ABSTRACT
This chapter discusses the first integral of the fractional Laplacian, identified as a fractional first derivative associated with the fractional Laplacian, and the second integral of the fractional Laplacian, identified as a Laplacian potential. Most important from a physical standpoint, the first integral of the fractional Laplacian provides us with an expression for a fractional diffusive flux encountered in physical applications. The chapter shows that higher-order fractional derivatives can be defined in terms of integral representations involving corresponding finite-difference stencils on a uniform grid. The Brinkman approximation amounts to replacing the fractional first derivative with a weighed average of the limiting distributions. The local law for the diffusive or conductive flux relies on a local homogenization principle based on the continuum approximation.
