Numerical methods for fractional differentiation

Preface......Page 6
Acknowledgements......Page 7
Contents......Page 8
About the Authors......Page 13
1 Review of Fractional Differentiation......Page 15
  1.1.1 The Gamma Function......Page 23
  1.1.2 The Beta Function......Page 25
  1.1.3 The Complementary Error Function......Page 28
  1.1.4 The Mittag–Leffler Function......Page 29
  1.1.5 Laplace Transformation and Convolution......Page 32
 1.2 Riemann–Liouville Fractional Differentiation......Page 33
 1.3 Caputo Fractional Derivative......Page 41
 1.4 Classical Fractional Derivatives......Page 44
 1.5 Partial Riemann–Liouville Fractional Derivative......Page 47
 1.6 Fractional Operators with Variable Order......Page 50
 1.7 Tempered Fractional Differentiation......Page 63
  1.7.1 Properties of Tempered Fractional Derivative and Integral......Page 68
  1.7.2 Laplace Transforms of the Tempered Fractional Calculus......Page 70
 1.8 Caputo–Fabrizio Fractional Differentiation......Page 74
  1.8.1 Caputo–Fabrizio Fractional Derivative in Caputo Sense......Page 75
  1.8.3 Fourier Transform of Fractional Gradient, Divergence and Laplacian......Page 76
  1.8.4 Caputo–Fabrizio Fractional Derivative  in Riemann–Liouville Sense......Page 78
 1.10 The Atangana–Baleanu Fractional Derivative  and Integral......Page 80
 1.11 The Riesz Potential and Riesz Fractional Derivatives......Page 84
  1.11.1 The Atangana–Gómez Fractional Derivative......Page 86
 References......Page 91
2 Finite Difference Approximations......Page 97
 2.1 Finite Difference Approximation Schemes......Page 98
  2.1.1 Taylor Series and Finite Difference Approximation......Page 99
  2.1.2 Higher Order Finite Difference Approximation......Page 101
 2.2 Error Analysis......Page 103
  2.2.1 Illustrative Example......Page 106
  2.2.2 Order of Accuracy and Consistency......Page 108
  2.2.3 Matrix Notation......Page 111
  2.3.1 Von Neumann (Fourier Series) Stability Analysis......Page 113
  2.3.2 Matrix Stability Analysis......Page 120
  2.3.3 Convergence......Page 124
  2.3.4 Symmetry......Page 128
 2.4 Numerical Results......Page 129
 2.5 Finite Difference Approximations Schemes  for Fractional Equations......Page 130
  2.5.1 Matrix Representation of the Finite Difference Schemes......Page 134
  2.5.2 Convergence Analysis of Fractional Finite Difference Schemes......Page 136
  2.5.3 Stability Analysis of Fractional Finite Difference Schemes......Page 137
 2.6 Numerical Approximation of Time-Fractional Sub-diffusion Process with the Second-Order Implicit Difference Method......Page 141
  2.6.1 Second-Order Implicit Difference Approximation......Page 142
 References......Page 149
 3.1 Numerical Approximation for Time Derivative......Page 152
 3.2 Numerical Approximation for Space First-Order Derivative......Page 155
 3.3 Numerical Approximation for Space Second-Order Derivative......Page 158
 3.4 Crank–Nicholson Scheme for Time-Fractional Differential Equations in Riemann–Liouville Sense......Page 160
 3.5 A New Definition of Fractional Time Derivative in Riemann–Liouville Sense......Page 166
 References......Page 173
 4.1 Numerical Approximation for Time Derivative......Page 174
 4.2 Numerical Approximation for Space First-Order Derivative......Page 176
  4.3.1 Fractional Euler and Adams Methods......Page 178
  4.3.2 The New Fractional Adams–Bashforth Scheme with Caputo Derivative......Page 179
  4.3.3 Existence and Uniqueness of Solutions......Page 181
 References......Page 186
 5.1 Numerical Approximation for the Caputo–Fabrizio Fractional Derivative in Caputo Sense......Page 187
