SIAM Journal on Scientific Computing, ISSN 1064-8275, 2018, Volume 40, Issue 4, pp. A2383 - A2405

A numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions. The scheme is based on reformulating the...

Numerical approximation | Finite element–spectral method | Fractional laplacian | finite element-spectral method | MATHEMATICS, APPLIED | fractional laplacian | numerical approximation | ELLIPTIC-OPERATORS | EXTENSION PROBLEM | Mathematics - Numerical Analysis | MATHEMATICS AND COMPUTING

Numerical approximation | Finite element–spectral method | Fractional laplacian | finite element-spectral method | MATHEMATICS, APPLIED | fractional laplacian | numerical approximation | ELLIPTIC-OPERATORS | EXTENSION PROBLEM | Mathematics - Numerical Analysis | MATHEMATICS AND COMPUTING

Journal Article

Discrete and Continuous Dynamical Systems- Series A, ISSN 1078-0947, 04/2016, Volume 36, Issue 4, pp. 1813 - 1845

Journal Article

Communications in Partial Differential Equations, ISSN 0360-5302, 08/2007, Volume 32, Issue 8, pp. 1245 - 1260

The operator square root of the Laplacian (− ▵) 1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the...

Fractional Laplacian | 35J70 | 26A33 | Degenerate elliptic equations | MATHEMATICS | MATHEMATICS, APPLIED | degenerate elliptic equations | DEGENERATE ELLIPTIC-EQUATIONS | fractional Laplacian | Mathematics - Analysis of PDEs

Fractional Laplacian | 35J70 | 26A33 | Degenerate elliptic equations | MATHEMATICS | MATHEMATICS, APPLIED | degenerate elliptic equations | DEGENERATE ELLIPTIC-EQUATIONS | fractional Laplacian | Mathematics - Analysis of PDEs

Journal Article

Advances in Calculus of Variations, ISSN 1864-8258, 10/2016, Volume 9, Issue 4, pp. 323 - 355

We consider the eigenvalue problem for the in an open bounded, possibly disconnected set , under homogeneous Dirichlet boundary conditions. After discussing...

47J10 | quasilinear nonlocal operators | 35R09 | Nonlocal eigenvalue problems | spectral optimization | 35P30 | Caccioppoli estimates | Quasilinear nonlocal operators | Spectral optimization | MATHEMATICS | MATHEMATICS, APPLIED | SPECTRUM | Eigen values | Functional Analysis | Analysis of PDEs | Mathematics

47J10 | quasilinear nonlocal operators | 35R09 | Nonlocal eigenvalue problems | spectral optimization | 35P30 | Caccioppoli estimates | Quasilinear nonlocal operators | Spectral optimization | MATHEMATICS | MATHEMATICS, APPLIED | SPECTRUM | Eigen values | Functional Analysis | Analysis of PDEs | Mathematics

Journal Article

Inventiones mathematicae, ISSN 0020-9910, 2/2008, Volume 171, Issue 2, pp. 425 - 461

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle...

Mathematics, general | Mathematics | MATHEMATICS | DEGENERATE | Studies | Mathematics - Analysis of PDEs

Mathematics, general | Mathematics | MATHEMATICS | DEGENERATE | Studies | Mathematics - Analysis of PDEs

Journal Article

Chaos, Solitons and Fractals: the interdisciplinary journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, ISSN 0960-0779, 01/2016, Volume 82, pp. 38 - 47

•Specifying positive physically admissible potentials leading by Hamilton’s variational principle to the fractional Laplacian matrix on the cyclic...

Fractional Laplacian matrix | Periodic fractional Laplacian | Discrete fractional calculus | Riesz fractional derivative | Power-law matrix functions | Lattice fractional Laplacian | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MULTIDISCIPLINARY | CALCULUS | PHYSICS, MATHEMATICAL | Mathematics | Spectral Theory | Mathematical Physics | Physics

Fractional Laplacian matrix | Periodic fractional Laplacian | Discrete fractional calculus | Riesz fractional derivative | Power-law matrix functions | Lattice fractional Laplacian | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MULTIDISCIPLINARY | CALCULUS | PHYSICS, MATHEMATICAL | Mathematics | Spectral Theory | Mathematical Physics | Physics

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 03/2020, Volume 404, p. 1

The fractional Laplacian in Rd, which we write as (−Δ)α/2 with α∈(0,2), has multiple equivalent characterizations. Moreover, in bounded domains, boundary...

