Fractional Laplacian Equation Research Articles

Direct Method of Moving Planes for Tempered Fractional Laplacian Equations

Direct Method of Moving Planes for Tempered Fractional Laplacian Equations

Existence and multiplicity of positive solutions for a critical fractional Laplacian equation with singular nonlinearity

Existence and multiplicity of positive solutions for a critical fractional Laplacian equation with singular nonlinearity

Frequency-dependent Q simulation and viscoacoustic reverse time migration based on the fractional Zener model

Seismic attenuation is a basic physical property of the earth, which significantly affects the characteristics of seismic wavefields. Accurately simulating wave propagation in the earth is essential to image subsurface structures. Some prevailing methods (e.g., the standard linear solid and fractional Laplacian equation) to describe seismic wave propagation in attenuating media are mainly based on the constant- Q model (CQM), which is valid at room temperature and pressure. However, laboratory measurements suggest that the quality factor Q is a function of frequencies in some regions. To simulate the frequency-dependent Q effect, we derive a viscoacoustic wave equation from the stress-strain relationship of the fractional Zener model (FZM) with variable fractional orders. During the implementation, we separate the real and imaginary parts of the modulus and introduce a low-rank decomposition method to solve the FZM equation. Because the amplitude dissipation and phase dispersion are decoupled, we establish a compensated reverse time migration ( Q-RTM) algorithm to mitigate adverse effects caused by seismic attenuation and improve the quality of seismic migration in frequency-dependent attenuating media. A two-layer model and the BP gas chimney model are used to perform Q-RTM tests. A low-pass filter with a Tukey window function is applied to suppress numerical instability during the compensation. Numerical results demonstrate that our FZM Q-RTM approach can produce high-resolution images with corrected reflector positions and amplitudes. Because the CQM equation ignores the frequency dependence of Q, it may lead to overcompensation in Q-RTM.

Comparative analysis for fractional Laplace and Helmholtz equations on sphere with mixed boundary conditions

Comparative analysis for fractional Laplace and Helmholtz equations on sphere with mixed boundary conditions

Dirichlet Problems for Fractional Laplace Equations with Singular Nonlinearity

Dirichlet Problems for Fractional Laplace Equations with Singular Nonlinearity

On the standing wave in coupled fractional Klein–Gordon equation

Abstract The aim of this paper is to deal with the standing wave problems in coupled nonlinear fractional Klein–Gordon equations. First, we establish the constrained minimizations for a single nonlinear fractional Laplace equation. Then we prove the existence of a standing wave with a ground state using a variational argument. Next, applying the potential well argument and the concavity method, we obtain the sharp criterion for blowing up and global existence. Finally, we show the instability of the standing wave.

Existence of solution for the (p, q)-fractional Laplacian equation with nonlocal Choquard reaction and exponential growth

In this paper, we study the existence of weak solution to -fractional Choquard equation in as follows where , , f has exponential growth. The function f and V are continuous which satisfy some suitable assumptions. In our best knowledge, sofar there is not any work about existence of weak solution to -fractional Choquard equation with exponential growth. Our results are even new in the case -Laplace equation which are complement the results of Fiscella and Pucci [ equations with critical exponential nonlinearities in . J Math Anal Appl. 2021;501(1), Article 123379].

Uniqueness and some related estimates for Dirichlet problem with fractional Laplacian

AbstractThe non‐uniqueness of solutions of basic Dirichlet‐type problems is a surprising feature of the fractional Laplace equation. This paper establishes a somewhat sharp uniqueness condition for the fractional Laplace equation. We derive the ‐estimate for fractional Laplacian operators to better understand this phenomenon. We introduce several naturally weighted fractional Sobolev spaces and establish embedding relationships among them. These existence‐uniqueness conditions and the spaces we introduce here are intrinsically related to the fractional Laplacian and provide basic information for studying related problems.

Neural Network Method for Integral Fractional Laplace Equations

Neural Network Method for Integral Fractional Laplace Equations

Properties of Solutions to Fractional Laplace Equation with Singular Term

Properties of Solutions to Fractional Laplace Equation with Singular Term

Weighted mixed-norm Lp estimates for equations in non-divergence form with singular coefficients: The Dirichlet problem

We study a class of non-divergence form elliptic and parabolic equations with singular first-order coefficients in an upper half space with the homogeneous Dirichlet boundary condition. In the simplest setting, the operators in the equations under consideration appear in the study of fractional heat and fractional Laplace equations. Intrinsic weighted Sobolev spaces are found in which the existence and uniqueness of strong solutions are proved under certain smallness conditions on the weighted mean oscillations of the coefficients in small parabolic cylinders. Our results are new even when the coefficients are constants and they cover the case where the weights may not be in the Ap-Muckenhoupt class.

