Fractional Laplacian Research Articles

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.

Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons’ conjecture

Abstract In this paper, we consider the general dual fractional parabolic problem ∂ t α u ( x , t ) + L u ( x , t ) = f ( t , u ( x , t ) ) in R n × R . ${\partial }_{t}^{\alpha }u\left(x,t\right)+\mathcal{L}u\left(x,t\right)=f\left(t,u\left(x,t\right)\right) \text{in} {\mathbb{R}}^{n}{\times}\mathbb{R}.$ We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must be monotone increasing and one-dimensional symmetric along that direction under an appropriate decreasing condition on f. Our result here actually solves a well-known problem known as Gibbons’ conjecture in the setting of the dual fractional parabolic equations. To overcome the difficulties caused by the nonlocal divergence type operator L $\mathcal{L}$ and the Marchaud time derivative ∂ t α ${\partial }_{t}^{\alpha }$ , we introduce several new ideas. First, we derive a general weighted average inequality corresponding to the nonlocal operator L $\mathcal{L}$ , which plays a fundamental bridging role in proving maximum principles in unbounded domains. Then we combine these two essential ingredients to carry out the sliding method to establish the Gibbons’ conjecture. It is worth noting that our results are novel even for a special case of L $\mathcal{L}$ , the fractional Laplacian (−Δ) s , and the approach developed in this paper will be adapted to a broad range of nonlocal parabolic equations involving more general Marchaud time derivatives and more general non-local elliptic operators.

Liouville type theorems involving fractional order systems

Abstract In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: ( − Δ ) α / 2 u ( x ) = f ( u ( x ) , v ( x ) ) , x ∈ R n , ( − Δ ) α / 2 v ( x ) = g ( u ( x ) , v ( x ) ) , x ∈ R n . $\begin{cases}{\left(-{\Delta}\right)}^{\alpha /2}u\left(x\right)=f\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n},\quad \hfill \\ {\left(-{\Delta}\right)}^{\alpha /2}v\left(x\right)=g\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n}.\quad \hfill \end{cases}$ Under nature structure conditions on f and g, we classify the positive solutions for the semi-linear elliptic system involving the fractional Laplacian by using the direct method of the moving spheres introducing by W. Chen, Y. Li, and R. Zhang (“A direct method of moving spheres on fractional order equations,” J. Funct. Anal., vol. 272, pp. 4131–4157, 2017). In the half space, we establish a Liouville type theorem without any assumption of integrability by combining the direct method of moving planes and moving spheres, which improves the result proved by W. Dai, Z. Liu, and G. Lu (“Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space,” Potential Anal., vol. 46, pp. 569–588, 2017).

The Dirichlet problem for Lévy-stable operators with L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-data

We prove Sobolev regularity for distributional solutions to the Dirichlet problem for generators of 2s-stable processes and exterior data, inhomogeneity in weighted L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-spaces. This class of operators includes the fractional Laplacian. For these rough exterior data the theory of weak variational solutions is not applicable. Our regularity estimate is robust in the limit s→1-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s\\rightarrow 1-$$\\end{document} which allows us to recover the local theory.

A Grid-Overlay Finite Difference Method for the Fractional Laplacian on Arbitrary Bounded Domains

A Grid-Overlay Finite Difference Method for the Fractional Laplacian on Arbitrary Bounded Domains

Multiplicity of Normalized Solutions for the Fractional Schrödinger Equation with Potentials

We are concerned with the existence and multiplicity of normalized solutions to the fractional Schrödinger equation (−Δ)su+V(εx)u=λu+h(εx)f(u)inRN,∫RN|u|2dx=a,, where (−Δ)s is the fractional Laplacian, s∈(0,1), a,ε>0, λ∈R is an unknown parameter that appears as a Lagrange multiplier, h:RN→[0,+∞) are bounded and continuous, and f is L2-subcritical. Under some assumptions on the potential V, we show the existence of normalized solutions depends on the global maximum points of h when ε is small enough.

Variable-Order Fractional Laplacian and its Accurate and Efficient Computations with Meshfree Methods

Variable-Order Fractional Laplacian and its Accurate and Efficient Computations with Meshfree Methods

Sign‐changing solutions of fractional Laplacian system with critical exponent

This paper deals with the following fractional Laplacian system with critical exponent: where is a bounded smooth open connected set in is the fractional critical Sobolev exponent, and is the first eigenvalue of fractional Laplacian under the condition in . We prove that, for each fixed and slightly smaller than , the above system with admits a sign‐changing solution in the following sense: one component changes sign, while the other one is positive. Our result includes the lower dimensional case . Compared with the classical Laplacian case, our problem is nonlocal and the first component of the solution is sign‐changing, some new difficulties arise and new arguments and estimates should be introduced.

