Fractional Laplacian

In mathematics, the fractional Laplacian is an operator which generalizes the notion of spatial derivatives to fractional powers.

Definition

For $0<s<1$, the fractional Laplacian of order s $(-\Delta )^{s}$ can be defined on functions $f:\mathbb {R} ^{d}\rightarrow \mathbb {R}$ as a Fourier multiplier given by the formula

${\mathcal {F}}((-\Delta )^{s}f))(\xi )=|\xi |^{2s}{\mathcal {F}}(f)(\xi )$

where the Fourier transform ${\mathcal {F}}(f)$ of a function $f:\mathbb {R} ^{d}\rightarrow \mathbb {R}$ is given by

${\mathcal {F}}(f)(\xi )=\int \limits _{\mathbb {R} ^{d}}{f(x)e^{-ix\cdot \xi }\,dx}.$

More concretely, the fractional Laplacian can be written as a singular integral operator defined by

$(-\Delta )^{s}f(x)=c_{d,s}\int \limits _{\mathbb {R} ^{d}}{{\frac {f(x)-f(y)}{|x-y|^{d+2s}}}\,dy}$

where $c_{d,s}={\frac {4^{s}\Gamma (d/2+s)}{\pi ^{d/2}|\Gamma (-s)|}}$. 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 $(-\Delta )^{s/2}$ (as defined above), where now $0<s<2$, so that the notion of order matches that of a (pseudo-)differential operator.

See also

• Fractional calculus
• Nonlocal operator
• Riemann-Liouville integral

External links

• "Fractional Laplacian". Nonlocal Equations Wiki, Department of Mathematics, The University of Texas at Austin.