The paper discusses new cubature formulas for the Riesz potential and fractional Laplacian (−Δ)α/2, 0<α<2, in framework of method approximate approximations. This approach, combined with separated representations, makes successful also high dimensions. We prove error estimates report on numerical results illustrating that our are accurate provide predicted convergence rate 2, 4, 6, 8 up to dime...

We provide a detailed description of the relationships between fractional Laplacian order 2s∈(0,n) on Rn and s-polyharmonic extension operator.

We study the extremal solution for the problem (−∆)u = λf(u) in Ω, u ≡ 0 in R \ Ω, where λ > 0 is a parameter and s ∈ (0, 1). We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions n < 4s. We also show that, for exponential and power-like ...

The absorption of acoustic wave propagation in a broad variety of lossy media is characterized by an empirical power law function of frequency, y ω α0 . It has long been noted that exponent y ranges from 0 to 2 for diverse media. Recently, the present author developed a fractional Laplacian wave equation to accurately model the power law dissipation, which can be further reduced to the fraction...

We prove nonexistence of nontrivial bounded solutions to some nonlinear problems involving nonlocal operators of the form Lu(x) = − ∑ aij∂iju+ PV ∫ Rn (u(x)− u(x+ y))K(y)dy. These operators are infinitesimal generators of symmetric Lévy processes. Our results apply to even kernels K satisfying that K(y)|y| is nondecreasing along rays from the origin, for some σ ∈ (0, 2) in case aij ≡ 0 and for ...

Absorption of acoustic wave propagation in a large variety of lossy media is characterized by an empirical power law function of frequency, 0j!j y . It has long been noted that the exponent y ranges from 0 to 2 for diverse media. Recently, the present author [J. Acoust. Soc. Am. 115 (2004) 1424] developed a fractional Laplacian wave equation to accurately model the power law dissipation, which ...