A static memory method for modelling time-fractional power law absorption

The attenuation of ultrasound propagation through tissue is known to follow a frequency power-law in the time domain. This can be modelled as a loss operator in the equation of state within the Euler equations that takes the form of a fraction time derivative operator. This can be treated through a second order accurate transfer between the fractional time derivative and a fractional Laplacian spacial operator in order to avoid storage of the full time history. This is used for example by the k-Wave toolbox. As an alternate finite history methods have been suggested. Building on recent work, here, we re-write the time fractional derivative as a finite sum using a recursion relation. This allows the time fractional derivative to be directly computed with a static memory requirement. For homogeneous media, the advantage of this over the fractional Laplacian methods may not initially be obvious with a base higher memory requirement and computational cost with only a small increase in accuracy. However, upon introducing a heterogeneous medium with regions of different power law attenuation; the increased computational cost of the time fractional method is contained exclusively within the pre-computation, while the increased cost for the fractional Laplacian is applied to each time step.

Clustering/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number μ

We consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number μ, approximated by finite differences. In a recent analysis, singular value clustering and eigenvalue clustering have been proposed for a τ preconditioning when the variable coefficient wave number μ is uniformly bounded. Here, we extend the analysis to the unbounded case by focusing on the case of a power singularity. Several numerical experiments concerning the spectral behavior and convergence of the related preconditioned GMRES are presented.

Qualitative properties of solutions for system involving the fractional Laplacian

In this paper, we consider the following non-linear system involving the fractional Laplacian 0.1 \begin{equation} \left\{\begin{array}{@{}ll} (-\Delta)^{s} u (x)= f(u,\,v), \\ (-\Delta)^{s} v (x)= g(u,\,v), \end{array} \right. \end{equation} in two different types of domains, one is bounded, and the other is an infinite cylinder, where $0< s<1$ . We employ the direct sliding method for fractional Laplacian, different from the conventional extension and moving planes methods, to derive the monotonicity of solutions for (0.1) in $x_n$ variable. Meanwhile, we develop a new iteration method for systems in the proofs. Hopefully, the iteration method can also be applied to solve other problems.

Quasilinear Lane–Emden type systems with sub-natural growth terms

Global pointwise estimates are obtained for quasilinear Lane–Emden-type systems involving measures in the “sublinear growth” rate. We give necessary and sufficient conditions for existence expressed in terms of Wolff’s potential. Our approach is based on recent advances due to Kilpeläinen and Malý in the potential theory. This method enables us to treat several problems, such as equations involving general quasilinear operators and fractional Laplacian.

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

Monte Carlo method for the Cauchy problem of fractional diffusion equation concerning fractional Laplacian

In this paper, we study a Cauchy problem of fractional diffusion equation concerning fractional Laplacian. We prove the existence and uniqueness of solution to the problem under investigation in Hölder space. Then we apply the Monte Carlo method to solving this Cauchy problem. For the problem with free force term, we derive an unbiased scheme which only produces the statistic error. For the problem with inhomogeneous force term, we establish the simple and high-order jump-adapted schemes by approximating the trajectory of symmetric stable process. The error estimates of these two jump-adapted schemes are given based on the Hölder regularity of the solution. Numerical experiments including even one hundred dimensional case confirm the theoretical analysis and show the numerical efficiency.

On (global) unique continuation properties of the fractional discrete Laplacian

We study various qualitative and quantitative (global) unique continuation properties for the fractional discrete Laplacian. We show that while the fractional Laplacian enjoys striking rigidity properties in the form of (global) unique continuation properties, the fractional discrete Laplacian does not enjoy these in general. While discretization thus counteracts the strong rigidity properties of the continuum fractional Laplacian, by discussing quantitative forms of unique continuation, we illustrate that these properties can be recovered if exponentially small (in the lattice size) correction terms are added. In particular, this allows us to deduce stability properties for a discrete, linear inverse problem for the fractional Laplacian. We complement these observations with a transference principle and the discussion of these properties on the discrete torus.

The radial symmetry of positive solutions for semilinear problems involving weighted fractional Laplacians

The radial symmetry of positive solutions for semilinear problems involving weighted fractional Laplacians

A priori estimates and Liouville type theorems for semilinear equations and systems with fractional Laplacian

In this paper, we firstly obtain a priori estimates of nonnegative solutions to fractional Laplacian equations and systems with critical Sobolev exponent. Secondly, by deriving the universal estimates of solutions, we establish the connections between Liouville type theorems and local properties of nonnegative solutions to fractional Laplacian equations and fractional Lane-Emden systems with subcritical Sobolev exponent. Our main results improve the work of Chen et al. (2016) [8] and Polácik et al. (2007) [26]. Specially, for fractional Lane-Emden systems, our results seem to be the first results on universal estimates to cover the full subcritical range. Furthermore, the novelty of our method lies in a completely new blow-up analysis coupled with a truncating technique and the construction of barrier function which is sufficiently different from the blow-up method used in [8,26].

Upper semicontinuity of the attractor for a nonlinear hyperbolic-parabolic coupled system with fractional Laplacian

In this paper we establish the existence and uniqueness of global solutions (in time), as well as the existence, regularity and stability (upper semicontinuity) of the attractor for the semigroup generated by the solutions of a two-dimensional nonlinear hyperbolic-parabolic coupled system with fractional Laplacian. In addition, we also obtain the existence of an exponential attractor and show that this attractor has a finite fractal dimension in a space containing the phase space of the dynamical system.

On a family of fully nonlinear integrodifferential operators : from fractional Laplacian to nonlocal Monge–Ampère

On a family of fully nonlinear integrodifferential operators : from fractional Laplacian to nonlocal Monge–Ampère