Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Weakly-singular formulation of the Fractional Laplacian operator

Oscar P. Bruno, Sabhrant Sachan

Abstract

This paper presents a new formulation of the fractional Laplacian operator $(-Δ)^s$ in $n$-dimensional space ($n \ge 1$). The proposed formulation expresses $(-Δ)^s$ as a composition of the classical Laplace differential operator and a weakly singular integral operator -- which can be used to reduce e.g. the Dirichlet problem for the fractional Laplacian to a weakly singular integral equation involving both volumetric and boundary integral operators. This reformulation is well suited for efficient and accurate numerical implementation. Although a full description of the associated high-order algorithm is deferred to a subsequent contribution, several numerical examples are included in this paper to demonstrate the high accuracy and computational efficiency achieved by the proposed approach.

Weakly-singular formulation of the Fractional Laplacian operator

Abstract

This paper presents a new formulation of the fractional Laplacian operator in -dimensional space (). The proposed formulation expresses as a composition of the classical Laplace differential operator and a weakly singular integral operator -- which can be used to reduce e.g. the Dirichlet problem for the fractional Laplacian to a weakly singular integral equation involving both volumetric and boundary integral operators. This reformulation is well suited for efficient and accurate numerical implementation. Although a full description of the associated high-order algorithm is deferred to a subsequent contribution, several numerical examples are included in this paper to demonstrate the high accuracy and computational efficiency achieved by the proposed approach.
Paper Structure (14 sections, 10 theorems, 150 equations, 6 figures, 4 tables)

This paper contains 14 sections, 10 theorems, 150 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weakly Singular Fractional Laplacian Formulation
  4. Tubular Neighborhood.
  5. Geometric Inequalities.
  6. Simple Motivating Discussion.
  7. Additional Notations.
  8. Inductive Statement.
  9. Proof for $m=0$.
  10. Preparation for the Inductive Step.
  11. Inductive Step.
  12. Numerical illustrations
  13. Appendix: Differentiation under the integral sign
  14. Appendix: Proof of Lemma \ref{['lem:und_int']}

Key Result

Theorem 2.1

Let $n\in\mathbb{N}$ and let $\Omega\in\mathbb{R}^n$ denote a bounded and open domain whose boundary is infinitely differentiable, and let $s\in (0,1)$. Then, for each function $f \in C^{t+0}(\overline{\Omega})$, $t \ge 0$, equation frac_lapl admits a unique solution $u \in C(\mathbb{R}^n)$, which m for a certain function $\phi \in C^{t+s-0}(\overline{\Omega}) \cap C^{t+2s-0}(\Omega)$. Further, $f

Figures (6)

  • Figure 1: A multiply-connected domain with two holes
  • Figure 2: Tubular neighborhood for the domain $\Omega$ in dimension $n=3$, around the point $z$, and parametrized over the rectangular domain $R_\varepsilon = [0,1]^{2}\times [0,\varepsilon]$.
  • Figure 3:
  • Figure 4:
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (20)

  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 10 more