Part Ⅰ Background 3
1 Heredity and Nonlocality 3
1.1 Heredity 3
1.1.1 Concept of heredity 3
1.1.2 A short excursus in history 4
1.2 Volterra's heredity theory 6
1.2.1 Volterra's heredity laws 6
1.2.2 Hereditary string 7
1.2.3 Hereditary oscillator 9
1.2.4 Energy principle 10
1.2.5 Hereditary electrodynamics 10
1.3 Hereditary kinetics 11
1.3.1 Mechanical origin of heredity 11
1.3.2 Hereditary Boltzmann equation 16
1.3.3 Fokker-Planck equation 18
1.3.4 Pauli and Van Hove equations 19
1.3.5 Hybrid kinetic equations 20
1.4 Hereditary hydrodynamics 22
1.4.1 Physical motivation 22
1.4.2 Polymeric liquids 24
1.4.3 Turbulent diffusion 27
1.4.4 Coarse-gained diffusion models 28
1.5 Hereditary viscoelasticity 29
1.5.1 Boltzmann's viscoelasticity model 29
1.5.2 Elastic solid:amesoscopic approach 30
1.5.3 One-dimensional harmonic lattice 31
1.5.4 Axiomatic approach to continuum mechanics 33
1.6 Hereditary thermodynamics 34
1.6.1 Mechanical approach 34
1.6.2 Hereditary heat-transfer 35
1.6.3 Extended irreversible thermodynamics 36
1.6.4 Axiomatic approach 38
1.6.5 Ecology and climatology 41
1.7 Nonlocal models 41
1.7.1 Many-electron atoms 41
1.7.2 Electron correlation in metals 43
1.7.3 Plasma 43
1.7.4 Vlasov's nonlocal statistical mechanics 45
1.7.5 Turbulence 47
1.7.6 Aggregation equations 49
1.7.7 Nonlocal models in nano-plasticity 50
1.7.8 Nonlocal wave equations 53
References 54
2 Selfsimilarity 59
2.1 Power functions 59
2.1.1 Standard power function 59
2.1.2 Properties of power functions 61
2.1.3 Memory 62
2.1.4 Fractals 64
2.2 Hydrodynamics 67
2.2.1 Newtonian fluids 67
2.2.2 Turbulence 68
2.2.3 Microscopic fluctuations 71
2.2.4 Non-Newtonian fluids 72
2.3 Polymers 75
2.3.1 The Nutting law 75
2.3.2 Relaxation of polymer chains 75
2.3.3 Interpenetrating polymer networks 77
2.4 Reaction-diffusion 78
2.4.1 Diffusion 78
2.4.2 Polymerization 79
2.4.3 Coagulation and fragmentation 80
2.5 Solids 81
2.5.1 Dielectrics 81
2.5.2 Semiconductors 83
2.5.3 Spinglasses 85
2.5.4 Jonscher's universal relaxation law 85
2.6 Optics 87
2.6.1 Luminescence decay 87
2.6.2 Anomalous exciton kinetics 88
2.6.3 Blinking fluorescence of quantum dots 88
2.7 Geophysics 89
2.7.1 Atmosphere and ocean turbulence 89
2.7.2 Groundwater 90
2.7.3 Earthquakes 91
2.7.4 Tsunami 91
2.7.5 Fractal approach 92
2.8 Astrophysics and cosmology 93
2.8.1 Solar wind 93
2.8.2 Interstellar magnetic fields 93
2.8.3 Scintillation statistics 94
2.8.4 Velocity and density statistics from spectral lines 94
2.8.5 Large-scale structure 95
2.8.6 Stochastic selfsimilarity 96
2.9 Some statistical mechanisms 96
2.9.1 Three simple examples 96
2.9.2 Activation mechanism 98
2.9.3 Tunneling 99
2.9.4 Multiple trapping 99
2.9.5 Averaging over a parameter 100
2.9.6 Fermi acceleration 101
References 102
3 Stochasticity 107
3.1 Brownian motion 107
3.1.1 Two kinds of motion 107
3.1.2 Dynamic selfsimilarity 108
3.1.3 Stochastic seffsimilarity 109
3.1.4 Selfsimilarity and stationarity 110
3.1.5 Brownian motion 111
3.1.6 Bm in a nonstationary nonhomogeneous environment 114
3.2 One-dimensional Lévy motion 119
3.2.1 Stable random variables 119
3.2.2 Stable characteristic functions 121
3.2.3 Stable probability densities 123
3.2.4 Discrete time Lévy motion 125
3.2.5 Generalized limit theorem 127
3.2.6 Continuous time Lévy motion 130
3.3 Multidimensional Lévy motion 131
