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,[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.

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See also

References

  1. Kwaśnicki, Mateusz (2017). "Ten equivalent definitions of the fractional Laplace operator". Fractional Calculus and Applied Analysis. 20. arXiv:1507.07356. doi:10.1515/fca-2017-0002.
  • "Fractional Laplacian". Nonlocal Equations Wiki, Department of Mathematics, The University of Texas at Austin.
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