Pozrikidis C. The Fractional Laplacian

Boca Raton: CRC Press, 2016. - 292p.The ordinary Laplacian is defined as the ordinary second derivative of a function of one variable or the sum of the ordinary second partial derivatives of a function of a higher number of variables in a physical or abstract Caresian space. Physically, the ordinary Laplacian describes an ordinary diffusion process in an isotropic medium mediated by non-idle random walkers who step into neighboring or nearby sites of an idealized grid, but are unable to perform long jumps.
In the physical sciences, the ordinary Laplacian appears as a contribution to a conservation law or evolution equation due to a diffusive species flux according to Fick’s law, a conductive thermal flux according to Fourier’s law, or a viscous stress according to the Newtonian constitutive equation. An implied assumption is that the rate of transport of a field of interest at a certain location is determined by an appropriate field variable at that location, independent of the global structure of the transported field.
The fractional Laplacian, also called the Riesz fractional derivative, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighboring or nearby sites, and also perform excursions to remote sites by way of L´evy flights. Literal or conceptual flights have been observed or alleged to occur in a variety of applications, including turbulent fluid motion and material transport in fractured media. In the context of mechanics, the fractional Laplacian describes the motion of a chain or array of particles that are connected by elastic springs not only to their nearest neighbors, but also to all other particles. The spring constant diminishes with the particle separation, while the particle array may describe an ordinary or fractal configuration.
A key physical concept underlying the notion of the fractional Laplacian is the fractional diffusive flux, arising as a generalization of the ordinary diffusive flux expressed by Fick’s law, the ordinary conductive flux expressed by Fourier’s law, or the expression for the viscous stress according to the Newtonian constitutive equation. The generalized flux associated with the fractional Laplacian provides us with expressions for the rate of transport at a certain location as an integral of an appropriate field variable over an appropriate domain of influence. The fractional diffusive flux at a certain location is affected by the state of the field in the entire space.