Analysis of an approximation to a fractional extension problem

Abstract

The purpose of this article is to study an approximation to an abstract Bessel-type problem, which is a generalization of the extension problem associated with fractional powers of the Laplace operator. Motivated by the success of such approaches in the analysis of time-stepping methods for abstract Cauchy problems, we adopt a similar framework herein. The proposed method differs from many standard techniques, as we approximate the true solution to the abstract problem, rather than solve an associated discrete problem. The numerical method is shown to be consistent, stable, and convergent in an appropriate Banach space. These results are built upon well understood results from semigroup theory. Numerical experiments are provided to demonstrate the theoretical results.

A series representation of the discrete fractional Laplace operator of arbitrary order

“…We now provide some clarifying remarks regarding the above result. Item (i) of Theorem 1.1 above is a direct consequence of combining Definition 3.11 and Padgett [34, Theorem 2.1] (applied for every n ∈ Z with s ) in the notation of Padgett [34,Theorem 2.1]). See the beginning of Section 2 for an explanation of this "applied with" notation (i.e., the symbol " ").…”

“…See the beginning of Section 2 for an explanation of this "applied with" notation (i.e., the symbol " "). The right-hand-side of (1.5) is not considered in detail, herein, as it is an elementary consequence of Padgett [34,Theorem 2.1]. Item (ii) of Theorem 1.1 follows directly from Lemma 4.4 and Lemma 5.1.…”

“…Interestingly, the constant c ∈ [0, ∞) in (1.3) depends only upon the parameter s ∈ (0, 1) and not upon d ∈ N. More importantly, this demonstrates that one may trade out the highly non-local problem given by (1.1) for the local problem given by (1.2) and (1.3). This technique has also been recently further generalized to cases of arbitrary non-negative operators defined on Banach spaces [2,16,31,32,34]. While the above formulations (i.e., (1.1), (1.2), and (1.3)) may be used to provide insights into the continuous fractional Laplace operator with order s ∈ (0, 1), they cannot be directly used to provide any insight into the discrete case or the case where s ∈ (0, ∞).…”

A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes

“…Furthermore, recently, Chen and Shen have solved numerically Poisson-type problems with the diagonalization matrix method and the enriched spectral method [10]. Padgett dealt with the same problem numerically in [11]. In addition, different numerical methods have been applied for approximating the fractional powers of the laplacian operator, see [12][13][14].…”

Intrinsic properties of strongly continuous fractional semigroups in normed vector spaces

“…Researchers in numerical analysis have recently benefited from a surge of activities which overlaps with pure mathematical analysis (see, for instance, [3,6,15,16,24,26,28] and publications cited therein). Inspired by this fact and the recent works of Littlejohn and Wellman concerning the investigation of self-adjoint operators in extended Hilbert spaces (cf., e.g., [19,20,21]), in this article we consider so-called strongly continuous fractional semigroups in arbitrary normed vector spaces.…”