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Fractional Laplacian - Wikipedia

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In mathematics, the fractional Laplacian is an operator that generalizes the notion of the Laplace operator to fractional powers of spatial derivatives. It is frequently used in the analysis of nonlocal partial differential equations, especially in geometry and diffusion theory. Applications include: Each of these replaces the classical Laplacian in a geometric PDE with the half-Laplacian ( − Δ ) 1 / 2 to account for nonlocal effects. In literature the definition of the fractional Laplacian often varies, but most of the time those definitions are equivalent. The following is a short overview proven by Kwaśnicki, M in.[4] Let 𝑝 ∈ [ 1 , ∞ ) and 𝑋 := 𝐿 𝑝 ( 𝑅 𝑛 ) or let 𝑋 := 𝐶 0 ( 𝑅 𝑛 ) or 𝑋 := 𝐶 𝑏 𝑢 ( 𝑅 𝑛 ) , where: Additionally, let 𝑠 ∈ ( 0 , 1 ) . If we further restrict to 𝑝 ∈ [ 1 , 2 ] , we get This definition uses the Fourier transform for 𝑓 ∈ 𝐿 𝑝 ( 𝑅 𝑛 ) . This definition can also be broadened through the Bessel potential to all 𝑝 ∈ [ 1 , ∞ ) . Th

Fractional Laplacian - Wikipedia Jump to content From Wikipedia, the free encyclopedia Nonlocal mathematical operator In mathematical analysis , the fractional Laplacian is an operator that generalizes the notion of the Laplace operator to fractional powers of spatial derivatives. It is frequently used in the analysis of nonlocal partial differential equations, especially in geometry and diffusion theory. Definition [ edit ] In the literature, the definition of the fractional Laplacian often varies, but most of the time those definitions are equivalent. The following is a short overview proven

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