In mathematics, the fractional Laplacian is an operator, which generalizes the notion of Laplacian 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 (as defined above), where now , so that the notion of order matches that of a (pseudo-)differential operator.
See also
Fractional calculus
Nonlocal operator
Riemann-Liouville integral
References
External links
"Fractional Laplacian". Nonlocal Equations Wiki, Department of Mathematics, The University of Texas at Austin.