物理及工程中的分数维微积分 第1卷 数学基础及其理论2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

物理及工程中的分数维微积分 第1卷 数学基础及其理论PDF格式电子书版下载

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Part Ⅰ Background3

1 Heredity and Nonlocality3

1.1 Heredity3

1.1.1 Concept of heredity3

1.1.2 A short excursus in history4

1.2 Volterra's heredity theory6

1.2.1 Volterra's heredity laws6

1.2.2 Hereditary string7

1.2.3 Hereditary oscillator9

1.2.4 Energy principle10

1.2.5 Hereditary electrodynamics10

1.3 Hereditary kinetics11

1.3.1 Mechanical origin of heredity11

1.3.2 Hereditary Boltzmann equation16

1.3.3 Fokker-Planck equation18

1.3.4 Pauli and Van Hove equations19

1.3.5 Hybrid kinetic equations20

1.4 Hereditary hydrodynamics22

1.4.1 Physical motivation22

1.4.2 Polymeric liquids24

1.4.3 Turbulent diffusion27

1.4.4 Coarse-gained diffusion models28

1.5 Hereditary viscoelasticity29

1.5.1 Boltzmann's viscoelasticity model29

1.5.2 Elastic solid:amesoscopic approach30

1.5.3 One-dimensional harmonic lattice31

1.5.4 Axiomatic approach to continuum mechanics33

1.6 Hereditary thermodynamics34

1.6.1 Mechanical approach34

1.6.2 Hereditary heat-transfer35

1.6.3 Extended irreversible thermodynamics36

1.6.4 Axiomatic approach38

1.6.5 Ecology and climatology41

1.7 Nonlocal models41

1.7.1 Many-electron atoms41

1.7.2 Electron correlation in metals43

1.7.3 Plasma43

1.7.4 Vlasov's nonlocal statistical mechanics45

1.7.5 Turbulence47

1.7.6 Aggregation equations49

1.7.7 Nonlocal models in nano-plasticity50

1.7.8 Nonlocal wave equations53

References54

2 Selfsimilarity59

2.1 Power functions59

2.1.1 Standard power function59

2.1.2 Properties of power functions61

2.1.3 Memory62

2.1.4 Fractals64

2.2 Hydrodynamics67

2.2.1 Newtonian fluids67

2.2.2 Turbulence68

2.2.3 Microscopic fluctuations71

2.2.4 Non-Newtonian fluids72

2.3 Polymers75

2.3.1 The Nutting law75

2.3.2 Relaxation of polymer chains75

2.3.3 Interpenetrating polymer networks77

2.4 Reaction-diffusion78

2.4.1 Diffusion78

2.4.2 Polymerization79

2.4.3 Coagulation and fragmentation80

2.5 Solids81

2.5.1 Dielectrics81

2.5.2 Semiconductors83

2.5.3 Spinglasses85

2.5.4 Jonscher's universal relaxation law85

2.6 Optics87

2.6.1 Luminescence decay87

2.6.2 Anomalous exciton kinetics88

2.6.3 Blinking fluorescence of quantum dots88

2.7 Geophysics89

2.7.1 Atmosphere and ocean turbulence89

2.7.2 Groundwater90

2.7.3 Earthquakes91

2.7.4 Tsunami91

2.7.5 Fractal approach92

2.8 Astrophysics and cosmology93

2.8.1 Solar wind93

2.8.2 Interstellar magnetic fields93

2.8.3 Scintillation statistics94

2.8.4 Velocity and density statistics from spectral lines94

2.8.5 Large-scale structure95

2.8.6 Stochastic selfsimilarity96

2.9 Some statistical mechanisms96

2.9.1 Three simple examples96

2.9.2 Activation mechanism98

2.9.3 Tunneling99

2.9.4 Multiple trapping99

2.9.5 Averaging over a parameter100

2.9.6 Fermi acceleration101

References102

3 Stochasticity107

3.1 Brownian motion107

3.1.1 Two kinds of motion107

3.1.2 Dynamic selfsimilarity108

3.1.3 Stochastic seffsimilarity109

3.1.4 Selfsimilarity and stationarity110

3.1.5 Brownian motion111

3.1.6 Bm in a nonstationary nonhomogeneous environment114

3.2 One-dimensional Lévy motion119

3.2.1 Stable random variables119

3.2.2 Stable characteristic functions121

3.2.3 Stable probability densities123

3.2.4 Discrete time Lévy motion125

