Fractional Laplacian Equation Research Articles

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.

Two disjoint and infinite sets of solutions for a concave-convex critical fractional Laplacian equation

In this paper, we consider the following problem involving fractional Laplacian operator $$\begin{aligned} \left\{ \begin{array}{ll} (-\varDelta )^{\alpha } u=|u|^{2_{\alpha }^{*}-2}u+\lambda |u|^{q-2}u, &{} \text{ in } \varOmega , \\ u=0, &{} \text{ on } \partial \varOmega , \end{array}\right. \end{aligned}$$ where $$\varOmega $$ is a smooth bounded domain in $${\mathbb {R}}^{N}$$ , $$\lambda >0$$ , $$0<\alpha <1$$ , $$1< q < 2$$ , $$2_{\alpha }^{*}=\frac{2 N}{N-2 \alpha }$$ and $$(-\varDelta )^{\alpha }$$ is the spectral fractional Laplacian. We prove that if $$N>2 \alpha \times \frac{q+1}{q-1}$$ , then the above problem has two disjoint and infinite sets of solutions. The present work may be seen as the extension of the result got by Thomas Bartsch and Michel Willem in [6] for subcritical Laplacian equations, to the case of critical fractional equations.

Mathematical Analysis on Conducting Sphere Embedded in Non Integer Dimensional Space

We have derived an analytical solution in low frequency using the idea of a fractional Laplacian equation. Fractional dimensional (FD) space has importance in describing the complex physics phenomena. Here, the Laplacian equation in spherical coordinated (r,θ,0) is expressed in fractional dimensional space using Gegenbauer polynomials. The analytical solution is obtained by the separation variable method. The general solution is a product of angular and radial solutions and is independent of ϕ due to azimuthal symmetry. The classical solution is retained by setting fractional parameter α=3. Further, numerical results are discussed for different values of α and compared with available literature.

On a new class of fractional calculus of variations and related fractional differential equations

This paper analyzes a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of calculus of variations and fractional differential equations as well as nonstandard Neumann boundary operators. This paper establishes the well-posedness and regularities for a class of fractional calculus of variations problems and their Euler-Lagrange (fractional differential) equations. This is achieved first for one-sided Dirichlet energy functionals which lead to one-sided fractional Laplace equations, then for more general energy functionals which give rise to more general fractional differential equations.

An inverse problem study related to a fractional diffusion equation

This article concerns the problem of reconstruction of the source term in a fractional diffusion equation from internal noisy measured data. The considered model corresponds to the two dimensional fractional spectral Laplacian differential equation. The inverse problem is reformulated as a minimization one and solved by applying a regularization technique to avoid numerical instabilities caused by the noisy perturbations of the measured data. We propose an efficient and accurate identification procedure inspired from the Landweber-type denoising algorithms. The efficiency of the proposed identification process is justified by some numerical investigations.

A stochastic fractional Laplace equation driven by colored noise on bounded domain, and its covariance functional

The paper provides conditions for the fractional Laplacian and its spectral representation on stationary Gaussian random fields to be well-defined. In addition, we study existence and uniqueness of the weak solution for a stochastic fractional elliptic equation driven by an additive colored noise over an open bounded set. Both spectral and variational approaches are used to provide a solution. Further, the functional covariance associated with the solution is derived.

A priori estimates of solutions to nonlinear fractional Laplacian equation

&lt;abstract&gt;&lt;p&gt;In this paper, we focus on the research of a priori estimates of several types of semi-linear fractional Laplacian equations with a critical Sobolev exponent. Employing the method of moving planes, we can achieve a priori estimates which are closely connected to the existence of solutions to nonlinear fractional Laplacian equations. Our result can extend a priori estimates of the second order elliptic equation to the fractional Laplacian equation and we believe that the method used here will be applicable to more general nonlocal problems.&lt;/p&gt;&lt;/abstract&gt;

Removability of singularities and superharmonicity for some fractional Laplacian equations

We study some qualitative properties (including removable singularities and superharmonicity) of non-negative solutions to $$ (-\Delta)^\gamma u=fu^p\quad\text{in }\mathbb R^n\setminus\Sigma $$ which are singular at $\Sigma$. Here $\gamma \in (0, \frac{n}{2})$. Among other things, we first prove that if $\Sigma$ is a compact set in $\mathbb R^n$ with Assouad dimension $\bf d$ (not necessarily an integer), ${\bf d}<n-2\gamma$, and $ u\in L_\gamma(\mathbb R^n)\cap L^p_{loc}({\mathbb R^n\setminus\Sigma})$ is a non-negative solution for some $$p>\frac{n-\bf d}{n-{\bf d}-2\gamma},$$ then $u\in L^p_{loc}(\mathbb R^n)$ and $u$ is a distributional solution in $\mathbb R^n$. Then we prove that $ (-\Delta)^\sigma u >0$ for all $ \sigma \in (0, \gamma)$, if $\Sigma=\phi$.

Multiple Solutions for a Class of Variable-Order Fractional Laplacian Equations with Concave-Convex Nonlinearity

This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in RN, s(⋅) ∈ C (RN × RN, (0,1)), (-Δ)s(⋅) is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.

