Related papers: Numerical Approximation of the Fractional Laplacia…
In this paper, we develop a numerical multiscale method to solve the fractional Laplacian with a heterogeneous diffusion coefficient. When the coefficient is heterogeneous, this adds to the computational costs. Moreover, the fractional…
In this paper, we propose a novel pseudospectral method to approximate accurately and efficiently the fractional Laplacian without using truncation. More precisely, given a bounded regular function defined over $\mathbb R$, we map the…
In this paper, we propose accurate and efficient finite difference methods to discretize the two- and three-dimensional fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ ($0 < \alpha < 2$) in hypersingular integral form. The proposed…
We calculate the fractional Laplacian for functions of the form $u(x)=(1-|x|^2)_+^p$ and $v(x)=x_d u(x)$. As an application, we estimate the first eigenvalues of the fractional Laplacian in a ball.
In this article, we develop a new method to approximate numerically the fractional Laplacian of functions defined on $\mathbb R$, as well as some more general singular integrals. After mapping $\mathbb R$ into a finite interval, we…
A spectral method is considered for approximating the fractional Laplacian and solving the fractional Poisson problem in 2D and 3D unit balls. The method is based on the explicit formulation of the eigenfunctions and eigenvalues of the…
We study the fractional Laplacian and the homogeneous Sobolev spaces on R^d , by considering two definitions that are both considered classical. We compare these different definitions, and show how they are related by providing an explicit…
The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a non-local operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the…
In this paper, we develop a fast and accurate pseudospectral method to approximate numerically the half Laplacian $(-\Delta)^{1/2}$ of a function on $\mathbb{R}$, which is equivalent to the Hilbert transform of the derivative of the…
This paper presents a somewhat exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric equation (CFGHE) about the fractional regular singular…
In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ for $\alpha \in (0, 2)$. The main advantage of our method is to easily increase numerical…
In this paper we introduce two new fractional versions of the Laplacian. The first one is based on the classical formula that writes the usual Laplacian as the sum of the eigenvalues of the Hessian. The second one comes from looking at the…
This paper is devoted to the Laplacian operator of fractional order $s\in (0,1)$ in several dimensions. We first establish a representation formula for the partial derivatives of the solutions of the homogeneous Dirichlet problem. Along the…
The fractional Hamiltonian analysis of 1+1 dimensional field theory is investigated and the fractional Ostrogradski's formulation is obtained. The fractional path integral of both simple harmonic oscillator with an acceleration-squares part…
For the fractional Laplacian of variable order, an efficient and accurate numerical evaluation in multi-dimension is a challenge for the nature of a singular integral. We propose a simple and easy-to-implement finite difference scheme for…
The variable-order fractional Laplacian plays an important role in the study of heterogeneous systems. In this paper, we propose the first numerical methods for the variable-order Laplacian $(-\Delta)^{\alpha({\bf x})/2}$ with $0 <…
We discuss a family of random fields indexed by a parameter $s\in \mathbb{R}$ which we call the fractional Gaussian fields, given by \[ \mathrm{FGF}_s(\mathbb{R}^d)=(-\Delta)^{-s/2} W, \] where $W$ is a white noise on $\mathbb{R}^d$ and…
The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional powers of second order partial differential…
A new method for approximating fractional derivatives of the Gaussian function and Dawson's integral are presented. Unlike previous approaches, which are dominantly based on some discretization of Riemann-Liouville integral using polynomial…
In this paper, we propose an accurate finite difference method to discretize the $d$-dimensional (for $d\ge 1$) tempered integral fractional Laplacian and apply it to study the tempered effects on the solution of problems arising in various…