Fractional Laplacian Equation Research Articles

Locally solving fractional Laplacian viscoacoustic wave equation using Hermite distributed approximating functional method

Accurate seismic modeling in realistic media serves as the basis of seismic full-waveform inversion and imaging. Recently, viscoacoustic seismic modeling incorporating attenuation effects has been performed by solving a fractional Laplacian viscoacoustic wave equation. In this equation, attenuation, being spatially heterogeneous, is represented partially by the spatially varying power of the fractional Laplacian operator previously approximated by the global Fourier method. We have developed a local-spectral approach, based on the Hermite distributed approximating functional (HDAF) method, to implement the fractional Laplacian in the viscoacoustic wave equation. Our approach combines local methods’ simplicity and global methods’ accuracy. Several numerical examples are developed to evaluate the feasibility and accuracy of using the HDAF method for the frequency-independent [Formula: see text] fractional Laplacian wave equation.

A Fractional Laplace Equation: Regularity of Solutions and Finite Element Approximations

This paper deals with the integral version of the Dirichlet homogeneous fractional Laplace equation. For this problem weighted and fractional Sobolev a priori estimates are provided in terms of the Hölder regularity of the data. By relying on these results, optimal order of convergence for the standard linear finite element method is proved for quasi-uniform as well as graded meshes. Some numerical examples are given showing results in agreement with the theoretical predictions.

On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling

In this paper, a class of systems of two coupled nonlinear fractional Laplacian equations are investigated. Under very weak assumptions on the nonlinear terms $f$ and $g$, we establish some results about the existence of positive vector solutions and vector ground state solutions for the fractional Laplacian systems by using variational methods. In addition, we also study the asymptotic behavior of these solutions as the coupling parameter $β$ tends to zero.

Resonant problems for fractional Laplacian

In this paper we consider the following fractional Laplacian equation$\left\{\begin{array}{ll} (-\Delta).s u=g(x, u) & x\in\Omega,\\ u=0, & x \in \mathbb{R}.N\setminus\Omega,\end{array} \right.$ where $ s\in (0, 1)$ is fixed, $\Omega$ is an open bounded set of $\mathbb{R}.N$, $N > 2s$, with smooth boundary, $(-\Delta).s$ is the fractional Laplace operator. By Morse theory we obtain the existence of nontrivial weak solutions when the problem is resonant at both infinity and zero.

Three nontrivial solutions for nonlinear fractional Laplacian equations

Abstract We study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three non-zero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.

Multiplicity existence for sublinear fractional Laplacian problems

The purpose of this paper is mainly to investigate the existence of weak solutions of fractional Laplacian equations with sublinear growth and oscillatory behavior. By using variational methods and abstract critical point theory, we want to establish the existence of multiplicity positive, multiplicity negative, and in particular, of multiplicity sign changing solutions. The multiplicity results depend on the real parameter . The key point is the choice of the framework to study the existence of weak solutions. In the suitable framework, by verifying that the conditions in abstract critical point theory are satisfied, we are able to fulfill our strategy.

Infinitely many solutions for fractional Laplacian problems with local growth conditions

In this paper, we deal with the fractional Laplacian equations $$(\mathrm{P})\quad \left \{ \textstyle\begin{array}{@{}l@{\quad}l} (-\Delta)^{s} u = f(x,u), & x\in \Omega, u(x)= 0, & x\in \mathbb{R}^{N}\backslash\Omega, \end{array}\displaystyle \right . $$ where $0< s<1<p<+\infty$ , $N\in\mathbb{N}, N>2s$ , $\Omega\subset \mathbb{R}^{N}$ is a bounded domain with smooth boundary. Under local growth conditions of $f(x,t)$ , infinitely many solutions for problem (P) are obtained via variational methods.

Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains

Approximations of solutions of fractional Laplacian equations on bounded domains are considered. Such equations allow global interactions between points separated by arbitrarily large distances. Two approximations are introduced. First, interactions are localized so that only points less than some specified distance, referred to as the interaction radius, are allowed to interact. The resulting truncated problem is a special case of a more general nonlocal diffusion problem. The second approximation is the spatial discretization of the related nonlocal diffusion problem. A recently developed abstract framework for asymptotically compatible schemes is applied to prove convergence results for solutions of the truncated and discretized problem to the solutions of the fractional Laplacian problems. Intermediate results also provide new convergence results for the nonlocal diffusion problem. Special attention is paid to limiting behaviors as the interaction radius increases and the spatial grid size decreases, regardless of how these parameters may or may not be dependent. In particular, we show that conforming Galerkin finite element approximations of the nonlocal diffusion equation are always asymptotically compatible schemes for the corresponding fractional Laplacian model as the interaction radius increases and the grid size decreases. The results are developed with minimal regularity assumptions on the solution and are applicable to general domains and general geometric meshes with no restriction on the space dimension and with data that are only required to be square integrable. Furthermore, our results also solve an open conjecture given in the literature about the convergence of numerical solutions on a fixed mesh as the interaction radius increases.

A direct method of moving planes for fractional Laplacian equations in the unit ball

In this paper, we employ a direct method of moving planes for the fractional Laplacian equation in the unit ball. Instead of using the conventional extension method introduced by Caffarelli and Silvestre [6], Chen, Li and Li developed a direct method of moving planes for the fractional Laplacian [8]. Inspired by this new method, in this paper we deal with the semilinear pseudo -differential equation in the unit ball directly. We first review key ingredients needed in the method of moving planes in a bounded domain, such as the narrow region principle for the fractional Laplacian. Then, by using this new method, we obtain the radial symmetry and monotonicity of positive solutions for some interesting semi-linear equations.

Boundary integral operator for the fractional Laplacian on the boundary of a bounded smooth domain

We introduce the boundary integral operator induced from the fractional Laplace equation on the boundary of a bounded smooth domain. For~$\frac 12\lt \alpha \lt 1$, we show the bijectivity of the boundary integral operator~$S_{2\alpha }:L^p(\partial \Omega )\to H^{2\alpha -1}_p(\partial \Omega )$ for $1 \lt p \lt \infty $. As an application, we demonstrate the existence of the solution of the Dirichlet boundary value problem of the fractional Laplace equation.

Symmetry of positive solutions for equations involving higher order fractional Laplacian

In this paper, we consider problems associated with the higher order fractional Laplacian. Through the method of moving planes, we derive rotational symmetry of positive solutions and show their dependence on the x n x_n variable only. We also establish the equivalence between a semilinear higher order fractional Laplacian equation and its corresponding integral equation, so as to further deduce a Liouville type theorem and obtain a priori estimates for positive solutions.

Boundedness of solutions to Ginzburg–Landau fractional Laplacian equation

In this paper, we give the boundedness of solutions to Ginzburg–Landau fractional Laplacian equation, which extends the Herve–Herve theorem into the nonlinear fractional Laplacian equation. We follow Brezis’ idea to use the Kato inequality. A related linear fractional Schrödinger equation is also studied.

An Existence Result for Fractional Kirchhoff-Type Equations

The aim of this paper is to study a class of nonlocal fractional Laplacian equations of Kirchhoff-type. More precisely, by using an appropriate analytical context on fractional Sobolev spaces, we establish the existence of one non-trivial weak solution for nonlocal fractional problems exploiting suitable variational methods.

A critical fractional Laplace equation in the resonant case

In this paper we complete the study of the following non-local fractional equation involving critical nonlinearities $$ \cases (-\Delta)^s u-\lambda u=|u|^{2^*-2}u & {\text{in }} \Omega,\\ u=0 & {\text{in }} \mathbb{R}^n\setminus \Omega, \endcases $$ started in the recent papers \cite{13}, \cite{17}-\cite{19}. Here $s\in (0,1)$ is a fixed parameter, $(-\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive constant, $2^*=2n/(n-2s)$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $\RR^n$, $n> 2s$, with Lipschitz boundary. Aim of this paper is to study this critical problem in the special case when $n\not=4s$ and $\lambda$ is an eigenvalue of the operator $(-\Delta)^s$ with homogeneous Dirichlet boundary datum. In this setting we prove that this problem admits a non-trivial solution, so that with the results obtained in \cite{13}, \cite{17}-\cite{19}, we are able to show that this critical problem admits a nontrivial solution provided \roster \item"$\bullet$" $n> 4s$ and $\lambda> 0$, \item"$\bullet$" $n=4s$ and $\lambda> 0$ is different from the eigenvalues of $(-\Delta)^s$, \item"$\bullet$" $2s< n< 4s$ and $\lambda> 0$ is sufficiently large. \endroster In this way we extend completely the famous result of Brezis and Nirenberg (see \cite{4}, \cite{5}, \cite{9}, \cite{23}) for the critical Laplace equation to the non-local setting of the fractional Laplace equation.

