SIAM Review, ISSN 0036-1445, 12/2012, Volume 54, Issue 4, pp. 667 - 696

A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a...

Tensors | Approximation | Flux density | Vector calculus | Wave equations | SURVEY and REVIEW | Boundary conditions | Laplacians | Mathematical functions | Sobolev spaces | Modeling | Fractional operator | Nonlocal operator | Superdiffusion | Nonlocal heat conduction | Anomalous diffusion | Monlocal diffusion | Peridynamics | Finite element methods | Fractional Laplacian | Fractional Sobolev spaces | fractional Sobolev spaces | MATHEMATICS, APPLIED | nonlocal diffusion | BOUNDARY-VALUE-PROBLEMS | EQUATIONS | LONG-RANGE FORCES | nonlocal heat conduction | SYMMETRIC JUMP-PROCESSES | SOBOLEV SPACES | TRANSPORT | fractional operator | nonlocal operator | finite element methods | fractional Laplacian | vector calculus | superdiffusion | DYNAMICS | FRACTIONAL ADVECTION-DISPERSION | anomalous diffusion | OPERATORS | peridynamics | Finite element method | Usage | Diffusion processes | Analysis | Calculus | Research | Methods | Studies | Mathematical models | Laplace transforms | Diffusion | Heat conductivity

Tensors | Approximation | Flux density | Vector calculus | Wave equations | SURVEY and REVIEW | Boundary conditions | Laplacians | Mathematical functions | Sobolev spaces | Modeling | Fractional operator | Nonlocal operator | Superdiffusion | Nonlocal heat conduction | Anomalous diffusion | Monlocal diffusion | Peridynamics | Finite element methods | Fractional Laplacian | Fractional Sobolev spaces | fractional Sobolev spaces | MATHEMATICS, APPLIED | nonlocal diffusion | BOUNDARY-VALUE-PROBLEMS | EQUATIONS | LONG-RANGE FORCES | nonlocal heat conduction | SYMMETRIC JUMP-PROCESSES | SOBOLEV SPACES | TRANSPORT | fractional operator | nonlocal operator | finite element methods | fractional Laplacian | vector calculus | superdiffusion | DYNAMICS | FRACTIONAL ADVECTION-DISPERSION | anomalous diffusion | OPERATORS | peridynamics | Finite element method | Usage | Diffusion processes | Analysis | Calculus | Research | Methods | Studies | Mathematical models | Laplace transforms | Diffusion | Heat conductivity

Journal Article

Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 06/2015, Volume 289, pp. 60 - 78

Nonlocal models are becoming increasingly popular in numerical simulations of important application problems, ranging from anomalous diffusion in heterogeneous...

Nonlocal convection–diffusion problems | Maximum principle | Upwind model | Stability analysis | Central model | Finite element approximations | Nonlocal convection-diffusion problems | STATES | DISPERSION | DISCONTINUITIES | EQUATIONS | LONG-RANGE FORCES | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | BAR | ENGINEERING, MULTIDISCIPLINARY | PERIDYNAMIC MODELS | ADVECTION | DOMAINS | VOLUME-CONSTRAINED PROBLEMS | Analysis | Numerical analysis | Finite element method | Fracture mechanics | Approximation | Computer simulation | Mathematical analysis | Oscillations | Constants | Mathematical models

Nonlocal convection–diffusion problems | Maximum principle | Upwind model | Stability analysis | Central model | Finite element approximations | Nonlocal convection-diffusion problems | STATES | DISPERSION | DISCONTINUITIES | EQUATIONS | LONG-RANGE FORCES | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | BAR | ENGINEERING, MULTIDISCIPLINARY | PERIDYNAMIC MODELS | ADVECTION | DOMAINS | VOLUME-CONSTRAINED PROBLEMS | Analysis | Numerical analysis | Finite element method | Fracture mechanics | Approximation | Computer simulation | Mathematical analysis | Oscillations | Constants | Mathematical models

Journal Article

International Journal for Numerical Methods in Fluids, ISSN 0271-2091, 01/2017, Volume 83, Issue 3, pp. 307 - 327