3.2.5 Generalized limit theorem127

3.2.6 Continuous time Lévy motion130

3.3 Multidimensional Lévy motion131

3.3.1 Multivariate symmetric stable vectors131

3.3.2 Sub-Gaussian random vectors133

3.3.3 Isotropic stable distributions as limit distributions134

3.3.4 Isotropic stable densities135

3.3.5 Lévy-Feldheim motion137

3.4 Fractional Brownian motion138

3.4.1 Differential Brownian motion process138

3.4.2 Integral Brownian motion process139

3.4.3 Fractional Brownian motion142

3.4.4 Fractional Gaussian noises144

3.4.5 Barnes-Allan model145

3.4.6 Fractional Lévy motion146

3.5 Fractional Poisson motion148

3.5.1 Renewal processes148

3.5.2 Selfsimilar renewal processes150

3.5.3 Three forms of fractal dust generator151

3.5.4 The nth arrival time distribution153

3.5.5 Limit fractional Poisson distributions154

3.5.6 An alternative models of fPp156

3.5.7 Compound Poisson process158

3.6 Lévy flights and Lévy walks160

3.6.1 Lévy Flights160

3.6.2 Asymptotic solution of the LF problem162

3.6.3 Continuous time random walk164

3.6.4 Some special cases166

3.6.5 Speed limit effect169

3.6.6 Moments of spatial distribution171

3.6.7 Exact solution for one-dimensional walk175

3.7 Diffusion on fractals178

3.7.1 Diffusion on the Sierpinski gasket178

3.7.2 Equation for diffusion on fractals179

3.7.3 Diffusion on comb-structures181

3.7.4 Some more on a one-dimensional fractal dust183

3.7.5 Flights on a single sample187

3.7.6 Averaging over the whole fractal ensemble189

References192

Part Ⅱ Theory199

4 Fractional Differentiation199

4.1 Riemann-Liouville fractional derivatives199

4.2 Properties of R-L fractional derivatives202

4.2.1 Elementary properties202

4.2.2 The law of exponents203

4.2.3 Inverse operators203

4.2.4 Differentiation of a power function203

4.2.5 Term-by-term differentiation205

4.2.6 Differentiation of a product206

4.2.7 Differentiation of an integral207

4.2.8 Generalized Taylor series207

4.2.9 Expression of fractional derivatives through the integers208

4.2.10 Indirect differentiation:the chain rule209

4.2.11 Asymptotic behavior as x→a209

4.2.12 Asymptotic behavior of af(v) (x)as x→∞210

4.2.13 The Marchaud derivative211

4.3 Compositions and superpositions of fractional operators213

4.3.1 Fractional operators213

4.3.2 The Gerasimov-Caputo derivative214

4.3.3 Hilfer's interpolation R-L and G-C fractional derivatives217

4.3.4 Weighted compositions of fractional operators218

4.3.5 Fractional derivatives of distributed orders218

4.4 Generalized functions approach219

4.4.1 Generalized functions219

4.4.2 Basic properties220

4.4.3 Regularization of power functions222

4.4.4 Marchaud derivative as a result of regularization224

4.5 Integral transformations224

4.5.1 The Laplace transformation224

4.5.2 The Mellin transform226

4.5.3 The Fourier transform229

4.6 Potentials and fractional derivatives230

4.6.1 The Riesz potentials on a straight line230

4.6.2 The Fourier transforms of the Riesz potentials232

4.6.3 The Riesz derivatives232

4.6.4 The Fourier transforms of the Riesz derivatives234

4.6.5 The Feller potential235

4.7 Fractional operators in multidimensional spaces237

4.7.1 The Riesz potentials and derivatives237

4.7.2 Directional derivatives and gradients240

4.7.3 Varous fractionalizing grad,div,and curl operators242

4.8 Concluding remarks245

4.8.1 Leibniz's definition245

4.8.2 Euler-Lacroix's definition246

