Fractional Laplacian Operator Research Articles

Existence of solutions for a nonhomogeneous sublinear fractional Schrödinger equation

ABSTRACT This paper is concerned with the following sublinear fractional Schrödinger equation: where , N>2s, is the fractional Laplacian operator, and V, a both change sign in , is a nonnegative perturbation such that . By using the s-harmonic extension technique and suitable variational methods, we prove the existence of at least two nontrivial solutions for the problem.

Numerical continuation for fractional PDEs: sharp teeth and bloated snakes

Partial differential equations (PDEs) involving fractional Laplace operators have been increasingly used to model non-local diffusion processes and are actively investigated using both analytical and numerical approaches. The purpose of this work is to study the effects of the spectral fractional Laplacian on the bifurcation structure of reaction–diffusion systems on bounded domains. In order to do this we use advanced numerical continuation techniques to compute the solution branches. Since current available continuation packages only support systems involving the standard Laplacian, we first extend the pde2path software to treat fractional PDEs (in the spectral definition). The new capabilities are then applied to the study of the Allen–Cahn equation, the Swift–Hohenberg equation and the Schnakenberg system (in which the standard Laplacian is replaced by the spectral fractional Laplacian). In particular, we investigate the changes in snaking bifurcation diagrams and in the spatial structure of non-trivial steady states upon variation of the order of the fractional Laplacian. Our results show that the fractional order induces significant qualitative and quantitative changes in the overall bifurcation structures, of which some are shared by the three systems. This contributes to a better understanding of the effects of fractional diffusion in generic reaction–diffusion systems.

Existence of positive solutions for a critical fractional Kirchhoff equation with potential vanishing at infinity

AbstractIn this paper, we consider the following fractional Kirchhoff equation with critical nonlinearity where , is the fractional Laplace operator with , V vanishes at infinity and . Under appropriate assumptions on f, we prove the existence of positive solution by using the variational method.

Solving a non-linear fractional convection-diffusion equation using local discontinuous Galerkin method

We propose a local discontinuous Galerkin method for solving a nonlinear convection-diffusion equation consisting of a fractional diffusion described by a fractional Laplacian operator of order 0<p<2, a nonlinear diffusion, and a nonlinear convection term. The algorithm is developed by the local discontinuous Galerkin method using Spline interpolations to achieve higher accuracy. In this method, we convert the main problem to a first-order system and approximate the outcome by the Galerkin method. In this study, in contrast to the direct Galerkin method using Legender polynomials, we demonstrate that the proposed method can be suitable for the general fractional convection-diffusion problem, remarkably improve stability and provide convergence order O(hk+1), when k indicates the degree of polynomials. Numerical results have illustrated the accuracy of this scheme and compare it for different conditions.

The Principal Eigenvalue Problems for Perturbed Fractional Laplace Operators

We study a variety of basic properties of the principal eigenvalue of a perturbed fractional Laplace operator and weakly coupled cooperative systems involving fractional Laplace operators. Our work extends a number of well-known properties regarding the principal eigenvalues of linear second-order elliptic operators with Dirichlet boundary condition to perturbed fractional Laplace operators. The establish results are also utilized to investigate the spatio-temporary dynamics of population models.

σ-evolution models with low regular time-dependent effective structural damping

In this work we investigate decay rates for Sobolev solutions of σ-evolution equations in Rn under effects of a damping term represented by the action of a fractional Laplacian operator and a time-dependent coefficient, b(t)(−Δ)θut. We assume the coefficient b=b(t) is of low regularity with the shape of a regular function g, that is, a1b(t)≤g(t)≤a2b(t) for large times. For this purpose, we developed a new technique to handle the lack of control of the derivative of b=b(t).

Positive Solutions for Non-Variational Fractional Elliptic Systems with Negative Exponents

In this paper, we study strongly coupled elliptic systems in non-variational form with negative exponents involving fractional Laplace operators. We investigate the existence, nonexistence, and uniqueness of the positive classical solution. The results obtained here are a natural extension of the results obtained by Ghergu [J. Funct. Anal. 258 (2010), 3295–3318] for the fractional case.

An efficient spectral-Galerkin method for fractional reaction-diffusion equations in unbounded domains

In this work, we apply a fast and accurate numerical method for solving fractional reaction-diffusion equations in unbounded domains. By using the Fourier-like spectral approach in space, this method can effectively handle the fractional Laplace operator, leading to a fully diagonal representation of the fractional Laplacian. To fully discretize the underlying nonlinear reaction-diffusion systems, we propose to use an accurate time marching scheme based on ETDRK4. Numerical examples are presented to illustrate the effectiveness of the proposed method.

