Abstract
The fractional Laplacian, also known as the Riesz fractional derivative operator, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighbouring or nearby sites, as well as perform excursions to remote sites by way of Lévy flights. The fractional Laplacian has many applications in the boundary behaviours of solutions to differential equations. The goal of this paper is to investigate the half-order Laplacian operator ( − Δ ) 1 2 in the distributional sense, based on the generalized convolution and Temple’s delta sequence. Several interesting examples related to the fractional Laplacian operator of order 1 / 2 are presented with applications to differential equations, some of which cannot be obtained in the classical sense by the standard definition of the fractional Laplacian via Fourier transform.
Highlights
Let s ∈ (0, 1) and ∆ = ∂2 /∂x12 + · · · + ∂2 /∂xn2
At the end of this work, we describe applications of such studies to solving the differential equations involving the half-order
In order to extend the fractional Laplacian (−∆)1/2 distributionally, we briefly introduce the following basic concepts of distributions
Summary
In order to study the half-order Laplacian operator in the distribution, we introduce an infinitely-differentiable function τ ( x ) satisfying the following conditions:. This suggests the following explicit definition for defining (−∆) 2 u( x ) This explicit definition directly evaluates the half-order fractional Laplacian of u( x ) as a function of x, without relating to any testing function in the Schwartz space. It seems infeasible to calculate directly the fractional Laplacian operator of some functions or distributions by Definition 2. We are going to provide another definition for dealing with (−∆) 2 u( x ) efficiently, based on a testing function with compact support This definition is implicit and only used to define the meaning of:.
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