4.8.3 The Fourier definitions246

4.8.4 The Liouville definitions247

4.8.5 Riemann's definition with complementary function247

4.8.6 From Sonin's to Nishimoto's fractional operators248

4.8.7 Local fractional derivatives249

4.8.8 The Jumarie nonstandard approach250

References251

5 Equations and Solutions257

5.1 Ordinary equations257

5.1.1 Initialization257

5.1.2 Reduction to an integral equation260

5.1.3 Solution of inhomogeneous R-L fractional equation261

5.1.4 Solution of the inhomogeneous G-C fractional equation262

5.1.5 Indicial polynomial method263

5.1.6 Power series method265

5.1.7 Series expansion of inverse differential operators266

5.1.8 Method of integral transformations267

5.1.9 Green's function method270

5.1.10 The Adomian decomposition method271

5.1.11 Equations with compositions of fractional operators276

5.1.12 Equations with superpositions of fractional operators278

5.1.13 Equations with varying coefficients279

5.1.14 Nonlinear ordinary equations281

5.2 Partial fractional equations285

5.2.1 Super-ballistic equation285

5.2.2 Subballistic equation287

5.2.3 Subdiffusion equation288

5.2.4 The normalization problem289

5.2.5 Subdiffusion on a half-axis291

5.2.6 The signalling problem292

5.2.7 The telegraph equation293

5.2.8 Multidimensional subdiffusion:the Schneider-Wyss solution295

5.2.9 One-dimensional symmetric superdiffusion297

5.2.10 Equations with Lévy-superposition of R-L operators298

5.2.11 Equations with the Feller,Riesz,and Marchaud operators300

5.2.12 Lévy-Feldheim motion equation302

5.2.13 Fractional Poisson motion303

5.2.14 Lévy-Poisson motion305

5.2.15 Fractional compound Poisson motion306

5.2.16 The link between solutions307

5.2.17 Subordinated Lévy motion309

5.2.18 Diffusion in a bounded domain311

5.2.19 Equation for diffusion on fractals312

5.2.20 Equation for flights on a fractal dust314

5.2.21 Equation for percolation316

5.2.22 Nonlinear equations317

References321

6 Numerical Methods329

6.1 Grünwald-Letnikov derivatives329

6.1.1 Fractional differences329

6.1.2 The G-L derivatives of integer orders331

6.1.3 The G-L derivatives of negative fractional orders332

6.1.4 The G-L derivatives on a semi-axis333

6.2 Finite-differences methods334

6.2.1 Numerical approximation of R-L and G-C derivatives334

6.2.2 Numerical approximation of G-L derivatives336

6.2.3 Estimation of accuracy337

6.2.4 Approximation of the Riesz-Feller derivatives339

6.2.5 Predictor-corrector method341

6.2.6 The linear scheme342

6.2.7 The quadratic and cubic schemes344

6.2.8 The collocation splinemethod344

6.2.9 The GMMP method345

6.2.10 The CL method346

6.2.11 The YA method347

6.2.12 Galerkin's method348

6.2.13 Equation with the Riesz fractional derivatives349

6.2.14 Equation with Riesz-Feller derivatives351

6.3 Monte Carlo technique352

6.3.1 The inverse function method352

6.3.2 Density estimation354

6.3.3 Simulation of stable random variables357

6.3.4 Simulation of fractional exponential distribution361

6.3.5 Fractional R-L integral362

6.3.6 Simulation of a fractal dust in d-dimensional space363

6.3.7 Multidimensional Riesz potential366

6.3.8 Bifractional diffusion equation367

6.4 Variations,Homotopy and Differential Transforms371

6.4.1 Variational iteration method371

6.4.2 Homotopy analysis method373

6.4.3 Differential transform method375

References378

Index383