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Part Ⅰ Fraction？l Continuous Models of Fractal Distributions1 Fractional Integration and Fractals 3

1.1 Riemann-Liouville fractional integrals 4

1.2 Liouville fractional integrals 6

1.3 Riesz fractional integrals 7

1.4 Metric and measure spaces 9

1.5 Hausdorff measure 10

1.6 Hausdorff dimension and fractals 14

1.7 Box-counting dimension 16

1.8 Mass dimension of fractal systems 19

1.9 Elementary models of fractal distributions 20

1.10 Functions and integrals on fractals 22

1.11 Properties of integrals on fractals 25

1.12 Integration over non-integer-dimensional space 26

1.13 Multi-variable integration on fractals 28

1.14 Mass distribution on fractals 29

1.15 Density of states in Euclidean space 31

1.16 Fractional integral and measure on the real axis 32

1.17 Fractional integral and mass on the real axis 34

1.18 Mass of fractal media 36

1.19 Electric charge of fractal distribution 38

1.20 Probability on fractals 39

1.21 Fractal distribution of particles 41

References 44

2 Hydrodynamics of Fractal Media 49

2.1 Introduction 49

2.2 Equation of balance of mass 50

2.3 Total time derivative of fractional integral 51

2.4 Equation of continuity for fractal media 54

2.5 Fractional integral equation of balance of momentum 55

2.6 Differential equations of balance of momentum 56

2.7 Fractional integral equation of balance of energy 57

2.8 Differential equation of balance of energy 58

2.9 Euler's equations for fractal media 60

2.10 Navier-Stokes equations for fractal media 62

2.11 Equilibrium equation for fractal media 63

2.12 Bernoulli integral for fractal media 64

2.13 Sound waves in fractal media 66

2.14 One-dimensional wave equation in fractal media 67

2.15 Conclusion 69

References 69

3 Fractal Rigid Body Dynamics 73

3.1 Introduction 73

3.2 Fractional equation for moment of inertia 74

3.3 Moment of inertia of fractal rigid body ball 76

3.4 Moment of inertia for fractal rigid body cylinder 78

3.5 Equations of motion for fractal rigid body 81

3.6 Pendulum with fractal rigid body 82

3.7 Fractal rigid body rolling down an inclined plane 84

3.8 Conclusion 85

References 86

4 Electrodynamics of Fractal Distributions of Charges and Fields 89

4.1 Introduction 89

4.2 Electric charge of fractal distribution 90

4.3 Electric current for fractal distribution 92

4.4 Gauss'theorem for fractal distribution 93

4.5 Stokes'theorem for fractal distribution 93

4.6 Charge conservation for fractal distribution 94

4.7 Coulomb's and Biot-Savart laws for fractal distribution 95

4.8 Gauss'law for fractal distribution 96

4.9 Ampere's law for fractal distribution 97

4.10 Integral Maxwell equations for fractal distribution 98

4.11 Fractal distribution as an effective medium 100

4.12 Electric multipole expansion for fractal distribution 101

4.13 Electric dipole moment of fractal distribution 103

4.14 Electric quadrupole moment of fractal distribution 104

4.15 Magnetohydrodynamics of fractal distribution 107

4.16 Stationary states in magnetohydrodynamics of fractal distributions 110

4.17 Conclusion 111

References 112

5 Ginzburg-Landau Equation for Fractal Media 115

5.1 Introduction 115

5.2 Fractional generalization of free energy functional 116

5.3 Ginzburg-Landau equation from free energy functional 117

5.4 Fractional equations from variational equation 118

5.5 Conclusion 121

References 121

6 Fokker-Planck Equation for Fractal Distributions of Probability 123

6.1 Introduction 123

6.2 Fractional equation for average values 124

6.3 Fractional Chapman-Kolmogorov equation 125

6.4 Fokker-Planck equation for fractal distribution 127

6.5 Stationary solutions of generalized Fokker-Planck equation 130

6.6 Conclusion 132

References 132

