Fractional Laplacian

In mathematics, the fractional Laplacian is an operator which generalizes the notion of spatial derivatives to fractional powers. In mathematics, the fractional Laplacian is an operator which generalizes the notion of spatial derivatives to fractional powers. For 0 < s < 1 {displaystyle 0<s<1} , the fractional Laplacian of order s ( − Δ ) s {displaystyle (-Delta )^{s}} can be defined on functions f : R d → R {displaystyle f:mathbb {R} ^{d} ightarrow mathbb {R} } as a Fourier multiplier given by the formula where the Fourier transform F ( f ) {displaystyle {mathcal {F}}(f)} of a function f : R d → R {displaystyle f:mathbb {R} ^{d} ightarrow mathbb {R} } is given by More concretely, the fractional Laplacian can be written as a singular integral operator defined by where c d , s = 4 s Γ ( d / 2 + s ) π d / 2 | Γ ( − s ) | {displaystyle 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 ( − Δ ) s / 2 {displaystyle (-Delta )^{s/2}} (as defined above), where now 0 < s < 2 {displaystyle 0<s<2} , so that the notion of order matches that of a (pseudo-)differential operator.