Inverse problems for the fractional Laplace equation with lower order nonlinear perturbations

&lt;p style='text-indent:20px;'&gt;We study the inverse problem for the fractional Laplace equation with multiple nonlinear lower order terms. We show that the direct problem is well-posed and the inverse problem is uniquely solvable. More specifically, the unknown nonlinearities can be uniquely determined from exterior measurements under suitable settings.&lt;/p&gt;

Classification of anti-symmetric solutions to nonlinear fractional Laplace equations

Classification of anti-symmetric solutions to nonlinear fractional Laplace equations

A Strong Maximum Principle for the Fractional Laplace Equation with Mixed Boundary Condition

Abstract In this work we prove a strong maximum principle for fractional elliptic problems with mixed Dirichlet–Neumann boundary data which extends the one proved by J. Dávila (cf. [11]) to the fractional setting. In particular, we present a comparison result for two solutions of the fractional Laplace equation involving the spectral fractional Laplacian endowed with homogeneous mixed boundary condition. This result represents a non–local counterpart to a Hopf’s Lemma for fractional elliptic problems with mixed boundary data.

Multiplicity of Solutions for a Fractional Laplacian Equation Involving a Perturbation

Multiplicity of Solutions for a Fractional Laplacian Equation Involving a Perturbation

Modeling viscoacoustic wave propagation using a new spatial variable-order fractional Laplacian wave equation

We have developed a new time-domain viscoacoustic wave equation for simulating wave propagation in anelastic media. The new wave equation is derived by inserting the complex-valued phase velocity described by the Kjartansson attenuation model into the frequency-wavenumber domain acoustic wave equation. Our wave equation includes one second-order temporal derivative and two spatial variable-order fractional Laplacian operators. The two fractional Laplacian operators describe the phase dispersion and amplitude attenuation effects, respectively. To facilitate the numerical solution for our wave equation, we use the arbitrary-order Taylor series expansion (TSE) to approximate the mixed-domain fractional Laplacians and achieve the decoupling of the wavenumber and the fractional order. Then, our viscoacoustic wave equation can be directly solved using the pseudospectral method. We adopt a hybrid pseudospectral/finite-difference method (HPSFDM) to stably simulate wave propagation in arbitrarily complex media. We validate the high accuracy of our approximate dispersion term and approximate the dissipation term in comparison with the accurate dispersion term and accurate dissipation term. The accuracy of the numerical solutions is evaluated by comparison with the analytical solutions in homogeneous media. Theory analysis and simulation results indicate that our viscoacoustic wave equation has higher precision than the traditional fractional viscoacoustic wave equation in describing constant- Q attenuation. For a model with Q &lt; 10, the calculation cost for solving the new wave equation with TSE HPSFDM is lower than that for solving the traditional fractional-order wave equation with TSE HPSFDM under the high numerical simulation precision. Furthermore, we examine the accuracy of HPSFDM in heterogeneous media using several numerical examples.

Multiplicity of nontrivial solutions for a class of fractional elliptic equations

We are concerned with the following fractional Laplacian equation (−Δ)su=a(x)|u(x)|q−2u(x)+λb(x)|u(x)|p−2u(x) in (−1,1),u>0 in (−1,1),u=0 in ℝ∖(−1,1), where s∈(0,1), λ>0. Using variational methods, we show existence and multiplicity of positive solutions.

On the Decomposition of Solutions: From Fractional Diffusion to Fractional Laplacian

Abstract This paper investigates the structure of solutions to the BVP of a class of fractional ordinary differential equations involving both fractional derivatives (R-L or Caputo) and fractional Laplacian with variable coefficients. This family of equations generalize the usual fractional diffusion equation and fractional Laplace equation. We provide a deep insight to the structure of the solutions shared by this family of equations. The specific decomposition of the solution is obtained, which consists of the “good” part and the “bad” part that precisely control the regularity and singularity, respectively. Other associated properties of the solution will be characterized as well.

Compact Sobolev-Slobodeckij embeddings and positive solutions to fractional Laplacian equations

AbstractIn this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (−Δ)sin ℝn, forn≥ 2, such as(0.1)(−Δ)su+E(x)u+V(x)uq−1=K(x)f(u)+u2s⋆−1.$$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=K(x) f(u)+u^{2_{s}^{\star}-1}.$$Here,s ∈(0, 1),q∈2,2s⋆$q \in\left[2,2_{s}^{\star}\right)$with2s⋆:=2nn−2s$2_{s}^{\star}:=\frac{2 n}{n-2 s}$being the fractional critical Sobolev exponent,E(x),K(x),V(x) &gt; 0 : ℝn→ ℝ are measurable functions which satisfy joint “vanishing at infinity” conditions in a measure-theoretic sense, andf(u) is a continuous function on ℝ of quasi-critical, super-q-linear growth withf(u) ≥ 0 ifu≥ 0. Besides, we study the existence of multiple positive solutions to an elliptic equation in ℝnsuch as(0.2)(−Δ)su+E(x)u+V(x)uq−1=λK(x)ur−1,$$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=\lambda K(x) u^{r-1},$$where 2 &lt;r&lt;q&lt; ∞(both possibly (super-)critical),E(x),K(x),V(x) &gt; 0 : ℝn→ ℝ are measurable functions satisfying joint integrability conditions, andλ&gt; 0 is a parameter. To study (0.1)-(0.2), we first describe a family of general fractional Sobolev-Slobodeckij spacesMs;q,p(ℝn) as well as their associated compact embedding results.