7 Statistical Mechanics of Fractal Phase Space Distributions 135

7.1 Introduction 135

7.2 Fractal distribution in phase space 136

7.3 Fractional phase volume for configuration space 136

7.4 Fractional phase volume for phase space 139

7.5 Fractional generalization of normalization condition 139

7.6 Continuity equation for fractal distribution in configuration space 141

7.7 Continuity equation for fractal distribution in phase space 142

7.8 Fractional average values for configuration space 144

7.9 Fractional average values for phase space 145

7.10 Generalized Liouville equation 146

7.11 Reduced distribution functions 147

7.12 Conclusion 148

References 150

Part Ⅱ Fractional Dynamics and Long-Range Interactions 150

8 Fractional Dynamics of Media with Long-Range Interaction 153

8.1 Introduction 153

8.2 Equations of lattice vibrations and dispersion law 155

8.3 Equations of motion for interacting particles 160

8.4 Transform operation for discrete models 162

8.5 Fourier series transform of equations of motion 163

8.6 Alpha-interaction of particles 166

8.7 Fractional spatial derivatives 170

8.8 Riesz fractional derivatives and integrals 174

8.9 Continuous limits of discrete equations 177

8.10 Linear nearest-neighbor interaction 180

8.11 Linear integer long-range alpha-interaction 181

8.12 Linear fractional long-range alpha-interaction 184

8.13 Fractional reaction-diffusion equation 187

8.14 Nonlinear long-range alpha-interaction 190

8.15 Fractional 3-dimensional lattice equation 194

8.16 Fractional derivatives from dispersion law 195

8.17 Fractal long-range interaction 198

8.18 Fractal dispersion law 203

8.19 Grünwald-Letnikov-Riesz long-range interaction 206

8.20 Conclusion 208

References 209

9 Fractional Ginzburg-Landau Equation 215

9.1 Introduction 215

9.2 Particular solution of fractional Ginzburg-Landau equation 216

9.3 Stability of plane-wave solution 220

9.4 Forced fractional equation 221

9.5 Conclusion 222

References 223

10 Psi-Series Approach to Fractional Equations 227

10.1 Introduction 227

10.2 Singular behavior of fractional equation 228

10.3 Resonance terms of fractional equation 229

10.4 Psi-series for fractional equation of rational order 230

10.5 Next to singular behavior 233

10.6 Conclusion 235

References 236

Part Ⅲ Fractional Spatial Dynamics 241

11 Fractional Vector Calculus 241

11.1 Introduction 241

11.2 Generalization of vector calculus 242

11.3 Fundamental theorem of fractional calculus 247

11.4 Fractional differential vector operators 250

11.5 Fractional integral vector operations 253

11.6 Fractional Green's formula 254

11.7 Fractional Stokes'formula 257

11.8 Fractional Gauss'formula 259

11.9 Conclusion 261

References 262

12 Fractional Exterior Calculus and Fractional Differential Forms 265

12.1 Introduction 265

12.2 Differential forms of integer order 266

12.3 Fractional exterior derivative 269

12.4 Fractional differential forms 274

12.5 Hodge star operator 279

12.6 Vector operations by differential forms 281

12.7 Fractional Maxwell's equations in terms of fractional forms 282

12.8 Caputo derivative in electrodynamics 284

12.9 Fractional nonlocal Maxwell's equations 285

12.10 Fractional waves 287

12.11 Conclusion 288

References 289

13 Fractional Dynamical Systems 293

13.1 Introduction 293

13.2 Fractional generalization of gradient systems 294

13.3 Examples of fractional gradient systems 301

13.4 Hamiltonian dynamical systems 305

13.5 Fractional generalization of Hamiltonian systems 307

13.6 Conclusion 311

References 312

14 Fractional Calculus of Variations in Dynamics 315

14.1 Introduction 315

14.2 Hamilton's equations and variations of integer order 315

14.3 Fractional variations and Hamilton's equations 317