Stable Lévy motion | Nonlocal model | Fractional Laplacian | Regularity | Anomalous diffusion | NUMERICAL-METHODS | BROWNIAN-MOTION | MONTE-CARLO METHODS | APPROXIMATION | Stable Levy motion | CAHN-HILLIARD | DIFFERENTIAL-EQUATIONS | POTENTIAL-THEORY | PHYSICS, MATHEMATICAL | EXTENSION PROBLEM | SPECTRAL METHOD | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | DIFFUSION | First principles | Boundary value problems | Approximation | Numerical methods | Boundary conditions | Poisson equation | Community relations | Spectra | Radial basis function | Operators (mathematics) | Domains | Numerical analysis | Collocation methods | Sampling methods | MATHEMATICS AND COMPUTING | nonlocal model | anomalous diffusion | regularity | stable Lévy motion

Stable Lévy motion | Nonlocal model | Fractional Laplacian | Regularity | Anomalous diffusion | NUMERICAL-METHODS | BROWNIAN-MOTION | MONTE-CARLO METHODS | APPROXIMATION | Stable Levy motion | CAHN-HILLIARD | DIFFERENTIAL-EQUATIONS | POTENTIAL-THEORY | PHYSICS, MATHEMATICAL | EXTENSION PROBLEM | SPECTRAL METHOD | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | DIFFUSION | First principles | Boundary value problems | Approximation | Numerical methods | Boundary conditions | Poisson equation | Community relations | Spectra | Radial basis function | Operators (mathematics) | Domains | Numerical analysis | Collocation methods | Sampling methods | MATHEMATICS AND COMPUTING | nonlocal model | anomalous diffusion | regularity | stable Lévy motion

Journal Article

Journal of Statistical Mechanics: Theory and Experiment, ISSN 1742-5468, 09/2014, Volume 2014, Issue 9, pp. P09032 - 11

This paper starts by introducing the Grunwald-Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the...

nonlinear dynamics | MECHANICS | PHYSICS, MATHEMATICAL | CALCULUS | Statistical mechanics | Two dimensional | Derivatives

nonlinear dynamics | MECHANICS | PHYSICS, MATHEMATICAL | CALCULUS | Statistical mechanics | Two dimensional | Derivatives

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 08/2019, Volume 390, pp. 306 - 322

This paper investigates the use of radial basis function (RBF) interpolants to estimate a function's fractional Laplacian of a given order through a mesh-free...

Wendland RBFs | Pseudospectral methods | Fractional Laplacian | Fractional calculus | Mesh-free methods | Fractional Poisson equation | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | APPROXIMATION | Fractional Poissonequation | EQUATIONS | DYNAMICS | PHYSICS, MATHEMATICAL | Computer science | Engineering | Algorithms | Radial basis function | Fourier transforms | Sobolev space | Meshless methods | Estimation | Dirichlet problem | Poisson equation | Spectral methods | Convergence

Wendland RBFs | Pseudospectral methods | Fractional Laplacian | Fractional calculus | Mesh-free methods | Fractional Poisson equation | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | APPROXIMATION | Fractional Poissonequation | EQUATIONS | DYNAMICS | PHYSICS, MATHEMATICAL | Computer science | Engineering | Algorithms | Radial basis function | Fourier transforms | Sobolev space | Meshless methods | Estimation | Dirichlet problem | Poisson equation | Spectral methods | Convergence

Journal Article

Proceedings of the Royal Society of Edinburgh Section A: Mathematics, ISSN 0308-2105, 2019, Volume 149, Issue 5, pp. 1 - 18

We give a unified approach to strong maximum principles for a large class of nonlocal operators of order s is an element of (0, 1) that includes the Dirichlet,...

maximum principle | Fractional Laplace operators | MATHEMATICS | MATHEMATICS, APPLIED | OPERATOR | REGULARITY | ELLIPTIC-EQUATIONS | Dirichlet problem | Principles

maximum principle | Fractional Laplace operators | MATHEMATICS | MATHEMATICS, APPLIED | OPERATOR | REGULARITY | ELLIPTIC-EQUATIONS | Dirichlet problem | Principles

Journal Article

Mathematics and Computers in Simulation, ISSN 0378-4754, 02/2020, Volume 168, pp. 122 - 134

In this paper, a Crank–Nicolson Fourier spectral Galerkin method is proposed for solving the cubic fractional Schrödinger equation. Firstly, we discuss the...