Summary In several settings, diffusive behavior is observed to not follow the rate of spread predicted by parabolic partial differential equations (PDEs) such...

anomalous diffusion | nonlocal models | finite element methods | reduced‐order modeling | reduced-order modeling | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | APPROXIMATION | PHYSICS, FLUIDS & PLASMAS | EQUATION | PROPER ORTHOGONAL DECOMPOSITION | Reduced order | Approximation | Partial differential equations | Mathematical analysis | Mathematical models | Derivatives | Reduced order models | Diffusion

anomalous diffusion | nonlocal models | finite element methods | reduced‐order modeling | reduced-order modeling | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | APPROXIMATION | PHYSICS, FLUIDS & PLASMAS | EQUATION | PROPER ORTHOGONAL DECOMPOSITION | Reduced order | Approximation | Partial differential equations | Mathematical analysis | Mathematical models | Derivatives | Reduced order models | Diffusion

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 03/2018, Volume 459, Issue 2, pp. 997 - 1015

In this paper, we focus on studying the existence of attractors in the phase spaces L2(Ω) and Lp(Ω) (among others) for time-dependent p-Laplacian equations...

Multi-valued dynamical systems | Nonlocal p-Laplacian equations | Asymptotic compactness | Pullback attractors | EXISTENCE | MATHEMATICS | NONAUTONOMOUS 2D-NAVIER-STOKES EQUATIONS | MATHEMATICS, APPLIED | GLOBAL ATTRACTORS

Multi-valued dynamical systems | Nonlocal p-Laplacian equations | Asymptotic compactness | Pullback attractors | EXISTENCE | MATHEMATICS | NONAUTONOMOUS 2D-NAVIER-STOKES EQUATIONS | MATHEMATICS, APPLIED | GLOBAL ATTRACTORS

Journal Article

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Blow-up analysis in quasilinear reaction–diffusion problems with weighted nonlocal source

Computers and Mathematics with Applications, ISSN 0898-1221, 02/2018, Volume 75, Issue 4, pp. 1288 - 1301

In this paper, we consider the blow-up of solutions to a class of quasilinear reaction–diffusion problems...

Nonlocal source | Reaction–diffusion problems | Blow-up | SYSTEM | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | BOUNDS | Reaction-diffusion problems | TIME

Nonlocal source | Reaction–diffusion problems | Blow-up | SYSTEM | MATHEMATICS, APPLIED | PARABOLIC EQUATIONS | BOUNDS | Reaction-diffusion problems | TIME

Journal Article

Discrete and Continuous Dynamical Systems- Series A, ISSN 1078-0947, 03/2018, Volume 38, Issue 3, pp. 1405 - 1425

Discrete and Continuous Dynamical Systems - A 38(3):1405-1425, 2018 We prove an energy inequality for nonlocal diffusion operators of the following type, and...

Energy methods | Entropy methods | Asymptotic behaviour | Nonlocal diffusion | Mathematics - Analysis of PDEs

Energy methods | Entropy methods | Asymptotic behaviour | Nonlocal diffusion | Mathematics - Analysis of PDEs

Journal Article

Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 04/2017, Volume 317, Issue C, pp. 746 - 770

The construction, analysis, and application of reduced-basis methods for uncertainty quantification problems involving nonlocal diffusion problems with random...

Uncertainty quantification | Reduced-basis methods | Finite element methods | Nonlocal diffusion | VECTOR CALCULUS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | STOCHASTIC COLLOCATION METHOD | ENGINEERING, MULTIDISCIPLINARY | PARTIAL-DIFFERENTIAL-EQUATIONS | Analysis | Methods | Differential equations

Uncertainty quantification | Reduced-basis methods | Finite element methods | Nonlocal diffusion | VECTOR CALCULUS | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | STOCHASTIC COLLOCATION METHOD | ENGINEERING, MULTIDISCIPLINARY | PARTIAL-DIFFERENTIAL-EQUATIONS | Analysis | Methods | Differential equations

Journal Article

Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 02/2016, Volume 299, pp. 401 - 420

In this paper, we study nodal-type quadrature rules for approximating hypersingular integrals and their applications to numerical solution of finite-part...

