Fractional Laplacian Equation Research Articles

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.

On the Dirichlet problem for fractional Laplace equation on a general domain

In this paper, we study the weak strong uniqueness of the Dirichlet type problems of fractional Laplace (Poisson) equations. We construct the Green’s function and the Poisson kernel. We then provide a somewhat sharp condition for the solution to be unique. We also show that the solution under such condition exists and must be given by our Green’s function and Poisson kernel. In doing these, we establish several basic and useful properties of the Green’s function and Poisson kernel. Based on these, we obtain some further a priori estimates of the solutions. Surprisingly those estimates are quite different from the ones for the local type elliptic equations such as Laplace equations. These are basic properties to the fractional Laplace equations and can be useful in the study of related problems.

Stable Q-compensated reverse time migration in TTI media based on a modified fractional Laplacian pure-viscoacoustic wave equation

Abstract The anisotropy and attenuation properties of real earth media can lead to amplitude reduction and phase dispersion as seismic waves propagate through it. Ignoring these effects will degrade the resolution of seismic imaging profiles, thereby affecting the accuracy of geological interpretation. To characterize the impacts of viscosity and anisotropy, we formulate a modified pure-viscoacoustic (PU-V) wave equation including the decoupled fractional Laplacian (DFL) for tilted transversely isotropic (TTI) media, which enables the generation of stable wavefields that are resilient to noise interference. Numerical tests show that the newly derived PU-V wave equation is capable of accurately simulating the viscoacoustic wavefields in anisotropic media with strong attenuation. Building on our TTI PU-V wave equation, we implement stable reverse time migration technique with attenuation compensation (Q-TTI RTM), effectively migrating the impacts of anisotropy and compensates for attenuation. In the Q-TTI RTM workflow, to remove the unstable high-frequency components in attenuation-compensated wavefields, we construct a stable attenuation-compensated wavefield modeling (ACWM) operator. The proposed stable ACWM operator consists of velocity anisotropic and attenuation anisotropic parameters, effectively suppressing the high-frequency artifacts in the attenuation-compensated wavefield. Synthetic examples demonstrate that our stable Q-TTI RTM technique can simultaneously and accurately correct for the influences of anisotropy and attenuation, resulting in the high-quality imaging results.

Direct Method of Moving Planes for Tempered Fractional Laplacian Equations

Direct Method of Moving Planes for Tempered Fractional Laplacian Equations

Solution of Fractional Differential Equations Involving Hilfer-Hadamard Fractional Derivatives

Objectives: The aim is to establish prerequisite properties for the Hilfer-Hadamard fractional derivatives and address boundary value problems related to fractional polar Laplace and fractional Sturm-Liouville equations involving Hilfer-Hadamard fractional derivatives. Methods: Existing definitions and findings are utilized to obtain the properties for fractional derivatives, and the Adomian decomposition method is employed to solve the fractional differential equations. Findings: Validity conditions for the law of exponents are determined, and the study investigates the fractional differential equations and their corresponding solutions, possessing the capacity to replace the traditional polar Laplace and Sturm-Liouville boundary value problems to effectively represent real-world phenomena. Novelty: The study introduces the substitution of two consecutively operated Hilfer-Hadamard fractional derivatives with a corresponding single Hilfer-Hadamard fractional derivative using the law of exponents. Additionally, the polar Laplace and Sturm-Liouville boundary value problems are extended to their respective fractional counterparts, expressed in a concise format using HilferHadamard fractional derivatives. Keywords: Adomian decomposition method, Hilfer-Hadamard fractional derivative, Fractional polar Laplace equation, Fractional Sturm-Liouville boundary value problem

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 ( p , q ) -fractional Choquard equation in R N as follows L p s u + L q s u + V ( x ) ( | u | p − 2 u + | u | q − 2 u ) = ( 1 | x | μ ∗ F ( u ) ) f ( u ) , where 2 ≤ N s = p < q , 0 ≤ μ < N , 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 ( p , q ) -fractional Choquard equation with exponential growth. Our results are even new in the case ( N , q ) -Laplace equation which are complement the results of Fiscella and Pucci [ ( p , N ) equations with critical exponential nonlinearities in R N . 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.

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}&amp; \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=0,\hspace{1.0em}&amp; \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&lt;\alpha&lt;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.

Neural Network Method for Integral Fractional Laplace Equations

Neural Network Method for Integral Fractional Laplace Equations

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 $0&lt; s&lt; 1$. 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 $T\rightarrow+\infty$.