Propagation of waves in fractal spaces

In this study, we study waves propagation in fractal spaces based on two independent variational approaches: the first one is based on the ‘product-like fractal measure’ introduced by Li and Ostoja-Starzewski in their analysis of nonlinear fractal dynamics in anisotropic porous media whereas the second and another is based on fractal calculus, which includes analytical functions with fractal support. The first approach is motivating since it describes physics in anisotropic media characterized by fractal dimensions. The second one is also of particular importance since it offers a new framework to reformulate the laws of physics in spaces with fractional dimensions. In both approaches, the fractional Laplace equation is derived and solved using plausible boundary conditions. This study proves the importance of fractals in wave motion We show that these models are able to describe the propagation of waves in a fractal Hausdorff dimensional space.

Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition

Abstract In this article, we study a class of nonlinear fractional Laplace problems with a parameter and superlinear nonlinearity ( − Δ ) s u = λ u + f ( x , u ) , in Ω , u = 0 , in R N \ Ω . \left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{ll}{\left(-\Delta )}^{s}u=\lambda u+f\left(x,u),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}\backslash \Omega \right.\end{array}\right. Multiplicity of nontrivial solutions is obtained when the parameter is near the eigenvalue of the fractional Laplace operator without Ambrosetti and Rabinowitz condition for the nonlinearity. Our methods are the combination of minimax method, bifurcation theory, and Morse theory.

更高分数阶Laplacian方程的径向解

更高分数阶Laplacian方程的径向解

Moving spheres for semilinear spectral fractional Laplacian equations in the half space

In this paper, we introduce a direct method of moving spheres for the spectral fractional Laplacian $ (-\Delta_D)^{\alpha/2} $ with $ 0<\alpha<2 $ on the half Euclidean space. As one expected, the key ingredient is the narrow region maximum principle, which can be obtained via the hide monotonicity of the kernel used in the definition of the spectral fractional Laplacian. Using this direct method of moving spheres, we establish monotonicity or symmetry results for nonlinear spectral Laplacian equations on the half Euclidean space.

Periodic solutions of fractional Laplace equations: Least period, axial symmetry and limit

We are concerned with periodic solutions of the fractional Laplace equation \begin{equation*} {(-\partial_{xx})^s}u(x)+F'(u(x))=0 \quad \mbox{in }\mathbb{R}, \end{equation*} where $0< s< 1$. The smooth function $F$ is a double-well potential with wells at $+1$ and $-1$. We show that the value of least positive period is $2{\pi}\times({1}/{-F''(0)})^{{1}/({2s})}$. The axial symmetry of odd periodic solutions is obtained by moving plane method. We also prove that odd periodic solutions $u_{T}(x)$ converge to a layer solution of the same equation as periods $T\rightarrow+\infty$.

A modified walk‐on‐sphere method for high dimensional fractional Poisson equation

AbstractWe develop walk‐on‐sphere method for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk‐on‐sphere method is based on probabilistic representation of the fractional Poisson equation. We propose efficient quadrature rules to evaluate integral representation in the ball and apply rejection sampling method to drawing from the computed probabilities in general domains. Moreover, we provide an estimate of the number of walks in the mean value for the method when the domain is a ball. We show that the number of walks is increasing in the fractional order and the distance of the starting point to the origin. We also give the relationship between the Green function of fractional Laplace equation and that of the classical Laplace equation. Numerical results for problems in 2–10 dimensions verify our theory and the efficiency of the modified walk‐on‐sphere method.

Monte Carlo fPINNs: Deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations

We introduce a sampling-based machine learning approach, Monte Carlo fractional physics-informed neural networks (MC-fPINNs), for solving forward and inverse fractional partial differential equations (FPDEs). As a generalization of the physics-informed neural networks (PINNs), MC-fPINNs utilize a Monte Carlo approximation strategy to compute the fractional derivatives of the DNN outputs, and construct an unbiased estimation of the physical soft constraints in the loss function. Our sampling approach can yield lower overall computational cost compared to fPINNs proposed in Pang et al.(2019), hence it can solve high dimensional FPDEs at reasonable cost. We validate the performance of MC-fPINNs via several examples, including high dimensional integral fractional Laplacian equations, parametric identification of time–space fractional PDEs, and fractional diffusion equation with random inputs. The results show that MC-fPINNs are flexible and quite effective in tackling high dimensional FPDEs.

Sliding Methods for a Class of Generalized Fractional Laplacian Equations

Sliding Methods for a Class of Generalized Fractional Laplacian Equations

Deep Ritz Method for the Spectral Fractional Laplacian Equation Using the Caffarelli--Silvestre Extension

In this paper, we propose a novel method for solving high-dimensional spectral fractional Laplacian equations. Using the Caffarelli-Silvestre extension, the $d$-dimensional spectral fractional equation is reformulated as a regular partial differential equation of dimension $d+1$. We transform the extended equation as a minimal Ritz energy functional problem and search for its minimizer in a special class of deep neural networks. Moreover, based on the approximation property of networks, we establish estimates on the error made by the deep Ritz method. Numerical results are reported to demonstrate the effectiveness of the proposed method for solving fractional Laplacian equations up to ten dimensions. Technically, in this method, we design a special network-based structure to adapt to the singularity and exponential decaying of the true solution. Also, A hybrid integration technique combining Monte Carlo method and sinc quadrature is developed to compute the loss function with higher accuracy.