ON THE GENERALIZED FRACTIONAL LAPLACIAN

The objective of this paper is, for the first time, to extend the fractional Laplacian (-Delta)(s)u(x) over the space C-k(R-n) (which contains S(R-n) as a proper subspace) for all s > 0 and s not equal = 1, 2, ..., based on the normalization in distribution theory, Pizzetti's formula and surface integrals in R-n. We further present two theorems showing that our extended fractional Laplacian is continuous at the end points 1, 2, .... Two illustrative examples are provided to demonstrate computational techniques for obtaining the fractional Laplacian using special functions, Cauchy's residue theorem and integral identities. An application to defining the Riesz derivative in the classical sense at odd numbers is also considered at the end.

Automated summary

The paper extends the fractional Laplacian to the space C-k(R-n) for all s > 0 and s not equal to 1, 2, ..., using normalization in distribution theory, Pizzetti's formula, and surface integrals. It presents two theorems showing the continuity of the extended fractional Laplacian at the end points and provides two illustrative examples using special functions, Cauchy's residue theorem, and integral identities. Additionally, an application to defining the Riesz derivative at odd numbers is considered.