Some overdetermined problems for the fractional Laplacian equation on the exterior domain and the annular domain

In this paper, by the method of moving planes, we extend the Serrin-type overdetermined problem to the fractional Laplacian equation on the exterior set of a multiply connected domain and the annular domain.

On Fractional Capacities Relative to Bounded Open Lipschitz Sets

For 1≤p≤∞ and n/p<α<1 + n/p, the fractional capacity Cα,p(K,Ω) of a compact subset K with respect to a bounded open Lipschitz set Ω⊃K of \(\mathbb R^{n}\) is systematically investigated and effectively applied to the fractional Sobolev space Wα,p(Ω) which plays an important role in calculus of variations, harmonic analysis, potential theory, partial differential equations, mathematical physics and so forth. In particular, we explore: the basic properties of Cα,p(K,Ω); the existence of a capacitary potential, and the relationship between Cα,p(K,Ω) and the fractional Laplace equation. the Cα,p(K,Ω)-based nature of a nonnegative Radon measure μ given on Ω that induces an embedding of Wα,p(Ω) into the Lebesgue space Lq(Ω,μ) as 1≤p≤q<∞ or the exponentially integrable Lebesgue space \(\exp (c|f|^{\frac {2n}{2n-\alpha }}) \in L^{1}({\Omega }, \mu )\) as p=2n/α or the Lebesgue space L∞(Ω,μ) as 2n/α<p≤∞; the geometric characterization of a fractional Sobolev embedding domain via Cα,p(K,Ω). some of the essential behaviours of K with Cα,p(K,Ω)=0.

A new definition of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction

This paper proposes a new implicit definition of the fractional Laplacian. Compared with the existing explicit definitions in literature, this novel definition has clear physical significance and is mathematically simple and numerically easy to calculate for multidimensional problems. In stark contrast to a quick increasing and extensive applications of time-fractional derivative to diverse scientific and engineering problems, little has been reported on space-fractional derivative modeling. This is largely because the existing definitions are only feasible for one-dimensional case and become mathematically too complicated and computationally very expensive when applied to higher dimensional cases. In this study, we apply the newly-defined fractional Laplacian for modeling the power law behaviors of three-dimensional nonlocal heat conduction. The singular boundary method (SBM), a recent boundary-only collocation discretization method, is employed to numerically solve the proposed fractional Laplacian heat equation. And the computational costs are observed moderate owing to the proposed new definition of fractional Laplacian and the boundary-only discretization, meshfree, and integration-free natures of the SBM technique. Numerical experiments show the validity of the proposed definition of fractional Laplacian.

Some observations on the Green function for the ball in the fractional Laplace framework

We consider a fractional Laplace equation and we give a self-contained elementary exposition of the representation formula for the Green function on the ball. In this exposition, only elementary calculus techniques will be used, in particular, no probabilistic methods or computer assisted algebraic manipulations are needed. The main result in itself is not new, however we believe that the exposition is original and easy to follow, hence we hope that this paper will be accessible to a wide audience of young researchers and graduate students that want to approach the subject, and even to professors that would like to present a complete proof in a PhD or Master Degree course.

Sumudu transform series expansion method for solving the local fractional Laplace equation in fractal thermal problems

In this article, the Sumudu transform series expansion method is used to handle the local fractional Laplace equation arising in the steady fractal heat-transfer problem via local fractional calculus.

The non-differentiable solution for local fractional Laplace equation in steady heat-conduction problem

In this paper, we investigate the local fractional Laplace equation in the steady heat-conduction problem. The solutions involving the non-differentiable graph are obtained by using the characteristic equation method (CEM) via local fractional derivative. The obtained results are given to present the accuracy of the technology to solve the steady heat-conduction in fractal media.