Fractional Laplacian Operator Research Articles

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.

Discretization of Fractional Fully Nonlinear Equations by Powers of Discrete Laplacians

AbstractWe study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $$\sigma \in (0,2)$$ σ ∈ ( 0 , 2 ) since they involve fractional Laplace operators $$(-\Delta )^{\sigma /2}$$ ( - Δ ) σ / 2 . They arise e.g. in control and game theory as dynamic programming equations – HJB and Isaacs equation – and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $$\sigma $$ σ . The accuracy of previous approximations of fractional fully nonlinear equations depend on $$\sigma $$ σ and are worse when $$\sigma $$ σ is close to 2. We show that the schemes are monotone, consistent, $$L^\infty $$ L ∞ -stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.

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.

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.

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.

Bayesian inversion of a fractional elliptic system derived from seismic exploration

In this paper, we concentrate on the Bayesian inversion of a dispersion‐dominated fractional Helmholtz (DDFH) equation, which has been introduced in studies concerning seismic exploration. To establish the inversion theory, we meticulously examine the DDFH equation. We transform it into a system comprising both fractional‐ and integer‐order elliptic equations, extending the conventional definition of the spectral fractional Laplace operator to accommodate non‐homogeneous boundary conditions. Subsequently, we establish the well‐posedness theory for scenarios involving both small and large wavenumbers. Our proof hinges upon the regularity attributes of select fractional elliptic equations and capitalizes fully on the structural peculiarities of the elliptic system, which distinguish it from classical cases. Thereafter, we focus on the inverse medium scattering problem pertinent to the DDFH equation, framed within the Bayesian statistical framework. We address two scenarios: one devoid of model reduction errors and another characterized by such errors—arising from the implementation of certain absorbing boundary conditions. More precisely, based on the properties of the forward operator, well‐posedness of the posterior measures have been proved in both cases, which provide an opportunity to quantify the uncertainties of this problem.

Tempered fractional Sobolev spaces

The fractional Laplacian operator Δs is the infinitesimal generator of the isotropic (2s)-stable Lévy process on Rn, which is the scaling limit of the Lévy flight with isotropic power law measure |x|−n−2s. However, their second and higher moments are divergent, leading to the difficulty in the modeling of practical physical processes. Replacing |x|−n−2s by the measure of an isotropic tempered power law with the tempering exponent λ (i.e., e−λ|x||x|−n−2s), the tempered fractional Laplacian operator (Δ+λ)s was introduced in [17] as the infinitesimal generator of the tempered Lévy process. In this paper, guided by the fractional Sobolev spaces Ws,p corresponding to the fractional Laplacian operator Δs, we deal with the tempered fractional Sobolev spaces Ws,λ,p associated with the tempered fractional Laplacian (Δ+λ)s. First, the definition of the tempered fractional Sobolev spaces Ws,λ,p is given via the Gagliardo approach, and some of their basic properties were studied. Subsequently, we focus on the Hilbert case Hs,λ based on the Fourier transform. In particular, we deal with its relation with the tempered fractional Sobolev space Ws,λ,2 and analyze their role in the trace theory, overcoming the challenges posed by the tempering exponent λ. Then we investigate the asymptotic behavior of λ→0+,s→1− and s→0+ that appear in the definition of the tempered fractional Laplacian operator (Δ+λ)s. Moreover, we show continuous and compact embeddings investigating the problem of the extension domains and the generalized Hölder regularity results. As an application of the tempered fractional Sobolev spaces, we prove that the process defined by the tempered Lévy process is a solution of some PDE with tempered fractional Lapalcian operator.

Single‐peak solution for a fractional slightly subcritical problem with non‐power nonlinearity

AbstractWe consider the following fractional problem involving slightly subcritical non‐power nonlinearity, where is a smooth bounded domain in , , , is the fractional critical Sobolev exponent and is a small parameter, is the spectral fractional Laplacian operator. We construct a positive bubbling solution, which concentrates at a nondegenerate critical point of the Robin function by Lyapunov–Schmidt reduction procedure.

