Fractional Laplacian Research Articles

Nonlocal critical growth elliptic problems with jumping nonlinearities

In this paper we study a nonlocal critical growth elliptic problem driven by the fractional Laplacian in the presence of a jumping nonlinearity. By using variational and topological methods and applying some new linking theorems recently proved by Perera and Sportelli in [19], we prove the existence of a nontrivial solution for the problem under consideration.The results we obtain here are the nonlocal counterparts of the ones obtained in [19] in the context of a local equation. Due to the nonlocal nature of our problem, some additional difficulties arise, and the arguments employed in the local setting need to be improved or reconceived. In fact, the proofs of our main theorems require some refined techniques and new regularity results for weak solutions of nonlocal problems that are of independent interest.We would like to point out that our results are specifically for a nonlocal problem with the fractional operator in integral form. However, we do not exclude the possibility that our results may have a counterpart for the spectral operator studied in [27]. Since nonlocal operators in integral form are being widely investigated in the current literature, especially in connection with geometric problems, we have restricted ourselves to elliptic equations driven by a fractional operator in integral form here.

Local existence results in Sobolev spaces for generalized magnetohydrodynamics equations

We consider generalized magnetohydrodynamics (MHD) equations in with fractional diffusions and for the velocity and magnetic field , respectively, where are nonnegative constants and is the fractional Laplacian of order . Using the smoothing effect of for any , we establish local existence results for the generalized MHD equations when the initial data belong to with possibly different orders and . Here, denotes the standard ‐based Sobolev space on of order . To prove our existence results, we also derive new product and commutator estimates in fractional Sobolev spaces which are of independent interest in themselves.

Analysis of Positive Weak Solutions for a Class of Fractional Laplacian Elliptic Systems of Type Kirchhoff

In this research work, a sub-supersolution approach is utilized to investigate the existence and nonexistence of weak positive solution for a class of fractional Laplacian Kirchhoff type elliptic systems in bounded domains with one parameter.

Spin-orbit-coupled fractional oscillators and trapped Bose-Einstein condensates.

We study the ensemble of pseudo-spin 1/2 ultracold bosons, performing Lévy flights, confined in a parabolic potential. The (pseudo-) spin-orbit coupling (SOC) is additionally imposed on these particles. We consider the structure and dynamics of macroscopic pseudospin qubits based on Bose-Einstein condensates, obtained from the above "fractional" bosons. Under "fractional" we understand the substitution of the ordinary second derivative (kinetic energy term) in the Gross-Pitaevskii equationby a so-called fractional Laplacian, characterized by the Lévy index μ. We show that the joint action of interparticle interaction, SOC, and Zeeman splitting in a synthetic magnetic field makes the dynamics of corresponding qubit highly nontrivial with evident chaotic features at both strong interactions and Lévy indices μ→1 when the Lévy trajectories of bosons with long jumps dominated over those derived from ordinary Gaussian distribution, corresponding to μ=2. Using analytical and numerical arguments, we discuss the possibilities to control the above qubit using the synergy of SOC, interaction strength, and "fractionality," characterized by the Lévy index μ.

Multidomain spectral approach to rational‐order fractional derivatives

AbstractWe propose a method to numerically compute fractional derivatives (or the fractional Laplacian) on the whole real line via Riesz fractional integrals. The compactified real line is divided into a number of intervals, thus amounting to a multidomain approach; after transformations in accordance with the underlying curve ensuring analyticity of the respective integrands, the integrals over the different domains are computed with a Clenshaw–Curtis algorithm. As an example, we consider solitary waves for fractional Korteweg‐de Vries equations and compare these to results obtained with a discrete Fourier transform.