3.3.1 Multivariate symmetric stable vectors 131
3.3.2 Sub-Gaussian random vectors 133
3.3.3 Isotropic stable distributions as limit distributions 134
3.3.4 Isotropic stable densities 135
3.3.5 Lévy-Feldheim motion 137
3.4 Fractional Brownian motion 138
3.4.1 Differential Brownian motion process 138
3.4.2 Integral Brownian motion process 139
3.4.3 Fractional Brownian motion 142
3.4.4 Fractional Gaussian noises 144
3.4.5 Barnes-Allan model 145
3.4.6 Fractional Lévy motion 146
3.5 Fractional Poisson motion 148
3.5.1 Renewal processes 148
3.5.2 Selfsimilar renewal processes 150
3.5.3 Three forms of fractal dust generator 151
3.5.4 The nth arrival time distribution 153
3.5.5 Limit fractional Poisson distributions 154
3.5.6 An alternative models of fPp 156
3.5.7 Compound Poisson process 158
3.6 Lévy flights and Lévy walks 160
3.6.1 Lévy Flights 160
3.6.2 Asymptotic solution of the LF problem 162
3.6.3 Continuous time random walk 164
3.6.4 Some special cases 166
3.6.5 Speed limit effect 169
3.6.6 Moments of spatial distribution 171
3.6.7 Exact solution for one-dimensional walk 175
3.7 Diffusion on fractals 178
3.7.1 Diffusion on the Sierpinski gasket 178
3.7.2 Equation for diffusion on fractals 179
3.7.3 Diffusion on comb-structures 181
3.7.4 Some more on a one-dimensional fractal dust 183
3.7.5 Flights on a single sample 187
3.7.6 Averaging over the whole fractal ensemble 189
References 192
Part Ⅱ Theory 199
4 Fractional Differentiation 199
4.1 Riemann-Liouville fractional derivatives 199
4.2 Properties of R-L fractional derivatives 202
4.2.1 Elementary properties 202
4.2.2 The law of exponents 203
4.2.3 Inverse operators 203
4.2.4 Differentiation of a power function 203
4.2.5 Term-by-term differentiation 205
4.2.6 Differentiation of a product 206
4.2.7 Differentiation of an integral 207
4.2.8 Generalized Taylor series 207
4.2.9 Expression of fractional derivatives through the integers 208
4.2.10 Indirect differentiation:the chain rule 209
4.2.11 Asymptotic behavior as x→a 209
4.2.12 Asymptotic behavior of af(v) (x)as x→∞ 210
4.2.13 The Marchaud derivative 211
4.3 Compositions and superpositions of fractional operators 213
4.3.1 Fractional operators 213
4.3.2 The Gerasimov-Caputo derivative 214
4.3.3 Hilfer's interpolation R-L and G-C fractional derivatives 217
4.3.4 Weighted compositions of fractional operators 218
4.3.5 Fractional derivatives of distributed orders 218
4.4 Generalized functions approach 219
4.4.1 Generalized functions 219
4.4.2 Basic properties 220
4.4.3 Regularization of power functions 222
4.4.4 Marchaud derivative as a result of regularization 224
4.5 Integral transformations 224
4.5.1 The Laplace transformation 224
4.5.2 The Mellin transform 226
4.5.3 The Fourier transform 229
4.6 Potentials and fractional derivatives 230
4.6.1 The Riesz potentials on a straight line 230
4.6.2 The Fourier transforms of the Riesz potentials 232
4.6.3 The Riesz derivatives 232
4.6.4 The Fourier transforms of the Riesz derivatives 234
4.6.5 The Feller potential 235
4.7 Fractional operators in multidimensional spaces 237
4.7.1 The Riesz potentials and derivatives 237
4.7.2 Directional derivatives and gradients 240
4.7.3 Varous fractionalizing grad,div,and curl operators 242
4.8 Concluding remarks 245
4.8.1 Leibniz's definition 245
4.8.2 Euler-Lacroix's definition 246
4.8.3 The Fourier definitions 246