Efficient modeling of fractional Laplacian viscoacoustic wave equation with fractional finite-difference method

The fractional viscoacoustic/viscoelastic wave equation, which accurately quantifies the frequency-independent anelastic effects, has been the focus of seismic industry in recent years. The pseudo-spectral (PS) method stands as one of the most widely used numerical methods for solving the fractional wave equation. However, the PS method often suffers from low accuracy and efficiency, particularly when modeling wave propagation in heterogeneous media. To address these issues, we propose a novel and efficient fractional finite-difference (FD) method for solving the wave equation with fractional Laplacian operators. This method develops an arbitrary high-order FD operator via the generating function of our fractional FD (F-FD) scheme, enhancing accuracy with L2-optimal FD coefficients. Similar to classic FD methods, our F-FD method is characterized by straightforward programming and excellent 3D extensibility. It surpasses the PS method by eliminating the need for Fast Fourier Transform (FFT) and inverse-FFT (IFFT) operations at each time step, offering significant benefits for 3D applications. Consequently, the F-FD method proves more adept for wave-equation-based seismic data processes like imaging and inversion. Compared with existing F-FD methods, our approach uniquely approximates the entire fractional Laplacian operator and stands as a local numerical algorithm, with an adjustable F-FD operator order based on model parameters for enhanced practicality. Accuracy analyses confirm that our method matches the precision of the PS method with a correctly ordered F-FD operator. Numerical examples show that the proposed method has good applicability for complex models. Finally, we have carried out reverse time migration on the Marmousi-2 model, and the imaging profiles indicate that the proposed method can be effectively applied to seismic imaging, demonstrating good practicability.

A novel physics-aware graph network using high-order numerical methods in weather forecasting model

In this paper, we present a novel architecture for accurate weather forecasting within the framework of Physics-aware Graph Networks (PaGN), which aims to improve the prediction of weather time series by integrating both data and physical equations defined on sparsely distributed spatial domains. This approach employs geometric deep learning by accounting for both supervised loss and physics loss. Solely in supervised learning, data-driven training demands a large amount of data, whereas integrating physics information can reduce this requirement, resulting in faster training. The physics awareness also provides a stronger inductive bias and better generalization, directing the machine learning process by clarifying the complexity of weather data. Thus, a more accurate discretization of physics equations can enhance the physics awareness of the complex weather dynamics. Meanwhile, weather data is essentially gathered at observatories that are not located densely, since covering the Earth with numerous observatories is prohibitively expensive. Given its reliance on graph structure, the proposed PaGN is well-suited for handling sparsely distributed data particularly defined on arbitrary geometric domain. One of the main contributions of this study is the use of high-order numerical methods approximating physics equations defined on graph neural networks for achieving accuracy improvements. The fractional graph Laplacian operator is further integrated with the physics equations to account for the non-local characteristics of complex weather data. For computational efficiency, the accuracy improved PaGN architecture is smoothly implemented in the embedding space. A series of numerical tests on weather datasets is performed to verify the improved accuracy, robustness, and applicability of the proposed methodology. We first carry out the numerical study on the synthetic datasets obtained from the physics equations with an extra forcing term. We then extensively investigate the accuracies of weather forecasting for various configurations in terms of the geophysical regions and the accuracy-improved PaGN approaches for diffusion and wave equations. Moreover, we emphasize that the fractional graph Laplacian operator incorporated into our PaGN model can further improve the accuracy for weather forecasting.

Phase-space entropy cascade and irreversibility of stochastic heating in nearly collisionless plasma turbulence.

