The Fractional Laplacian

The fractional Laplacian in one dimension
Random walkers with constant steps
Ordinary diffusion
Random jumpers
Central limit theorem and stable distributions
Power-law probability jump lengths
A principal-value integral
Wires and springs
The fractional Laplacian
Fourier transform
Effect of fractional order
Numerical computation of the fractional Laplacian
Green’s function of the fractional Laplace equation
Fractional Poisson equation in a restricted domain
Green’s function of unsteady fractional diffusion

Numerical discretization in one dimension
Computation of a principal-value integral
Fractional Laplacian differentiation matrix
Fractional Poisson equation
Evolution under fractional diffusion
Differentiation by spectral expansion

Further concepts in one dimension
Fractional first derivative
Properties of the fractional first derivative
Laplacian potential
Fractional derivatives from finite-difference stencils
Fractional third derivative
Fractional fourth derivative

Periodic functions
Sine, cosines, and the complete Fourier series
Cosine Fourier series
Sine Fourier series
Green’s functions
Integral representation of the periodic Laplacian
Numerical discretization
Periodic differentiation matrix
Differentiation by spectral expansion
Embedding of the fractional Poisson equation

The fractional Laplacian in three dimensions
Stipulation on the Fourier transform
Integral representation
Fractional gradient
Laplacian potential
Green’s function of the fractional Laplace equation
The Riesz potential
Triply periodic Green’s function
Fractional Poisson equation
Evolution under fractional diffusion
Periodic functions and arbitrary domains
Fractional Stokes flow

The fractional Laplacian in two dimensions
Stipulation on the Fourier transform
Integral representation
Fractional gradient
Laplacian potential
Green’s function of the fractional Laplace equation
The Riesz potential
Doubly periodic Green’s function
Fractional Poisson equation
Evolution due to fractional diffusion
Periodic functions and arbitrary domains

Appendix A: Selected definite integrals
Appendix B: The Gamma function
Appendix C: The Gaussian distribution
Appendix D: The fractional Laplacian in arbitrary dimensions
Appendix E: Fractional derivatives
Appendix F: Aitken extrapolation of an infinite sum

References

Index