Nodal-type quadrature rules | Collocation method | Hypersingular integrals | Nonlocal diffusion | APPROXIMATIONS | CAVITY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | PERIDYNAMIC MODELS | QUADRATURE | RULES | FINITE-PART INTEGRALS

Nodal-type quadrature rules | Collocation method | Hypersingular integrals | Nonlocal diffusion | APPROXIMATIONS | CAVITY | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | PERIDYNAMIC MODELS | QUADRATURE | RULES | FINITE-PART INTEGRALS

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 03/2018, Volume 75, Issue 5, pp. 1685 - 1704

Spectral/pseudo-spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations. In this...

Nonlocal diffusion equations | Discontinuity | Radial basis functions | Spectral/pseudo-spectral methods | Cardinal function | MATHEMATICS, APPLIED | ELEMENT-METHOD | APPROXIMATION | PHASE-TRANSITIONS | STABILITY | BOUNDARY-VALUE-PROBLEMS | CONVOLUTION MODEL | TRAVELING-WAVES | PERIDYNAMIC MODELS | EQUATION | DOMAINS

Nonlocal diffusion equations | Discontinuity | Radial basis functions | Spectral/pseudo-spectral methods | Cardinal function | MATHEMATICS, APPLIED | ELEMENT-METHOD | APPROXIMATION | PHASE-TRANSITIONS | STABILITY | BOUNDARY-VALUE-PROBLEMS | CONVOLUTION MODEL | TRAVELING-WAVES | PERIDYNAMIC MODELS | EQUATION | DOMAINS

Journal Article

Journal of Scientific Computing, ISSN 0885-7474, 3/2019, Volume 78, Issue 3, pp. 1438 - 1466

In this paper we first present stability and error analysis of the fully discrete numerical schemes for general dissipative systems, in which the implicit...

65L06 | Computational Mathematics and Numerical Analysis | 65R20 | Theoretical, Mathematical and Computational Physics | 65L07 | Strong stability | Mathematics | Asymptotic compatibility | Fully discrete | Nonlocal diffusion | 45A05 | Algorithms | Implicit Runge–Kutta | Mathematical and Computational Engineering | 65M60 | MATHEMATICS, APPLIED | STABILITY | Implicit Runge-Kutta

65L06 | Computational Mathematics and Numerical Analysis | 65R20 | Theoretical, Mathematical and Computational Physics | 65L07 | Strong stability | Mathematics | Asymptotic compatibility | Fully discrete | Nonlocal diffusion | 45A05 | Algorithms | Implicit Runge–Kutta | Mathematical and Computational Engineering | 65M60 | MATHEMATICS, APPLIED | STABILITY | Implicit Runge-Kutta

Journal Article

Applied Mathematics & Optimization, ISSN 0095-4616, 4/2016, Volume 73, Issue 2, pp. 227 - 249

The problem of identifying the diffusion parameter appearing in a nonlocal steady diffusion equation is considered. The identification problem is formulated as...

Fractional operator | Parameter estimation | Systems Theory, Control | Theoretical, Mathematical and Computational Physics | Mathematics | Finite element methods | Nonlocal diffusion | Optimization | Nonlocal operator | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Vector calculus | Numerical and Computational Physics | Control theory | MATHEMATICS, APPLIED | APPROXIMATION | EQUATIONS | Studies | Diffusion | Mathematical analysis | Approximation | Optimal control | Mathematical models | Estimates | Galerkin methods | DIFFUSION EQUATIONS | ERRORS | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | APPROXIMATIONS | DIFFUSION | ONE-DIMENSIONAL CALCULATIONS | FINITE ELEMENT METHOD | VARIATIONAL METHODS | KERNELS

