Fractional Laplacian Research Articles

A comparison method for the fractional Laplacian and applications

We study the boundary behavior of solutions to fractional Laplacian. As the first result, the isolation of the first eigenvalue of the fractional Lane-Emden equation is proved in the bounded open sets with Wiener regular boundary. Then, a generalized Hopf's lemma and a global boundary Harnack inequality are proved for the fractional Laplacian.

Reaction-diffusion for fish populations with realistic mobility

Nonlinear reaction-diffusion equations, with Fisher logistic growth and constant diffusion coefficient, have been used in fisheries research to estimate sustainable harvesting rates and critical domain sizes of no-take areas. However, constant diffusivity in a population density corresponds to standard Brownian motion of individuals, with a normal distribution for displacement over a fixed time interval. For available good data sets on mobile fish populations, the distribution is certainly not normal. The data can be fitted with a long-tailed stable Lévy distribution that results from diffusion by fractional Laplacian. Exact multidimensional solutions are developed here for realistic Fisher–Kolmogorov–Petrovski-Piscounov models with diffusion by fractional Laplacian. These can also account for a delay in the reaction term. It is then shown how to modify critical domain sizes of protected areas with Dirichlet and Robin boundary conditions for populations.

A convergence result for the derivation of front propagation in nonlocal phase field models

We prove that the mean curvature of a smooth surface in \R^{n} , n\geq 2 , arises as the limit of a sequence of functions that are intrinsically related to the difference between an n - and 1 -dimensional fractional Laplacian of a phase transition. Depending on the order of the fractional Laplace operator, we recover the fractional mean curvature or the classical mean curvature of the surface. Moreover, we show that this is an essential ingredient for deriving the evolution of fronts in fractional reaction-diffusion equations such as those for atomic dislocations in crystals.

A Novel Approach to the Fractional Laplacian via Generalized Spherical Means

Although at least ten equivalent definitions of the fractional Laplacian exist in unbounded domains, we introduce an additional equivalent definition based on the generalized spherical mean-value operator—a Fourier multiplier operator involving the normalized Bessel function. Specifically, we demonstrate that this new definition allows us to reduce any n-dimensional fractional Laplacian to a one-dimensional operator, which simplifies computation and enhances efficiency. We propose two methods for computing the generalized spherical means of a given function: one by solving standard wave equations and the other by solving Darboux’s equations.

Enhancing Learning to Solve Multicomponent Fractional Viscoelastic Equations with U-net Fourier Neural Operators

Abstract The research of viscoelastic media is currently a hot topic in the interpretation and processing of seismic data. To accurately simulate the propagation of seismic waves in viscoelastic media, the fractional viscoelastic equation has emerged as an indispensable method. However, solving this equation numerically has proven to be challenging due to the complexity introduced by its fractional Laplacian operators. Recently, deep learning, especially Fourier neural operators (FNO), has shown excellent performance in learning to fast solve partial differential equations. Traditional FNO methods may face crosstalk problem and make it difficult to achieve satisfactory accuracy, when solving the multi-component fractional order viscoelastic equation. To solve this problem, we introduce a novel approach based on U-net Fourier neural operator (U-FNO). As an enhanced learning method to the traditional FNO-based method, the U-FNO-based method integrates a U-Fourier layer following the standard Fourier layer as a form of regularization, thereby achieving superior prediction accuracy for multi-component equations. Specifically, both the Fourier layers and U-Fourier layers in U-FNO are trained with the solutions of the equation from previous time steps as inputs. This training process enables the U-FNO to efficiently produce more accurate solutions for subsequent wavefield. Numerical simulations reveal that the U-FNO-based method efficiently learns to solve the fractional viscoelastic wave equation independent of fractional Laplacian operators. Additionally, U-FNO-based method offers superior prediction accuracy in comparison with the traditional FNO-based method.

