ABSTRACT

This chapter provides physical motivation for the fractional Laplacian of a function of one variable in the context of random walks underlying diffusion, provides a rigorous definition for the fractional Laplacian in terms of the Fourier transform or a principal-value integral, and discusses the Green’s function of the fractional Laplace equation and of the unsteady fractional diffusion equation. The evolution of the variance of the particle position is not obvious. To gain insights, we may follow the particle motion by numerical simulation. In the numerical implementation, the direction of displacement of each particle is assessed by a random-number generator. The position of a particle after n steps is the sum of n independent identically distributed positive or negative discrete displacements. If the distribution of the primary variables does not possess a finite variance, σ2, then the Lévy-Khinchin theorem may apply.