Fractional operator | Parameter estimation | Systems Theory, Control | Theoretical, Mathematical and Computational Physics | Mathematics | Finite element methods | Nonlocal diffusion | Optimization | Nonlocal operator | Mathematical Methods in Physics | Calculus of Variations and Optimal Control; Optimization | Vector calculus | Numerical and Computational Physics | Control theory | MATHEMATICS, APPLIED | APPROXIMATION | EQUATIONS | Studies | Diffusion | Mathematical analysis | Approximation | Optimal control | Mathematical models | Estimates | Galerkin methods | DIFFUSION EQUATIONS | ERRORS | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | APPROXIMATIONS | DIFFUSION | ONE-DIMENSIONAL CALCULATIONS | FINITE ELEMENT METHOD | VARIATIONAL METHODS | KERNELS

Journal Article

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Numerical methods for a class of nonlocal diffusion problems with the use of backward SDEs

Computers and Mathematics with Applications, ISSN 0898-1221, 06/2016, Volume 71, Issue 11, pp. 2479 - 2496

We propose a novel numerical approach for nonlocal diffusion equations Du et al. (2012) with integrable kernels, based on the relationship between the backward...

Compound Poisson process | Nonlocal diffusion equations | Superdiffusion | [formula omitted]-scheme | Backward stochastic differential equation with jumps | Adaptive approximation | Compound Poisson process-scheme | SCHEME | MATHEMATICS, APPLIED | APPROXIMATION | Compound Poisson process theta-scheme | VOLUME-CONSTRAINED PROBLEMS | Computer science | Analysis | Methods | Differential equations | Linear systems | Approximation | Mathematical analysis | Mathematical models | Stochasticity | Diffusion | Convergence

Compound Poisson process | Nonlocal diffusion equations | Superdiffusion | [formula omitted]-scheme | Backward stochastic differential equation with jumps | Adaptive approximation | Compound Poisson process-scheme | SCHEME | MATHEMATICS, APPLIED | APPROXIMATION | Compound Poisson process theta-scheme | VOLUME-CONSTRAINED PROBLEMS | Computer science | Analysis | Methods | Differential equations | Linear systems | Approximation | Mathematical analysis | Mathematical models | Stochasticity | Diffusion | Convergence

Journal Article

Central European Journal of Physics, ISSN 1895-1082, 10/2013, Volume 11, Issue 10, pp. 1255 - 1261

In this paper, the nonlocal diffusion in one-dimensional continua is investigated by means of a fractional calculus approach. The problem is set on finite...

Environmental Physics | fractional calculus | Physical Chemistry | heat conduction | Geophysics/Geodesy | Biophysics and Biological Physics | Physics, general | nonlocal media | Physics | ELASTICITY | PHYSICS, MULTIDISCIPLINARY | CALCULUS | EQUATIONS | IMPEDANCE SPECTROSCOPY | OPERATORS | WAVE-PROPAGATION

Environmental Physics | fractional calculus | Physical Chemistry | heat conduction | Geophysics/Geodesy | Biophysics and Biological Physics | Physics, general | nonlocal media | Physics | ELASTICITY | PHYSICS, MULTIDISCIPLINARY | CALCULUS | EQUATIONS | IMPEDANCE SPECTROSCOPY | OPERATORS | WAVE-PROPAGATION

Journal Article

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Nonlocal Convection-Diffusion Problems on Bounded Domains and Finite-Range Jump Processes

Computational Methods in Applied Mathematics, ISSN 1609-4840, 10/2017, Volume 17, Issue 4, pp. 707 - 722

A nonlocal convection-diffusion model is introduced for the master equation of Markov jump processes in bounded domains. With minimal assumptions on the model...

60J75 | Nonlocal Diffusion | Nonlocal Vector Calculus | 35A15 | 60J60 | 60G51 | 34B10 | Markov Processes | Lévy Processes | 35L65 | Master Equation | 45A05 | Nonlocal Operators | 45K05 | 35B40 | Variational Forms | MATHEMATICS, APPLIED | VECTOR CALCULUS | APPROXIMATION | DISPERSION | EQUATIONS | MODEL | ADVECTION | Levy Processes | VOLUME-CONSTRAINED PROBLEMS | Unsteady state | Markov chains | Well posed problems | Convection-diffusion equation

