*Article* **Remarks on the Generalized Fractional Laplacian Operator**

#### **Chenkuan Li 1,\*, Changpin Li 2, Thomas Humphries 1 and Hunter Plowman 1**


Received: 25 February 2019; Accepted: 25 March 2019; Published: 29 March 2019

**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 (−<sup>Δ</sup>) 12 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.

**Keywords:** distribution; fractional Laplacian; Riesz fractional derivative; delta sequence; convolution

**MSC:** 46F10; 26A33

In recent years, the fractional Laplacian operator has gained considerable attention due to its applications in many disciplines, such as partial differential equations, long-range interactions, anomalous diffusions and non-local quantum theories. There is also the physical meaning of the fractional Laplacian operator in bounded domains through its associated stochastic processes. However, the half-order Laplacian operator (−<sup>Δ</sup>) 12 , often appearing in various literature works and applications, needs to be studied carefully as the first-order Riesz derivative is undefined in the classical sense. The goal of this work is to use a new distributional approach to defining operator (−<sup>Δ</sup>) 12 in the generalized sense by Temple's delta sequence, as well as present fresh techniques in computing examples of the fractional Laplacian operator of order 1/2 and applications to solving partial differential equations related to this operator.