Power-Law Frequency-Dependent Q Simulation and Reverse-Time Migration in Transversely Isotropic Acoustic Media

ABSTRACT Velocity and attenuation (Q) anisotropy are widely distributed in the Earth’s interior, significantly affecting the kinematic and dynamic characteristics of seismic-wave propagations. Previous studies developed to simulate these effects are mainly restricted to the constant-Q assumption. However, seismic attenuation in high-temperature and high-pressure regions is demonstrated to be frequency-dependent and usually follows a power-law formulation. To simulate this Q effect in transversely isotropic (TI) attenuating media, we derive a new pure-viscoacoustic wave equation with decoupled fractional Laplacians, which can simultaneously simulate amplitude dissipation and velocity dispersion effects. Based on the wavenumber relationship between the observation and physical coordinate systems, the tilted TI (TTI) wave equation is further derived. Compared with the pseudoviscoacoustic wave equation, the proposed pure-viscoacoustic equation can simulate stable P wavefields in complex geological structures without S-wave artifacts. To solve this new equation, two low-rank decompositions are introduced to approximate the real and imaginary parts and avoid the separation of wavenumbers and dip angles, making it much simpler in programming and implementation. We further use this equation to perform Q-compensated reverse-time migration to generate high-resolution migration images in anisotropic attenuating media. Numerical examples demonstrate the effectiveness of the proposed method for pure-viscoacoustic wavefield simulations and migrations in TTI attenuating media with power-law frequency-dependent Q effects.

Semiclassical states for fractional Schrödinger equations with critical nonlinearities

In this paper, we consider the following fractional Schrödinger equation where is a parameter, is the fractional Laplacian and is a superlinear and subcritical nonlinearity. Under a local condition imposed on the potential function, combining the penalization method and the concentration‐compactness principle, we prove the existence of a positive solution for the above equation.

Magnetic fractional Poincaré inequality in punctured domains

We study Poincaré-Wirtinger type inequalities in the framework of magnetic fractional Sobolev spaces. In the local case, Lieb et al. (2003) [19] showed that, if a bounded domain Ω is the union of two disjoint sets Γ and Λ, then the Lp-norm of a function calculated on Ω is dominated by the sum of magnetic seminorms of the function, calculated on Γ and Λ separately. We show that the straightforward generalisation of their result to nonlocal setup does not hold true in general. We provide an alternative formulation of the problem for the nonlocal case. As an auxiliary result, we also show that the set of eigenvalues of the magnetic fractional Laplacian is discrete.

The role of antisymmetric functions in nonlocal equations

We use a Hopf-type lemma for antisymmetric super-solutions to the Dirichlet problem for the fractional Laplacian with zero-th order terms, in combination with the method of moving planes, to prove symmetry for the semilinear fractional parallel surface problem. That is, we prove that non-negative solutions to semilinear Dirichlet problems for the fractional Laplacian in a bounded open set Ω ⊂ R n \Omega \subset \mathbb {R}^n must be radially symmetric if one of their level surfaces is parallel to the boundary of Ω \Omega ; in turn, Ω \Omega must be a ball. Furthermore, we discuss maximum principles and the Harnack inequality for antisymmetric functions in the fractional setting and provide counter-examples to these theorems when only ‘local’ assumptions are imposed on the solutions. The construction of these counter-examples relies on an approximation result that states that ‘all antisymmetric functions are locally antisymmetric and s s -harmonic up to a small error’.

Sign-changing solutions for Kirchhoff-type variable-order fractional Laplacian problems

In this paper, we are concerned with the Kirchhoff-type variable-order fractional Laplacian problems involving critical exponents and logarithmic nonlinearity. By using the constraint variational method, we show the existence of one least energy sign-changing solution. Moreover, we show that this energy is strictly larger than twice the ground energy.

On the Non-degeneracy of the Robin Function for the Fractional Laplacian on Symmetric Domains

On the Non-degeneracy of the Robin Function for the Fractional Laplacian on Symmetric Domains

Fractional diffusion for Fokker–Planck equation with heavy tail equilibrium: An à la Koch spectral method in any dimension

In this paper, we extend the spectral method developed (Dechicha and Puel (2023)) to any dimension d ⩾ 1, in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form ( 1 + | v | 2 ) − β 2 , in the range β ∈ ] d , d + 4 [. The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation (Nonlinearity 28 (2015) 545). The strategy in this paper is the same as in dimension 1 but the tools are different, since dimension 1 was based on ODE methods. As a direct consequence of our construction, we obtain the fractional diffusion limit for the kinetic Fokker–Planck equation, for the correct density ρ : = ∫ R d f d v, with a fractional Laplacian κ ( − Δ ) β − d + 2 6 and a positive diffusion coefficient κ.