Conservative laws | Fractional Schrödinger equation | Numerical examples | Fourier spectral Galerkin method | Convergence analysis | COMPUTER SCIENCE, SOFTWARE ENGINEERING | SCHEME | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | ELEMENT-METHOD | Fractional Schrodinger equation | FINITE-DIFFERENCE METHOD | Water waves | Energy conservation | Analysis | Environmental law | Differential equations | Quantum theory | Methods

Conservative laws | Fractional Schrödinger equation | Numerical examples | Fourier spectral Galerkin method | Convergence analysis | COMPUTER SCIENCE, SOFTWARE ENGINEERING | SCHEME | MATHEMATICS, APPLIED | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | ELEMENT-METHOD | Fractional Schrodinger equation | FINITE-DIFFERENCE METHOD | Water waves | Energy conservation | Analysis | Environmental law | Differential equations | Quantum theory | Methods

Journal Article

Archive for Rational Mechanics and Analysis, ISSN 0003-9527, 2013, Volume 208, Issue 1, pp. 109 - 161

In this work we establish trace Hardy and trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants for domains satisfying suitable geometric...

MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | CONSTANT | GAMMA-CONVERGENCE | HALF-SPACE | CENSORED STABLE PROCESSES | DOMAINS | Half spaces | Exponents | Mathematical analysis | Inequalities | Series (mathematics) | Constants | Convexity | Archives

MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | CONSTANT | GAMMA-CONVERGENCE | HALF-SPACE | CENSORED STABLE PROCESSES | DOMAINS | Half spaces | Exponents | Mathematical analysis | Inequalities | Series (mathematics) | Constants | Convexity | Archives

Journal Article

Annales de l'Institut Henri Poincaré / Analyse non linéaire, ISSN 0294-1449, 03/2017, Volume 34, Issue 2, pp. 439 - 467

We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their...

Large solutions | Dirichlet problem | Spectral fractional Laplacian | Boundary blow-up solutions | MATHEMATICS, APPLIED | BEHAVIOR | REGULARITY | DIRICHLET LAPLACIANS | ELLIPTIC-EQUATIONS | HEAT KERNEL | KILLED BROWNIAN-MOTION | GREEN-FUNCTION | OPERATORS | DOMAINS | Analysis of PDEs | Mathematics

Large solutions | Dirichlet problem | Spectral fractional Laplacian | Boundary blow-up solutions | MATHEMATICS, APPLIED | BEHAVIOR | REGULARITY | DIRICHLET LAPLACIANS | ELLIPTIC-EQUATIONS | HEAT KERNEL | KILLED BROWNIAN-MOTION | GREEN-FUNCTION | OPERATORS | DOMAINS | Analysis of PDEs | Mathematics

Journal Article

Acta Physica Polonica B, ISSN 0587-4254, 05/2018, Volume 49, Issue 5, pp. 921 - 942

We address Levy-stable stochastic processes in bounded domains, with a focus on a discrimination between inequivalent proposals for what a boundary...

BROWNIAN-MOTION | EIGENVALUES | STATES | PHYSICS, MULTIDISCIPLINARY | TIME | EIGENFUNCTIONS | SPECTRAL PROPERTIES

BROWNIAN-MOTION | EIGENVALUES | STATES | PHYSICS, MULTIDISCIPLINARY | TIME | EIGENFUNCTIONS | SPECTRAL PROPERTIES

Journal Article

15.
Full Text
Computing fractional laplacians on complex-geometry domains: Algorithms and simulations

SIAM Journal on Scientific Computing, ISSN 1064-8275, 2017, Volume 39, Issue 4, pp. A1320 - A1344

We consider a fractional Laplacian defined in bounded domains by the eigen-decomposition of the integer-order Laplacian, and demonstrate how to compute very...

Spectral element method | Eigenvalue problem | Fractional diffusion | Fractional Laplacian | Fractional phase-field equations | MATHEMATICS, APPLIED | SPECTRAL METHODS | POLYGONAL DOMAINS | spectral element method | fractional phase-field equations | NUMERICAL APPROXIMATION | STOKES EQUATIONS | fractional diffusion | DIFFUSION-EQUATIONS | fractional Laplacian | ELEMENT METHODS | eigenvalue problem | SCHEMES

Spectral element method | Eigenvalue problem | Fractional diffusion | Fractional Laplacian | Fractional phase-field equations | MATHEMATICS, APPLIED | SPECTRAL METHODS | POLYGONAL DOMAINS | spectral element method | fractional phase-field equations | NUMERICAL APPROXIMATION | STOKES EQUATIONS | fractional diffusion | DIFFUSION-EQUATIONS | fractional Laplacian | ELEMENT METHODS | eigenvalue problem | SCHEMES

Journal Article

Discrete and Continuous Dynamical Systems- Series A, ISSN 1078-0947, 10/2015, Volume 35, Issue 10, pp. 4905 - 4929

We show that the fractional Laplacian can be viewed as a Dirichlet-to-Neumann map for a degenerate hyperbolic problem, namely, the wave equation with an...