We consider a nearly collisionless plasma consisting of a species of "test particles" in one spatial and one velocity dimension, stirred by an externally imposed stochastic electric field-a kinetic analog of the Kraichnan model of passive advection. The mean effect on the particle distribution function is turbulent diffusion in velocity space-known as stochastic heating. Accompanying this heating is the generation of fine-scale structure in the distribution function, which we characterize with the collisionless (Casimir) invariant C_{2}∝∫∫dxdv〈f^{2}〉-a quantity that here plays the role of (negative) entropy of the distribution function. We find that C_{2} is transferred from large scales to small scales in both position and velocity space via a phase-space cascade enabled by both particle streaming and nonlinear interactions between particles and the stochastic electric field. We compute the steady-state fluxes and spectrum of C_{2} in Fourier space, with k and s denoting spatial and velocity wave numbers, respectively. In our model, the nonlinearity in the evolution equationfor the spectrum turns into a fractional Laplacian operator in k space, leading to anomalous diffusion. Whereas even the linear phase mixing alone would lead to a constant flux of C_{2} to high s (towards the collisional dissipation range) at every k, the nonlinearity accelerates this cascade by intertwining velocity and position space so that the flux of C_{2} is to both high k and high s simultaneously. Integrating over velocity (spatial) wave numbers, the k-space (s-space) flux of C_{2} is constant down to a dissipation length (velocity) scale that tends to zero as the collision frequency does, even though the rate of collisional dissipation remains finite. The resulting spectrum in the inertial range is a self-similar function in the (k,s) plane, with power-law asymptotics at large k and s. Our model is fully analytically solvable, but the asymptotic scalings of the spectrum can also be found via a simple phenomenological theory whose key assumption is that the cascade is governed by a "critical balance" in phase space between the linear and nonlinear timescales. We argue that stochastic heating is made irreversible by this entropy cascade and that, while collisional dissipation accessed via phase mixing occurs only at small spatial scales rather than at every scale as it would in a linear system, the cascade makes phase mixing even more effective overall in the nonlinear regime than in the linear one.

The well-posedness of semilinear fractional dissipative equations on [formula omitted]

In this paper, we study the well-posedness of time-space fractional dissipative equations. Combining the Hörmander-multipliers theory and asymptotic property of the Mittag-Leffler functions, we give a useful method to estimate the solution operator which is independent of the C0-semigroup generated by the fractional Laplace operator (−Δ)β2 and the Mainardi's Wright type function. Furthermore, we discuss the global/local well-posedness in some appropriate time-space Lebesgue space and Besov spaces. We also obtain several results about blow-up alternative and asymptotic behavior. The approaches are based on the Hörmander-multipliers theory, Gagliardo-Nirenberg inequalities, fixed point techniques, Sobolev embedding and some methods in harmonic analysis.

Fundamental solutions and critical Lane-Emden exponents for nonlinear integral operators in cones

In this article we study the fundamental solutions or “α-harmonic functions” for some nonlinear positive homogeneous nonlocal elliptic problems in conical domains, such asF(u)=0inCω,u=0inRN∖Cω, where ω is a proper C2 domain in SN−1 for N≥2, Cω:={x:x≠0,|x|−1x∈ω} is the cone-like domain related to ω, and F is an extremal fully nonlinear integral operator. We prove the existence of two fundamental solutions that are homogeneous and do not change signs in the cone; one is bounded at the origin and the other at infinity.As an application, we use the fundamental solutions obtained to prove Liouville type theorems in cones for supersolutions of the Lane-Emden-Fowler equation in the formF(u)+up=0inCω,u=0inRN∖Cω. We also prove a generalized Hopf type lemma in domains with corners. Most of our results are new even when F is the fractional Laplacian operator.

On the existence of positive solutions of some nonlocal elliptic problems involving fractional p–Laplacian operator

Our aim in this paper is to analyze the existence of solutions to a nonlocal elliptic problem involving the fractional Laplacian operator. These operators are utilized to solve an equation defined within a bounded domain in . The operator is a fractional Laplacian , where with the condition and is a non-negative function almost everywhere with respect to the variable , and are real numbers. The article establishes two existence theorems for weak solutions using Tychonoff and Schauder fixed-point theorems. These theorems are formulated based on different hypotheses regarding the parameters and .