4.8.4 The Liouville definitions 247
4.8.5 Riemann's definition with complementary function 247
4.8.6 From Sonin's to Nishimoto's fractional operators 248
4.8.7 Local fractional derivatives 249
4.8.8 The Jumarie nonstandard approach 250
References 251
5 Equations and Solutions 257
5.1 Ordinary equations 257
5.1.1 Initialization 257
5.1.2 Reduction to an integral equation 260
5.1.3 Solution of inhomogeneous R-L fractional equation 261
5.1.4 Solution of the inhomogeneous G-C fractional equation 262
5.1.5 Indicial polynomial method 263
5.1.6 Power series method 265
5.1.7 Series expansion of inverse differential operators 266
5.1.8 Method of integral transformations 267
5.1.9 Green's function method 270
5.1.10 The Adomian decomposition method 271
5.1.11 Equations with compositions of fractional operators 276
5.1.12 Equations with superpositions of fractional operators 278
5.1.13 Equations with varying coefficients 279
5.1.14 Nonlinear ordinary equations 281
5.2 Partial fractional equations 285
5.2.1 Super-ballistic equation 285
5.2.2 Subballistic equation 287
5.2.3 Subdiffusion equation 288
5.2.4 The normalization problem 289
5.2.5 Subdiffusion on a half-axis 291
5.2.6 The signalling problem 292
5.2.7 The telegraph equation 293
5.2.8 Multidimensional subdiffusion:the Schneider-Wyss solution 295
5.2.9 One-dimensional symmetric superdiffusion 297
5.2.10 Equations with Lévy-superposition of R-L operators 298
5.2.11 Equations with the Feller,Riesz,and Marchaud operators 300
5.2.12 Lévy-Feldheim motion equation 302
5.2.13 Fractional Poisson motion 303
5.2.14 Lévy-Poisson motion 305
5.2.15 Fractional compound Poisson motion 306
5.2.16 The link between solutions 307
5.2.17 Subordinated Lévy motion 309
5.2.18 Diffusion in a bounded domain 311
5.2.19 Equation for diffusion on fractals 312
5.2.20 Equation for flights on a fractal dust 314
5.2.21 Equation for percolation 316
5.2.22 Nonlinear equations 317
References 321
6 Numerical Methods 329
6.1 Grünwald-Letnikov derivatives 329
6.1.1 Fractional differences 329
6.1.2 The G-L derivatives of integer orders 331
6.1.3 The G-L derivatives of negative fractional orders 332
6.1.4 The G-L derivatives on a semi-axis 333
6.2 Finite-differences methods 334
6.2.1 Numerical approximation of R-L and G-C derivatives 334
6.2.2 Numerical approximation of G-L derivatives 336
6.2.3 Estimation of accuracy 337
6.2.4 Approximation of the Riesz-Feller derivatives 339
6.2.5 Predictor-corrector method 341
6.2.6 The linear scheme 342
6.2.7 The quadratic and cubic schemes 344
6.2.8 The collocation splinemethod 344
6.2.9 The GMMP method 345
6.2.10 The CL method 346
6.2.11 The YA method 347
6.2.12 Galerkin's method 348
6.2.13 Equation with the Riesz fractional derivatives 349
6.2.14 Equation with Riesz-Feller derivatives 351
6.3 Monte Carlo technique 352
6.3.1 The inverse function method 352
6.3.2 Density estimation 354
6.3.3 Simulation of stable random variables 357
6.3.4 Simulation of fractional exponential distribution 361
6.3.5 Fractional R-L integral 362
6.3.6 Simulation of a fractal dust in d-dimensional space 363
6.3.7 Multidimensional Riesz potential 366
6.3.8 Bifractional diffusion equation 367
6.4 Variations,Homotopy and Differential Transforms 371
6.4.1 Variational iteration method 371
6.4.2 Homotopy analysis method 373
6.4.3 Differential transform method 375
References 378
Index 383