- Definition
- See also
- References
- External links
In mathematics, the fractional Laplacian is an operator which generalizes the notion of 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,[1] 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
- Riemann-Liouville integral
References
1. ^Kwasnicki, Mateusz. "Ten equivalent definitions of the fractional Laplace operator". https://arxiv.org/pdf/1507.07356.pdf"
External links
- "[https://www.ma.utexas.edu/mediawiki/index.php/Fractional_Laplacian Fractional Laplacian]". Nonlocal Equations Wiki, Department of Mathematics, The University of Texas at Austin.
1 : Fractional calculus