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

The existence of optimal solutions for nonlocal partial systems involving fractional Laplace operator with arbitrary growth

Abstract In this paper, we investigate nonlocal partial systems that incorporate the fractional Laplace operator. Our primary focus is to establish a theorem concerning the existence of optimal solutions for these equations. To achieve this, we utilize two fundamental tools: information obtained from an iterative reconstruction algorithm and a variant of the Phragmén–Lindelöf principle of concentration and compactness tailored for fractional systems. By employing these tools, we provide valuable insights into the nature of nonlocal partial systems and their optimal solutions.

Multiplicity of solutions for variable-order fractional Kirchhoff problem with singular term

In this paper, we consider a class of singular variable-order fractional Kirchhoff problem of the form: where is a bounded domain, is the variable-order fractional Laplacian operator, [u] s(·) is the Gagliardo seminorm and is a continuous and symmetric function. We assume that λ is a non-negative parameter, with and . We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem.

Fractional-order velocity-dilatation-rotation viscoelastic wave equation and numerical solution based on a constant-Q model

Viscoelastic wave equations based on the constant- Q (CQ) model can accurately describe the amplitude dissipation and phase distortion of waves in anelastic media. However, only three velocity or displacement components can be obtained directly by solving such equations. Starting from the time-domain second-order displacement viscoelastic wave equation, we derived the decoupled P- and S-wave displacement vector viscoelastic wave equation by using the polarization difference of P- and S-wave propagation in isotropic media. The equation can be transformed into the velocity-dilatation-rotation viscoelastic wave equation containing the first-order temporal derivative and fractional Laplacian operators, which can be solved directly by using the staggered-grid finite-difference and pseudospectral methods. We use the low-rank decomposition method to approximate the derived mixed space-wavenumber domain fractional Laplacian operators for modeling wave propagation in heterogeneous attenuating media. We also demonstrated the precision of our equation by comparing the numerical solutions with the analytical solutions. Furthermore, compared with the conventional velocity-stress viscoelastic wave equation, experimental results demonstrate that our equation can separate the pure P and S waves from the mixed wavefield during wavefield continuation. In addition, it can be separated into an equation containing predominantly an amplitude attenuation or a phase distortion term.

Optimal control problem with nonlinear fractional system constraint applied to image restoration

This study aims to investigate a novel nonlinear optimization problem that incorporate a partial differential equation (PDE) constraint for image denoising in the context of mixed noise removal. Based on ‐norm and decomposition approach, we develop a nonlinear system that involves the fractional Laplacian operator. Based on Schauder's fixed point theorem, we establish the existence and uniqueness of weak solution for the direct problem. Furthermore, we study the well‐posedness of the optimal control, and we also prove the existence of the weak solutions for the adjoint problem by using Galerkin's method. In order to numerically compute the solution of the proposed model, we introduce the numerical discretization scheme and the primal–dual algorithm used to solve our problem. Finally, we provide comparative numerical experiments to evaluate the efficiency and effectiveness of our proposed model.

Chimera states in a lattice of superdiffusively coupled neurons

Chimera states are a truly remarkable dynamical phenomenon that occur in systems of coupled oscillators. In this regime, regions of synchronized and unsynchronized elements are formed in the system. For many applied problems, especially in neuroscience, these states offer a rich potential for research. However, the plethora of models and the lack of a ”single simple principle” that leads to the development of chimeras makes it very difficult to understand their nature. In this work, we propose a three-component reaction-superdiffusion system based on a unified mechanism founded on the properties of the fractional Laplace operator and the nonlinear Hindmarsh-Rose model functions. In the proposed system, the non-local type of interaction forming the coupling between the elements depends significantly on the fractional Laplace operator exponents of the corresponding components. It is shown that in the framework of the superdiffusion type of interaction, chimera states are realized in the system. At the same time, many qualitative (shape, visual degree of inhomogeneity and area size) and quantitative characteristics of chimeras (synchronization factor, strength of incoherence, local order parameter, number of elements with a potential value exceeding a given one) depend significantly on the exponents of the fractional Laplace operator. In addition to classical chimeras and target-waves chimeras, the results of numerical simulations show the presence of mutually sustaining reaction processes of different scales in the system.