14.4 Lagrange's equations and variations of integer order 319

14.5 Fractional variations and Lagrange's equations 321

14.6 Helmholtz conditions and non-Lagrangian equations 323

14.7 Fractional variations and non-Hamiltonian systems 326

14.8 Fractional stability 328

14.9 Conclusion 330

References 331

15 Fractional Statistical Mechanics 335

15.1 Introduction 335

15.2 Liouville equation with fractional derivatives 336

15.3 Bogolyubov equation with fractional derivatives 340

15.4 Vlasov equation with fractional derivatives 343

15.5 Fokker-Planck equation with fractional derivatives 345

15.6 Conclusion 349

References 350

Part Ⅳ Fractional Temporal Dynamics 357

16 Fractional Temporal Electrodynamics 357

16.1 Introduction 357

16.2 Universal response laws 358

16.3 Linear electrodynamics of medium 360

16.4 Fractional equations for laws of universal response 362

16.5 Fractional equations of the Curie-von Schweidler law 364

16.6 Fractional Gauss'laws for electric field 366

16.7 Universal fractional equation for electric field 369

16.8 Universal fractional equation for magnetic field 370

16.9 Fractional damping of magnetic field 372

16.10 Conclusion 373

References 374

17 Fractional Nonholonomic Dynamics 377

17.1 Introduction 377

17.2 Nonholonomic dynamics 378

17.3 Fractional temporal derivatives 385

17.4 Fractional dynamics with nonholonomic constraints 388

17.5 Constraints with fractional derivatives 394

17.6 Equations of motion with fractional nonholonomic constraints 396

17.7 Example of fractional nonholonomic constraints 398

17.8 Fractional conditional extremum 401

17.9 Hamilton's approach to fractional nonholonomic constraints 403

17.10 Conclusion 405

References 406

18 Fractional Dynamics and Discrete Maps with Memory 409

18.1 Introduction 409

18.2 Discrete maps without memory 410

18.3 Caputo and Riemann-Liouville fractional derivatives 415

18.4 Fractional derivative in the kicked term and discrete maps 418

18.5 Fractional derivative in the kicked term and dissipative discrete maps 422

18.6 Fractional equation with higher order derivatives and discrete map 425

18.7 Fractional generalization of universal map for 1＜α≤2 429

18.8 Fractional universal map for α＞2 434

18.9 Riemann-Liouville derivative and universal map with memory 436

18.10 Caputo fractional derivative and universal map with memory 441

18.11 Fractional kicked damped rotator map 445

18.12 Fractional dissipative standard map 447

18.13 Fractional Hénon map 449

18.14 Conclusion 450

References 451

Part Ⅴ Fractional Quantum Dynamics 457

19 Fractional Dynamics of Hamiltonian Quantum Systems 457

19.1 Introduction 457

19.2 Fractional power of derivative and Heisenberg equation 458

19.3 Properties of fractional dynamics 460

19.4 Fractional quantum dynamics of free particle 462

19.5 Fractional quantum dynamics of harmonic oscillator 463

19.6 Conclusion 464

References 465

20 Fractional Dynamics of Open Quantum Systems 467

20.1 Introduction 467

20.2 Fractional power of superoperator 468

20.3 Fractional equation for quantum observables 471

20.4 Fractional dynamical semigroup 473

20.5 Fractional equation for quantum states 475

20.6 Fractional non-Markovian quantum dynamics 477

20.7 Fractional equations for quantum oscillator with friction 478

20.8 Quantum self-reproducing and self-cloning 482

20.9 Conclusion 486

References 487

21 Quantum Analogs of Fractional Derivatives 491

21.1 Introduction 491

21.2 Weyl quantization of differential operators 492

21.3 Quantization of Riemann-Liouville fractional derivatives 494

21.4 Quantization of Liouville fractional derivative 496

21.5 Quantization of nondifferentiable functions 497

21.6 Conclusion 500

References 501

Index 503

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