Theory and numerical approximations of fractional integrals and derivatives

Contents......Page 5
List of Figures......Page 7
List of Tables......Page 9
Preface......Page 11
 1.1 Riemann-Liouville integral......Page 14
 1.2 Fractional integrals of other types......Page 25
 2.1 Riemann-Liouville derivative......Page 29
 2.2 Some remarks on the Riemann-Liouville derivative......Page 43
 2.3 Caputo derivative......Page 46
 2.4 Some remarks on the Caputo derivative......Page 55
 2.5 Riesz derivative......Page 56
 2.6 The fractional Laplacian......Page 64
 2.7 Fractional derivatives of other types......Page 74
 2.8 Definite conditions for fractional differential equations......Page 81
 3.1 Numerical methods based on polynomial interpolation......Page 94
 3.2 Fractional linear multistep method......Page 99
 3.3 Spectral approximations......Page 102
 3.4 Diffusive approximation......Page 108
 4.1 L1, L2, and L2C methods......Page 111
 4.2 High-order methods based on polynomial interpolation......Page 125
 4.3 Fractional linear multistep method......Page 176
 4.4 Spectral approximations......Page 184
 4.5 Diffusive approximation......Page 189
 5.1 L1, L2, and L2C methods......Page 191
 5.2 Approximation based on spline interpolation......Page 193
 5.3 Grünwald-Letnikov type approximations......Page 200
 5.4 Fractional backward difference formulae with modifications......Page 209
 5.5 Fractional average central difference method......Page 237
 5.7 Numerical method based on finite-part integrals......Page 242
6 Numerical Riesz differentiation......Page 245
 6.1 Indirect approximations to the fractional diffusion operator......Page 247
 6.2 Direct approximations to the fractional diffusion operator......Page 259
 6.3 Indirect approximations to the fractional convection operator......Page 274
 6.4 Direct approximations to the fractional convection operator......Page 284
7 Numerical fractional Laplacian......Page 288
 7.1 Approximations based on regularization and interpolation......Page 292
 7.2 Approximation based on the weighted trapezoidal rule......Page 296
 7.3 Some remarks on the numerical fractional Laplacian......Page 299
Bibliography......Page 301
Index......Page 313