Performance Analysis and Parallel Scalability of Numerical Methods for Fractional-in-Space Diffusion Problems with Adaptive Time Stepping

We study numerical methods and algorithms for time-dependent fractional-in-space diffusion problems. The considered anomalous diffusion is modelled by the fractional Laplacian (−Δ)α, 0<α<1, following the integral definition. Fractional diffusion is non-local, and the finite element method (FEM) discretization in space leads to a dense stiffness matrix. It is well known that numerically solving such non-local boundary value problems is expensive. Difficulties increase significantly when the problem is time-dependent. The aim of the article is to develop computationally efficient methods and algorithms. There are two main features of our approach. Hierarchical semi-separable (HSS) compression is applied for an approximate solution of the arising linear systems. For time discretization, we use an adaptive forward–backward Euler scheme. The properties of the composite algorithm thus obtained are investigated. In particular, the block representation of HSS compression allowed us to upgrade the HSS solver to efficiently handle varying diagonally perturbed transition matrices corresponding to changing time steps. The contribution of the paper is threefold. The methods are completely constructive, which allows for a clearly structured description of the algorithms. A theoretical estimate of the computational complexity is presented. It shows the advantages of the adaptive time stepping in combination with the HSS solver. Theoretical results are supported by representative numerical experiments. Both sequential and parallel scalability and efficiency are analyzed. The presented results provide convincing proof of the concept of the proposed methods and algorithms.

Parabolic logistic equation with harvesting involving the fractional Laplacian

This paper deals with a class of parabolic reaction-diffusion equations driven by the fractional Laplacian as the diffusion operator over a bounded domain with zero Dirichlet external condition. Using a comparison principle and monotone iteration method, we establish existence and uniqueness results. We apply the existence result to the logistic growth problems with constant yield harvesting by constructing an ordered pair of positive sub- and supersolution of the corresponding elliptic problem.

Sign-changing solutions for a class of fractional Choquard equation with the Sobolev critical exponent in [formula omitted]

In this paper, we study the following fractional Choquard equation with critical exponent(−Δ)σu+u=(Iα⁎|u|p)|u|p−2u+|u|2σ⁎−2uinR3, where σ∈(34,1), α∈(2σ,3) and α+2σ3−2σ<p<2α,σ⁎:=3+α3−2σ, 2α,σ⁎ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, 2σ⁎:=63−2σ is the fractional Sobolev critical exponent and the operator (−Δ)σ stands for the fractional Laplacian of order σ. Based on the above assumptions, we establish the existence of positive ground state solutions, and if p∈(α+2σ3−2σ,2α3−2σ), we also have the corresponding regular property. Subsequently, by introducing some other additional hypotheses on σ, α, p and with the help of quantitative deformation lemma, we employ constrained minimization arguments on the sign-changing Nehari manifold to obtain the existence of ground state sign-changing solutions.

Laplacian edge detection method based on a new approximation of the second-order fractional derivative in the Caputo-Fabrizio sense

Edge detection is an important issue in image processing and computer vision problems. It includes all mathematical methods that aim to identify the discontinuous points in the image. This process leads to construct curves that indicate the boundaries of an object and therefore extract feature information. In recent years, this task has received great attention from researchers and scientists, as we find in the literature many methods with their different algorithms. Up to now, there are two methods, the gradient and Laplacian methods. The first depends on the first derivative, where pixels with a high gradient are considered edges, whereas the second depends on the second derivative, such as the pixel whose second derivative is zero, it is marked as an edge point. Although these methods have achieved remarkable success, they are sensitive to noise and sometimes produce false edges. For this reason, in this paper, we propose an efficient edge detection method based on the fractional Laplacian operator in the Caputo-Fabrizio sense. In this method we develop a new approximation formula to the second-order with the Caputo-Fabrizio fractional derivative. Then, we construct a new fractional mask to compute the Laplacian convolution. To check the effectiveness of our proposed method, we have provided some test images that are affected by a Gaussian noise. The numerical results and visual observation show that our proposed edge detection method gives better performance compared to the classical Laplacian method in terms of the peak signal to noise ratio (psnr) and fine details extraction.