Wave equation | Oscillatory integrals | Schrödinger group | Fractional Laplacian | Bessel functions | MATHEMATICS | MATHEMATICS, APPLIED | wave equation | oscillatory integrals | Schrodinger group | OPERATORS

Wave equation | Oscillatory integrals | Schrödinger group | Fractional Laplacian | Bessel functions | MATHEMATICS | MATHEMATICS, APPLIED | wave equation | oscillatory integrals | Schrodinger group | OPERATORS

Journal Article

Computational Mathematics and Mathematical Physics, ISSN 0965-5425, 03/2017, Volume 57, Issue 3, pp. 373 - 386

In this paper we study obstacle problems for the Navier (spectral) fractional Laplacian (-Delta(Omega)) (s) of order s is an element of (0,1) in a bounded...

free boundary problems | spectral fractional Laplacian | Variational inequalities | OBSTACLE PROBLEM | MATHEMATICS, APPLIED | REGULARITY | EQUATIONS | PHYSICS, MATHEMATICAL | OPERATORS | EXTENSION PROBLEM | Studies | Boundary value problems | Mathematical models | Mathematics | Obstacles | Spectra | Computation | Inequalities | Fluid flow | Mathematics - Analysis of PDEs

free boundary problems | spectral fractional Laplacian | Variational inequalities | OBSTACLE PROBLEM | MATHEMATICS, APPLIED | REGULARITY | EQUATIONS | PHYSICS, MATHEMATICAL | OPERATORS | EXTENSION PROBLEM | Studies | Boundary value problems | Mathematical models | Mathematics | Obstacles | Spectra | Computation | Inequalities | Fluid flow | Mathematics - Analysis of PDEs

Journal Article

Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 9/2011, Volume 42, Issue 1, pp. 21 - 41

We establish existence and non-existence results to the Brezis–Nirenberg type problem involving the square root of the Laplacian in a bounded domain with zero...

35J65 | Calculus of Variations and Optimal Control; Optimization | Systems Theory, Control | 35J60 | Theoretical, Mathematical and Computational Physics | Analysis | 35B99 | 35B33 | 35R11 | Mathematics | 58E30 | FRACTIONAL LAPLACIAN | OBSTACLE PROBLEM | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | REGULARITY | POSITIVE SOLUTIONS | EQUATIONS | BOUNDARY | Boundary conditions | Dirichlet problem | Partial differential equations | Mathematical analysis | Calculus of variations | Roots

35J65 | Calculus of Variations and Optimal Control; Optimization | Systems Theory, Control | 35J60 | Theoretical, Mathematical and Computational Physics | Analysis | 35B99 | 35B33 | 35R11 | Mathematics | 58E30 | FRACTIONAL LAPLACIAN | OBSTACLE PROBLEM | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | REGULARITY | POSITIVE SOLUTIONS | EQUATIONS | BOUNDARY | Boundary conditions | Dirichlet problem | Partial differential equations | Mathematical analysis | Calculus of variations | Roots

Journal Article

Discussiones Mathematicae Graph Theory, ISSN 1234-3099, 05/2020, Volume 40, Issue 2, pp. 585 - 600

We consider Laplacian fractional revival between two vertices of a graph . Assume that it occurs at time between vertices 1 and 2. We prove that for the...

spectral decomposition | 05C50 | 81P45 | quantum information transfer | Laplacian matrix | fractional revival | 15A18 | MATHEMATICS | PERFECT STATE TRANSFER

spectral decomposition | 05C50 | 81P45 | quantum information transfer | Laplacian matrix | fractional revival | 15A18 | MATHEMATICS | PERFECT STATE TRANSFER

Journal Article

Advances in Difference Equations, ISSN 1687-1839, 12/2013, Volume 2013, Issue 1, pp. 1 - 12

In this paper, we study the p-Laplacian model involving the Caputo fractional derivative with Dirichlet-Neumann boundary conditions. Using a fixed point...

Ordinary Differential Equations | Functional Analysis | Analysis | Difference and Functional Equations | Mathematics, general | Mathematics | Partial Differential Equations | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | POSITIVE SOLUTIONS | BOUNDARY-VALUE-PROBLEMS | DIFFERENTIAL-EQUATIONS | POINTS

Ordinary Differential Equations | Functional Analysis | Analysis | Difference and Functional Equations | Mathematics, general | Mathematics | Partial Differential Equations | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | POSITIVE SOLUTIONS | BOUNDARY-VALUE-PROBLEMS | DIFFERENTIAL-EQUATIONS | POINTS

Journal Article

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