Fractional stochastic heat equation with mixed operator and driven by fractional-type noise

<p>We investigated a novel stochastic fractional partial differential equation (FPDE) characterized by a mixed operator that integrated the standard Laplacian, the fractional Laplacian, and the gradient operator. The equation was driven by a random noise, which admitted a covariance measure structure with respect to the time variable and behaved as a Wiener process in space. Our analysis included establishing the existence of a solution in the general case and deriving an explicit form for its covariance function. Additionally, we delved into a specific case where the noise was modeled as a generalized fractional Brownian motion (gfBm) in time, with a particular emphasis on examining the regularity of the solution's sample paths.</p>

On the logarithmic fractional Schrödinger–Poisson system with saddle-like potential

In this paper, we use variational methods to prove the existence of a positive solution for the following class of logarithmic fractional Schrödinger–Poisson system: { ϵ 2 s ( − Δ ) s u + V ( x ) u − ϕ ( x ) u = u log ⁡ u 2 in R 3 , ϵ 2 t ( − Δ ) t ϕ = | u | 2 in R 3 , where ϵ > 0 , s , t ∈ ( 0 , 1 ) , ( − Δ ) α is the fractional Laplacian and V is a saddle-like potential.

A fast finite difference scheme for the time-space fractional diffusion equation

This paper proposes a fast numerical scheme for time-space fractional diffusion equation in two spatial dimensions, where the temporal fractional partial derivative is in the Caputo sense of order α ϵ (0; 1) and the spatial fractional derivative is given by fractional Laplacian (−Δ)β/2 in two space dimensions with β ϵ (1, 2). The temporal Caputo derivative is discretized by a fast evaluation based on the L2-1σ formula. The spatial discretization adopts fractional centered difference formula in two dimensions. The proposed fully discrete scheme is proved to be stable and convergent with 2nd order accuracy both in time and space. The feasibility, stability, and convergence of the proposed scheme are demonstrated by the numerical example.

On a viscoelastic Kirchhoff equation with fractional Laplacian

On a viscoelastic Kirchhoff equation with fractional Laplacian

An Easy-To-Use Tool to Solve Differential Equations with the Fractional Laplacian

Numerical simulation of fractional-order partial differential equations is a challenging task and the majority of computing environments does not provide support for these problems. In this paper we describe how to exploit some of the Matlab features (a programming language not supporting fractional calculus in a naive way) to solve partial differential equations with the spectral fractional Laplacian. For shortness we focus on fractional Poisson equations but the proposed approach can be extended, with just some technical difficulties, to more involved problems. This approach cannot be considered as a highly efficient and accurate way to solve fractional partial differential equations, but as an easy-to-use tool for non specialists in numerical computation to obtain solutions without having to produce sophisticated numerical codes.

Li-Yau and Harnack inequalities via curvature-dimension conditions for discrete long-range jump operators including the fractional discrete Laplacian

Li-Yau and Harnack inequalities via curvature-dimension conditions for discrete long-range jump operators including the fractional discrete Laplacian

A Numerical Scheme for Time-Space Fractional diffusion Models

Time-space fractional models are proved to be highly efficient in characterizing anomalous diffusion in many intricate systems. This study introduces a numerical scheme employing a matrix transform technique to discretize the fractional spectral Laplacian and subsequently applying generalized exponential time differencing to the semi-discrete system. The resulting second-order scheme is implemented efficiently using rational approximations for the two-parameter Mittag-Leffler function and the partial fraction decomposition of these approximations. The advantage of this approach is its applicability to different types of homogeneous boundary conditions including Robin boundary conditions. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed numerical scheme.

On the structure of the diffusion distance induced by the fractional dyadic Laplacian

On the structure of the diffusion distance induced by the fractional dyadic Laplacian