Weighted Analytic Regularity for the Integral Fractional Laplacian in Polyhedra

Weighted Analytic Regularity for the Integral Fractional Laplacian in Polyhedra

Time-periodic traveling waves and propagating terraces for multistable equations with a fractional Laplacian: An abstract dynamical systems approach

This paper is devoted to studying the existence of traveling wave solutions for reaction-diffusion equations with fractional Laplacians in time-periodic environments. By developing a dynamical systems approach, we establish the existence of propagating terraces for equations with multistable nonlinearities, which consequently implies the existence of bistable traveling waves. Furthermore, we investigate whether traveling waves exist in the multistable case. We provide an affirmative answer to this question for a special type of nonlinearity in high-frequency oscillating environments, and determine the homogenized limit of the traveling waves as the period approaches 0.

A high-efficiency Q-compensated pure-viscoacoustic reverse time migration for tilted transversely isotropic media

The attenuation and anisotropy characteristics of real earth media give rise to amplitude loss and phase dispersion during seismic wave propagation. To address these effects on seismic imaging, viscoacoustic anisotropic wave equations expressed by the fractional Laplacian have been derived. However, the huge computational expense associated with multiple Fast Fourier transforms for solving these wave equations makes them unsuitable for industrial applications, especially in three dimensions. Therefore, we first derived a cost-effective pure-viscoacoustic wave equation expressed by the memory-variable in tilted transversely isotropic (TTI) media, based on the standard linear solid model. The newly derived wave equation featuring decoupled amplitude dissipation and phase dispersion terms, can be easily solved using the finite-difference method (FDM). Computational efficiency analyses demonstrate that wavefields simulated by our newly derived wave equation are more efficient compared to the previous pure-viscoacoustic TTI wave equations. The decoupling characteristics of the phase dispersion and amplitude dissipation of the proposed wave equation are illustrated in numerical tests. Additionally, we extend the newly derived wave equation to implement Q-compensated reverse time migration (RTM) in attenuating TTI media. Synthetic examples and field data test demonstrate that the proposed Q-compensated TTI RTM effectively migrate the effects of anisotropy and attenuation, providing high-quality imaging results.

A K-Space-Based Temporal Compensating Scheme for a First-Order Viscoacoustic Wave Equation with Fractional Laplace Operators

Inherent constant Q attenuation can be described using fractional Laplacian operators. Typically, the fractional Laplacian viscoacoustic or viscoelastic wave equations are addressed utilizing the staggered-grid pseudo-spectral (SGPS) method. However, this approach results in time numerical dispersion errors due to the low-order finite difference approximation. In order to address these time-stepping errors, a k-space-based temporal compensating scheme is established to solve the first-order viscoacoustic wave equation. This scheme offers the advantage of being nearly free from grid dispersion for homogeneous media and enhances simulation stability. Numerical examples indicate that the proposed k-space scheme aligns well with analytical solutions for homogeneous media. Additionally, this method demonstrates excellent applicability for complex models and is more efficient due to its ability to adopt a larger time step compared with conventional staggered-grid pseudo-spectral methods.

Maximal $$L_p$$-regularity for x-dependent fractional heat equations with Dirichlet conditions

AbstractWe prove optimal regularity results in $$L_p$$ L p -based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2a ($$0&lt;a&lt;1$$ 0 &lt; a &lt; 1 ) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian $$(-\Delta )^a$$ ( - Δ ) a , as well as elliptic operators $$(-\nabla \cdot A(x)\nabla +b(x))^a$$ ( - ∇ · A ( x ) ∇ + b ( x ) ) a . The proofs are based on general results on maximal $$L_p$$ L p -regularity and its relation to $$\mathcal {R}$$ R -boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.