Exponential Turnpike property for fractional parabolic equations with non-zero exterior data

We consider averages convergence as the time-horizon goes to infinity of optimal solutions of time-dependent optimal control problems to optimal solutions of the corresponding stationary optimal control problems. Control problems play a key role in engineering, economics and sciences. To be more precise, in climate sciences, often times, relevant problems are formulated in long time scales, so that, the problem of possible asymptotic behaviors when the time-horizon goes to infinity becomes natural. Assuming that the controlled dynamics under consideration are stabilizable towards a stationary solution, the following natural question arises: Do time averages of optimal controls and trajectories converge to the stationary optimal controls and states as the time-horizon goes to infinity? This question is very closely related to the so-called turnpike property that shows that, often times, the optimal trajectory joining two points that are far apart, consists in, departing from the point of origin, rapidly getting close to the steady-state (the turnpike) to stay there most of the time, to quit it only very close to the final destination and time. In the present paper we deal with heat equations with non-zero exterior conditions (Dirichlet and nonlocal Robin) associated with the fractional Laplace operator (- Δ) s (0 &lt; s &lt; 1). We prove the turnpike property for the nonlocal Robin optimal control problem and the exponential turnpike property for both Dirichlet and nonlocal Robin optimal control problems.

Well-posedness for a fourth-order equation of Moore–Gibson–Thompson type

In this paper, we completely characterize, only in terms of the data, the well-posedness of a fourth order abstract evolution equation arising from the Moore–Gibson–Thomson equation with memory. This characterization is obtained in the scales of vector-valued Lebesgue, Besov and Triebel–Lizorkin function spaces. Our characterization is flexible enough to admite as examples the Laplacian and the fractional Laplacian operators, among others. We also provide a practical and general criteria that allows Lp–Lq-well posedness.

Positive and Negative Solutions of a Class of Fractional Schrödinger Equation

In this paper, we study the existence of positive and negative solutions for a class of fractional Schrodinger equations. Firstly, we give the definition of fractional Laplace operator and the conditions satisfied by nonlinear terms. This paper introduces the previous progress in this field, and gives the definitions of space and energy functional and the positive and negative parts of function. Then we introduce the main results of this paper. Next, we give the embedding relationship between workspace and <i>L^p</i> space and give the definition of inner product and norm of space. In order to obtain the existence of positive and negative solutions of the equation, we give the definitions of functions <i>u</i>⁺, <i>u</i>⁻ and functional weak solutions. This paper mainly uses mountain pass lemma to prove. Firstly, according to the embedding relationship of workspace and the condition of nonlinear term <i>f</i>, it is proved that functional <I>I</I> satisfies mountain road structure. Secondly, we need to prove that functional <I>I</I> satisfies the (<i>C_c</i>) condition, we first prove that the sequence <i>u_n</i> is bounded, then prove that UN has convergent subsequence by the definition of inner product and holder inequality. Therefore, we prove that functional <I>I</I> satisfies the (<i>C_c</i>) condition. Then, we define functional <I>I</I>^± and its inner product form to verify that functional <I>I</I>^± also has mountain path structure and satisfies (<i>C_c</i>) condition. Finally, taking <i>u</i>⁺ and <i>u</i>⁻ as experimental functions respectively, it is verified that they are the solutions of functional <I>I</I>. It is obtained that both <i>u</i>⁺ and <i>u</i>⁻ are the solutions of functional <I>I</I>. Therefore, we get the conclusion.

Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

In this paper, we investigate a initial value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace operator of order 0 &lt; ν ≤ 1 and the nonlinear memory source term. For 0 &lt; ν &lt; 1, the problem will be considered on a bounded domain of ℝd. By some Sobolev embeddings and the properties of the Mittag-Leffler function, we will give some results on the existence and the uniqueness of mild solution for problem (1.1) below. When ν = 1, we will introduce some Lp − Lq estimates, and based on them we derive the global existence of a mild solution in the whole space ℝd.

Nonexistence Results for Ultra-Parabolic Equations and Systems Involving Fractional Laplacian Operator in Heisenberg Group

Nonexistence Results for Ultra-Parabolic Equations and Systems Involving Fractional Laplacian Operator in Heisenberg Group

Numerical study of anomalous diffusion of light in semicrystalline polymer structures

From the spread of pollutants in the atmosphere to the transmission of nutrients across cell membranes, anomalous diffusion processes are ubiquitous in natural systems. The ability to understand and control the mechanisms guiding such processes across various scales has important application to research in materials science, finance, medicine, and energetics. Here we present a numerical study of anomalous diffusion of light through a semi-crystalline polymer structure where transport is guided by random disorder and nonlocal interactions. The numerical technique examines diffusion properties in one-dimensional (1D) space via the spectrum of an Anderson-type Hamiltonian with a discrete fractional Laplacian operator (-{\Delta})^s, 0<s<2 and a random distribution of disorder. The results show enhanced transport for s<1 (super-diffusion) and enhanced localization for s>1 (sub-diffusion) for most examined cases. An important finding of the present study is that transport can be enhanced at key spatial scales in the sub-diffusive case, where all states are normally expected to be localized for a (1D) disordered system.