60J75 | Nonlocal Diffusion | Nonlocal Vector Calculus | 35A15 | 60J60 | 60G51 | 34B10 | Markov Processes | Lévy Processes | 35L65 | Master Equation | 45A05 | Nonlocal Operators | 45K05 | 35B40 | Variational Forms | MATHEMATICS, APPLIED | VECTOR CALCULUS | APPROXIMATION | DISPERSION | EQUATIONS | MODEL | ADVECTION | Levy Processes | VOLUME-CONSTRAINED PROBLEMS | Unsteady state | Markov chains | Well posed problems | Convection-diffusion equation

Journal Article

Journal of Differential Equations, ISSN 0022-0396, 06/2012, Volume 252, Issue 12, pp. 6429 - 6447

We find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T(u)=−∫RdK(x,y)(u(y)−u(x))dy. Here we consider a kernel...

Eigenvalues | Nonlocal diffusion | MATHEMATICS | EVOLUTION | MODEL | EQUATION | PERIDYNAMICS

Eigenvalues | Nonlocal diffusion | MATHEMATICS | EVOLUTION | MODEL | EQUATION | PERIDYNAMICS

Journal Article

Abstract and Applied Analysis, ISSN 1085-3375, 5/2014, Volume 2014, pp. 1 - 11

A new two-time level implicit technique based on cubic trigonometric B-spline is proposed for the approximate solution of a nonclassical diffusion problem with...

MATHEMATICS | THERMOELASTICITY | SPECIFICATIONS | COLLOCATION METHOD | NONLOCAL BOUNDARY-CONDITIONS | PROPERTY | HEAT-EQUATION | EFFICIENT TECHNIQUES | PARABOLIC EQUATION SUBJECT | Trigonometrical functions | Mathematical models | Numerical analysis | Spline theory | Research | Mathematical research | Problems | Models | Calculus | Finite element analysis

MATHEMATICS | THERMOELASTICITY | SPECIFICATIONS | COLLOCATION METHOD | NONLOCAL BOUNDARY-CONDITIONS | PROPERTY | HEAT-EQUATION | EFFICIENT TECHNIQUES | PARABOLIC EQUATION SUBJECT | Trigonometrical functions | Mathematical models | Numerical analysis | Spline theory | Research | Mathematical research | Problems | Models | Calculus | Finite element analysis

Journal Article

Journal d'Analyse Mathématique, ISSN 0021-7670, 4/2014, Volume 122, Issue 1, pp. 375 - 401

In this paper, we obtain bounds for the decay rate in the L r (ℝ d )-norm for the solutions of a nonlocal and nonlinear evolution equation, namely, $$u_t...

Abstract Harmonic Analysis | Mathematics | Functional Analysis | Dynamical Systems and Ergodic Theory | Analysis | Partial Differential Equations | EIGENVALUE | DISPERSAL | MATHEMATICS | LAPLACIAN EVOLUTION EQUATION | MODEL | PERIDYNAMICS | Resveratrol

Abstract Harmonic Analysis | Mathematics | Functional Analysis | Dynamical Systems and Ergodic Theory | Analysis | Partial Differential Equations | EIGENVALUE | DISPERSAL | MATHEMATICS | LAPLACIAN EVOLUTION EQUATION | MODEL | PERIDYNAMICS | Resveratrol

Journal Article

Turkish Journal of Mathematics, ISSN 1300-0098, 05/2013, Volume 37, Issue 3, pp. 466 - 482

This paper is concerned with the blow-up of solutions to some nonlocal inhomogeneous dispersal equations subject to homogeneous Neumann boundary conditions. We...

Blow-up solutions | Comparison principle | Nonlocal dispersal | Bounds on blow-up time | TRAVELING-WAVE-FRONTS | DISPERSAL | MATHEMATICS | MODELS | BOUNDARY-CONDITIONS | nonlocal dispersal | SYSTEMS | TERM | EQUATION | bounds on blow-up time | comparison principle

Blow-up solutions | Comparison principle | Nonlocal dispersal | Bounds on blow-up time | TRAVELING-WAVE-FRONTS | DISPERSAL | MATHEMATICS | MODELS | BOUNDARY-CONDITIONS | nonlocal dispersal | SYSTEMS | TERM | EQUATION | bounds on blow-up time | comparison principle

Journal Article

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