On the Cauchy problem for logarithmic fractional Schrödinger equation

We consider the fractional Schrödinger equation with a logarithmic nonlinearity, when the power of the Laplacian is between zero and one. We prove global existence results in three different functional spaces: the Sobolev space corresponding to the quadratic form domain of the fractional Laplacian, the energy space, and a space contained in the operator domain of the fractional Laplacian. For this last case, a finite momentum assumption is made, and the key step consists in estimating the Lie commutator between the fractional Laplacian and the multiplication by a monomial.

Fractional Laplacian Elliptic Systems with Multiple Parameters

Fractional Laplacian Elliptic Systems with Multiple Parameters

Heat Kernel Estimates of Fractional Schrödinger Operators with Hardy Potential on Half-line

AbstractWe provide sharp two-sided estimates of the heat kernel of the Dirichlet fractional Laplacian on the half-line perturbed by a Hardy potential.

Source-independent Q-compensated viscoacoustic least-squares reverse time migration

The strong viscosity of the subsurface introduces amplitude absorption and phase-velocity dispersion. Incorrect compensation of the inherent attenuation (the strength of the seismic attenuation can be quantified by the inverse of the quality factor Q, which is defined as [Formula: see text] times the ratio of the stored energy to the lost energy in a single cycle of deformation) can significantly affect imaging quality. Although Q-least-squares reverse time migration ( Q-LSRTM) allows for the compensation of attenuation effects during the iterations, the traditional [Formula: see text]-norm minimization, which is highly sensitive to the source wavelet, poses a challenge in accurately estimating the source wavelet from the field data. Thus, we develop a source-independent Q-LSRTM, in which a convolutional objective function is introduced to replace the [Formula: see text]-norm constraint to mitigate the source wavelet effect. According to the Born approximation, we first linearize the constant-order decoupled fractional Laplacian viscoacoustic wave equation to derive the demigration operator and then construct the corresponding adjoint equation and gradient based on the convolutional objective function, iteratively estimating the reflectivity images. Our method relaxes the sensitivity to the wavelet compared with the conventional [Formula: see text]-norm scheme due to the convolutional objective function, which has the ability to construct the same new source for simulated and observed data. Numerical tests on a layered model, the Marmousi model, and field data demonstrate that our source-independent Q-LSRTM enables us to obtain high-quality reflectivity images even when using incorrect source wavelets.

Entropy solutions to the fully nonlocal diffusion equations

AbstractWe consider the fully nonlocal diffusion equations with nonnegative ‐data. Based on the approximation and energy methods, we prove the existence and uniqueness of nonnegative entropy solutions for such problems. In particular, our results are valid for the time‐space fractional Laplacian equations.

Liouville theorem of the regional fractional Lane–Emden equations

The purpose of this paper is to study the Liouville property for the Lane–Emden equation involving the regional fractional Laplacian ( − Δ ) Ω s u + Vu = h 1 u p + h 2 in Ω , u = 0 on ∂Ω , where s ∈ ( 0 , 1 ) , p>0, h 1 , h 2 are nonnegative functions and Ω ⊂ R N − 1 × [ 0 , + ∞ ) with N ≥ 2 , is an unbounded domain satisfying Ω t := { x ′ ∈ R N − 1 : ( x ′ , t ) ∈ Ω } with t ≥ 0 having an increasing monotonicity, that is, Ω t ⊂ Ω t ′ for t ′ ≥ t . The potential V ( x ′ , t ) decays as t → + ∞ . The properties of the limit domain Ω ∞ := lim t → ∞ Ω t in R N − 1 play an important role to obtain the nonexistence of positive solutions for semilinear elliptic equations with the regional fractional Laplacian. For s ∈ ( 0 , 1 2 ] , we provide a surprising nonexistence result if Ω ∞ is bounded. This particular phenomenon occurs because of the peculiar properties of the regional fractional Laplacian with the order s ∈ ( 0 , 1 2 ] .