Approximate and Mean Approximate Controllability Properties for Hilfer Time-Fractional Differential Equations

We study the approximate and mean approximate controllability properties of fractional partial differential equations associated with the so-called Hilfer type time-fractional derivative and a non-negative selfadjoint operator AB with compact resolvent on L2(Ω), where ${\Omega }\subset \mathbb {R}^{N}$ (N ≥ 1) is a bounded open set. More precisely, we show that if 0 ≤ ν ≤ 1 and 0 < μ ≤ 1, then the system $$ \mathbb{D}_{t}^{\mu,\nu} u+A_{B}u=f\chi_{\omega}\quad\text{ in }~{\Omega}\times (0,T),\qquad (\mathbb{I}_{t}^{(1-\nu)(1-\mu)}u)(\cdot,0)=u_{0}\quad \text{ in }~{\Omega}, $$ is approximately controllable in any time T > 0, u0 ∈ L2(Ω) and any nonempty open set ω ⊂Ω and χω is the characteristic function of ω. In addition, if the operator AB has the unique continuation property, then the system is also mean (memory) approximately controllable. The operator AB can be the realization in L2(Ω) of a symmetric, non-negative uniformly elliptic second order operator with Dirichlet or Robin boundary conditions, or the realization in L2(Ω) of the fractional Laplace operator (−Δ)s (0 < s < 1) with the Dirichlet exterior condition, u = 0 in $\mathbb {R}^{N}\setminus {\Omega }$ , or the nonlocal Robin exterior condition, $\mathcal {N}^{s}u+\beta u=0$ in $\mathbb {R}^{N}\setminus \overline {\Omega }$ .

On explicit form of the FEM stiffness matrix for the integral fractional Laplacian on non-uniform meshes

We derive exact form of the piecewise-linear finite element stiffness matrix on general non-uniform meshes for the integral fractional Laplacian operator in one dimension, where the derivation is accomplished in the Fourier transformed space. With such an exact formulation at our disposal, we are able to numerically study some intrinsic properties of the fractional stiffness matrix on some commonly used non-uniform meshes (e.g., the graded mesh), in particular, to examine their seamless transition to those of the usual Laplacian.

Maximum Principle for the Space-Time Fractional Conformable Differential System Involving the Fractional Laplace Operator

In this paper, the authors consider a IBVP for the time-space fractional PDE with the fractional conformable derivative and the fractional Laplace operator. A fractional conformable extremum principle is presented and proved. Based on the extremum principle, a maximum principle for the fractional conformable Laplace system is established. Furthermore, the maximum principle is applied to the linear space-time fractional Laplace conformable differential system to obtain a new comparison theorem. Besides that, the uniqueness and continuous dependence of the solution of the above system are also proved.

Existence and multiplicity results for p(⋅)&q(⋅) fractional Choquard problems with variable order

This paper is concerned with the existence and multiplicity of solutions for the fractional variable order Choquard type problem where and are two fractional Laplace operators with variable order and with different variable exponents and . Here is a bounded smooth domain with at least , λ is a real parameter, β, μ and k are continuous variable parameters, while F is the primitive function of a suitable f. Under some appropriate conditions on β and k, through variational methods, we prove existence and multiplicity of solutions for the above problem.

Pattern formation in superdiffusion predator–prey‐like problems with integer‐ and noninteger‐order derivatives

This paper focuses on the modeling and application of fractional derivative to model the interactions between two different species in which the one named predator depends on the other called prey solely for survival. The interaction between predator and prey has been one of the most intriguing and interesting subjects in applied mathematical biology and ecology. In the models, the classical reaction–diffusion equations subject to the Neumann boundary conditions are formulated on a finite but large domainx ∈ [0, L]by replacing the second‐order spatial derivatives with the fractional Laplacian operator of order1 &lt; α ≤ 2, which is classified as superdiffusion process. We examine the resulting coupled reaction–diffusion models for linear stability analysis and derive conditions under which the spatial patterns is evolved. In a view to understand our theoretical findings, the species spatial interactions is described in one and two dimensions. Through numerical experiments, we observe that a number of patterns can arise, including Turing spots, spiral‐like structures, and seemingly complex spatiotemporal distributions.

A variable‐order fractional p(·)‐Kirchhoff type problem in ℝN

This paper is concerned with the existence and multiplicity of solutions for the following variable s(·)‐order fractional p(·)‐Kirchhoff type problem where N &gt; p(x, y)s(x, y) for any , is a variable s(·)‐order p(·)‐fractional Laplace operator with and , for , and M is a continuous Kirchhoff‐type function, g(x, v) is a Carathéodory function, and μ &gt; 0 is a parameter. Under the weaker conditions, we obtain that there are at least two distinct solutions for the above problem by applying the generalized abstract critical point theorem. Moreover, we also show the existence of one solution and infinitely many solutions by using the mountain pass lemma and fountain theorem, respectively. In particular, the new compact embedding result of the space into will be used to overcome the lack of compactness in . The main feature and difficulty of this paper is the presence of a double non‐local term involving two variable parameters.