 141
5.5 Markovian Processes ...................................... 143
5.5.1 General Properties ................................ 143
5.5.2 Characteristic Functional of the Markovian
Process ........................................... 151
6 Random Fields ............................................ 155
6.1 Gaussian Vector Random Field ............................. 156
6.2 Statistical Topography of Random Processes and Fields .... 160
6.3 On the Criterion of Stochastic Structure Formation in
Random Media ............................................. 162
7 Correlation Splitting .................................... 165
7.1 General Remarks .......................................... 165
7.2 Gaussian Process ......................................... 168
7.3 Gaussian Random Fields ................................... 170
7.4 Poisson Process .......................................... 172
7.5 Telegrapher's Random Process ............................. 173
7.6 Generalized Telegrapher's Random Process ................. 177
7.7 General-Form Markovian Processes ......................... 178
7.8 Delta-Correlated Random Processes ........................ 181
7.8.1 General Remarks ................................... 181
7.8.2 Asymptotic Meaning of Delta-Correlated Processes
and Fields ........................................ 184
Part III: Stochastic Equations
8 General Approaches to Analyzing Stochastic Dynamic
Systems .................................................. 193
8.1 Ordinary Differential Equations .......................... 193
8.2 Partial Differential Equations ........................... 198
8.2.1 Passive Tracer Transfer in Random Field
of Velocities ..................................... 198
8.2.2 Parabolic Equation of Quasi-Optics ................ 200
8.2.3 Random Forces in the Theory of Hydrodynamic
Turbulence ........................................ 201
8.3 Stochastic Integral Equations (Methods of Quantum Field
Theory) .................................................. 203
8.3.1 Linear Integral Equations ......................... 203
8.3.2 Nonlinear Integral Equations ...................... 211
8.4 Completely Solvable Stochastic Dynamic Systems ........... 218
8.4.1 Ordinary Differential Equations ................... 218
8.4.2 Partial Differential Equations .................... 232
8.5 Delta-Correlated Fields and Processes .................... 237
8.5.1 General Remarks ................................... 237
8.5.2 One-Dimensional Nonlinear Differential Equation ... 240
8.5.3 Linear Operator Equation .......................... 243
8.5.4 Partial Differential Equations .................... 252
9 Stochastic Equations with the Markovian Fluctuations
of Parameters ............................................ 261
9.1 General Remarks .......................................... 261
9.2 Telegrapher's Processes .................................. 263
9.2.1 System of Linear Operator Equations ............... 264
9.2.2 One-Dimensional Nonlinear Differential Equation ... 269
9.2.3 Particle in the One-Dimensional Potential Field ... 271
9.2.4 Ordinary Differential Equation of the n-th Order .. 272
9.2.5 Statistical Interpretation of Telegrapher's
Equation .......................................... 274
9.3 Generalized Telegrapher's Process ........................ 275
9.3.1 Stochastic Linear Equation ........................ 276
9.3.2 One-Dimensional Nonlinear Differential Equation ... 280
9.3.3 Ordinal Differential Equation of the n-th Order ... 281
9.4 Gaussian Markovian Processes ............................. 283
9.4.1 General Remarks ................................... 283
9.4.2 Stochastic Linear Equation ........................ 284
9.4.3 Ordinal Differential Equation of the n-th Order ... 285
9:4.4 The Square of the Gaussian Markovian Process ...... 288
9.5 Markovian Processes with Finite-Dimensional Phase Space .. 289
9.5.1 Two-State Process ................................. 290
9.6 Causal Stochastic Integral Equations ..................... 291
9.6.1 General Remarks ................................... 291
9.6.2 Telegrapher's Random Process ...................... 293
9.6.3 Generalized Telegrapher's Random Process .......... 295
9.6.4 Gaussian Markovian Process ........................ 297
Part IV: Asymptotic and Approximate Methods for Analyzing
Stochastic Equations
10 Gaussian Random Field Delta-Correlated in Time
(Ordinary Differential Equations) ........................ 305
10.1 The Fokker-Planck Equation ............................... 305
10.2 Transitional Probability Distributions ................... 308
10.3 Applicability Range of the Fokker-Planck Equation ........ 310
10.3.1 Langevin Equation ................................. 311
10.4 The Simplest Markovian Random Processes .................. 315
10.4.1 System of Linear Equations ........................ 315
10.4.2 Wiener Random Process ............................. 316
10.4.3 Wiener Random Process with Shear .................. 318
10.4.4 Lognormal Random Process, Intermittency and
Dynamic Localization .............................. 320
10.5 Logarithmic-Normal Random Fields, Intermittency and
Clustering ............................................... 329
10.5.1 Logarithmic-Normal Random Fields .................. 329
10.5.2 Statistical Topography of the Lognormal Random
Fields ............................................ 331
10.6 Causal Integral Equations ................................ 334
10.6.1 General Remarks ................................... 335
10.6.2 Statistical Averaging ............................. 336
11 Methods for Solving and Analyzing the Fokker-Planck
Equation ................................................. 341
11.1 Integral Transformations ................................. 341
11.2 Steady-State Solutions of the Fokker-Planck Equation ..... 344
11.2.1 One-Dimensional Nonlinear Differential Equation ... 344
11.2.2 Hamiltonian Systems ............................... 345
11.2.3 Systems of Hydrodynamic Type ...................... 347
11.3 Boundary-Value Problems for the Fokker-Planck Equation
(Hopping Phenomenon) ..................................... 355
11.3.1 Hopping Phenomenon in Regular Systems ............. 355
11.3.2 Hopping Phenomena in Singular Systems ............. 359
11.4 Asymptotic and Approximate Methods of Solving the
Fokker-Plank Equation .................................... 363
11.4.1 Asymptotic Expansion .............................. 363
11.4.2 Method of Cumulant Expansions ..................... 364
11.4.3 Method of Fast Oscillation Averaging .............. 365
12 Diffusion and Higher Approximations ...................... 377
12.1 General Remarks .......................................... 377
12.2 Diffusion Approximation .................................. 378
12.2.1 Dynamics of a Particle ............................ 379
12.3 Higher Approximations .................................... 383
13 Some Other Approximate Approaches to the Problems
of Statistical Hydrodynamics ............................. 391
13.1 Quasi-Elastic Properties of Isotropic and Stationary
Noncompressible Turbulent Media .......................... 391
13.2 Sound Radiation by Vortex Motions ........................ 396
13.2.1 Sound Radiation by Vortex Lines ................... 398
13.2.2 Sound Radiation by Vortex Rings ................... 401
References ............................................... 407
Index ......................................................... 415
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