 5.2 Numerical Approximation for Time Derivative......Page 192
 5.3 Numerical Approximation for Space First-Order Derivative......Page 195
 5.4 Numerical Approximation for Space Second-Order Derivative......Page 199
  5.4.1 Three-Step Adams–Bashforth Scheme  with Caputo–Fabrizio Fractional Derivative......Page 201
  5.4.2 Stability Analysis......Page 202
  5.4.3 Applications......Page 204
 References......Page 206
 6.1 Atangana–Baleanu Fractional Derivative in Caputo Sense......Page 207
 6.2 Uniqueness and Existence of Solution via Chaotic Process......Page 210
 6.3 Numerical Experiments......Page 212
 Reference......Page 213
 7.1 Numerical Approximation of Fractional Ordinary Differential Equation with the Caputo Derivative......Page 214
  7.1.1 Numerical Schemes and Stability Analysis......Page 215
 7.2 Modelling the Spread of Viruses in Computer  via the Caputo Fractional Derivative......Page 220
 7.3 Modelling the Spread of River Blindness Disease with the Caputo Fractional Derivative......Page 225
 7.4 Modelling of Nonlinear Interpersonal Relationship with Time-Fractional Derivative......Page 229
 7.5 Modelling of El Niño Chaotic Dynamical System with the Caputo, Caputo–Fabrizio and Atangana–Baleanu Fractional Derivatives......Page 233
 7.6 A Novel Fractional Model for the Lassa Hemorrhagic Fever......Page 246
 7.7 Modelling of Ebola Hemorrhagic Fever: Fractional Derivative Approach......Page 251
 References......Page 259
 8.1 Space-Fractional Diffusion Equation with New (Atangana–Gomez) Fractional Derivative in Riemann–Liouville Sense......Page 261
 8.2 Space-Fractional Diffusion Equation  with the Riemann–Liouville Derivative......Page 265
  8.2.1 Fourier Transform Methods......Page 266
  8.2.2 Finite Difference Methods......Page 268
  8.2.3 Predictor–Corrector Method of Approximation......Page 269
  8.2.4 Fourier Spectral Method for Space-Fractional Reaction–Diffusion......Page 271
 8.3 Application of Caputo–Fabrizio Derivative to Nonlinear Reaction–Diffusion......Page 273
  8.3.2 Stability Analysis via Fixed Point Theorem......Page 274
 8.4 Application of Caputo–Fabrizio Derivative to Transmission Line Model with Losses......Page 278
  8.4.1 Stability Analysis of the Numerical Scheme  with the Caputo–Fabrizio Derivative......Page 279
 8.5 Application of the Caputo–Fabrizio Derivative in Caputo Sense to Time-Fractional Advection–Diffusion Equation......Page 282
  8.5.1 Stability Anlalysis of the Numerical Method with the Caputo–Fabrizio Derivative for Caputo Type......Page 284
  8.5.2 Convergence Analysis of the Numerical Scheme with the Caputo–Fabrizio Derivative for Caputo Type......Page 286
 8.6 Applications of Fractional Derivatives to Diffusion–Advection Equation......Page 287
  8.6.1 Riemann–Liouville Approach......Page 288
  8.6.2 Caputo–Fabrizio–Riemann Approach......Page 289
  8.6.3 Atangana–Baleanu–Riemann Approach......Page 290
 8.7 Application to Partial Fractional Differential Equation......Page 292
  8.7.1 The Fisher Equation......Page 293
  8.7.2 The Gray–Scott Model......Page 310
 8.8 Application of Riesz Fractional Derivative to Schrödinger Equation......Page 320
  8.8.1 One-Dimensional Fractional-in-Space Schrödinger Equation......Page 321
  8.8.2 Two-Dimensional Fractional-in-Space Schrödinger Equation......Page 322
  8.8.3 Numerical Methods of Discretization......Page 324
  8.9.1 One-Dimensional Results for Fractional Schrödinger Equation......Page 328
  8.9.2 Two-Dimensional Results for Fractional Schrödinger Equation......Page 329
  8.9.3 Three-Dimensional Results for Fractional Schrödinger Equation......Page 332
 References......Page 334