Fractional Laplacian
In mathematics, the fractional Laplacian is an operator which generalizes the notion of spatial derivatives to fractional powers.Definition
For, the fractional Laplacian of order s can be defined on functions as a Fourier multiplier given by the formula
where the Fourier transform of a function is given by
More concretely, the fractional Laplacian can be written as a singular integral operator defined by
where. These two definitions, along with several other definitions, are equivalent.
Some authors prefer to adopt the convention of defining the fractional Laplacian of order s as , where now, so that the notion of order matches that of a differential operator.
