Nonlocal models for sandpiles

2010 ◽  
pp. 191-222
Author(s):  
Fuensanta Andreu-Vaillo ◽  
José Mazón ◽  
Julio Rossi ◽  
J. Julián Toledo-Melero
Keyword(s):  
Nonlocal Models

Asymptotically compatible discretization of multidimensional nonlocal diffusion models and approximation of nonlocal Green’s functions

2018 ◽  
Vol 39 (2) ◽  
pp. 607-625 ◽  
Author(s):  
Qiang Du ◽  
Yunzhe Tao ◽  
Xiaochuan Tian ◽  
Jiang Yang
Keyword(s):  
Green's Functions ◽  
Numerical Solutions ◽  
Green’S Functions ◽  
Diffusion Equations ◽  
Nonlocal Diffusion ◽  
Optimal Order ◽  
Numerical Approximations ◽  
Splitting Technique ◽  
Singular Measures ◽  
Nonlocal Models

AbstractNonlocal diffusion equations and their numerical approximations have attracted much attention in the literature as nonlocal modeling becomes popular in various applications. This paper continues the study of robust discretization schemes for the numerical solution of nonlocal models. In particular, we present quadrature-based finite difference approximations of some linear nonlocal diffusion equations in multidimensions. These approximations are able to preserve various nice properties of the nonlocal continuum models such as the maximum principle and they are shown to be asymptotically compatible in the sense that as the nonlocality vanishes, the numerical solutions can give consistent local limits. The approximation errors are proved to be of optimal order in both nonlocal and asymptotically local settings. The numerical schemes involve a unique design of quadrature weights that reflect the multidimensional nature and require technical estimates on nonconventional divided differences for their numerical analysis. We also study numerical approximations of nonlocal Green’s functions associated with nonlocal models. Unlike their local counterparts, nonlocal Green’s functions might become singular measures that are not well defined pointwise. We demonstrate how to combine a splitting technique with the asymptotically compatible schemes to provide effective numerical approximations of these singular measures.

External optimal control of fractional parabolic PDEs

2020 ◽  
Vol 26 ◽  
pp. 20 ◽  
Author(s):  
Harbir Antil ◽  
Deepanshu Verma ◽  
Mahamadi Warma
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Convergence Rates ◽  
Complete Analysis ◽  
Very Weak Solutions ◽  
Parabolic Pdes ◽  
Nonlocal Models ◽  
Parabolic Case ◽  
Potential Benefits ◽  
Control Source

In [Antil et al. Inverse Probl. 35 (2019) 084003.] we introduced a new notion of optimal control and source identification (inverse) problems where we allow the control/source to be outside the domain where the fractional elliptic PDE is fulfilled. The current work extends this previous work to the parabolic case. Several new mathematical tools have been developed to handle the parabolic problem. We tackle the Dirichlet, Neumann and Robin cases. The need for these novel optimal control concepts stems from the fact that the classical PDE models only allow placing the control/source either on the boundary or in the interior where the PDE is satisfied. However, the nonlocal behavior of the fractional operator now allows placing the control/source in the exterior. We introduce the notions of weak and very-weak solutions to the fractional parabolic Dirichlet problem. We present an approach on how to approximate the fractional parabolic Dirichlet solutions by the fractional parabolic Robin solutions (with convergence rates). A complete analysis for the Dirichlet and Robin optimal control problems has been discussed. The numerical examples confirm our theoretical findings and further illustrate the potential benefits of nonlocal models over the local ones.

scholarly journals Preface to the Focused Issue on Fractional Derivatives and General Nonlocal Models

2019 ◽  
Vol 1 (4) ◽  
pp. 503-504 ◽  
Author(s):  
Qiang Du ◽  
Jan S. Hesthaven ◽  
Changpin Li ◽  
Chi-Wang Shu ◽  
Tao Tang
Keyword(s):  
Fractional Derivatives ◽  
Nonlocal Models

scholarly journals Discrete and nonlocal models of Engesser and Haringx elastica

2017 ◽  
Vol 130 ◽  
pp. 571-585 ◽  
Author(s):  
Attila Kocsis ◽  
Noël Challamel ◽  
György Károlyi
Keyword(s):  
Nonlocal Models

Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress-driven nonlocal models

2018 ◽  
Vol 154 ◽  
pp. 20-32 ◽  
Author(s):  
R. Barretta ◽  
S. Ali Faghidian ◽  
R. Luciano ◽  
C.M. Medaglia ◽  
R. Penna
Keyword(s):  
Strain Gradient ◽  
Free Vibrations ◽  
Nonlocal Models

scholarly journals Nonlocal gradient operators with a nonspherical interaction neighborhood and their applications

2020 ◽  
Vol 54 (1) ◽  
pp. 105-128 ◽  
Author(s):  
Hwi Lee ◽  
Qiang Du
Keyword(s):  
Linear Elasticity ◽  
Linear Models ◽  
Differential Operators ◽  
Stokes Equations ◽  
Integral Operators ◽  
Helmholtz Decomposition ◽  
Nonlocal Models ◽  
Extra Assumption

Nonlocal gradient operators are prototypical nonlocal differential operators that are very important in the studies of nonlocal models. One of the simplest variational settings for such studies is the nonlocal Dirichlet energies wherein the energy densities are quadratic in the nonlocal gradients. There have been earlier studies to illuminate the link between the coercivity of the Dirichlet energies and the interaction strengths of radially symmetric kernels that constitute nonlocal gradient operators in the form of integral operators. In this work we adopt a different perspective and focus on nonlocal gradient operators with a non-spherical interaction neighborhood. We show that the truncation of the spherical interaction neighborhood to a half sphere helps making nonlocal gradient operators well-defined and the associated nonlocal Dirichlet energies coercive. These become possible, unlike the case with full spherical neighborhoods, without any extra assumption on the strengths of the kernels near the origin. We then present some applications of the nonlocal gradient operators with non-spherical interaction neighborhoods. These include nonlocal linear models in mechanics such as nonlocal isotropic linear elasticity and nonlocal Stokes equations, and a nonlocal extension of the Helmholtz decomposition.

scholarly journals Nonlocal and local models for taxis in cell migration: a rigorous limit procedure

2020 ◽  
Vol 81 (6-7) ◽  
pp. 1251-1298 ◽  
Author(s):  
Maria Eckardt ◽  
Kevin J. Painter ◽  
Christina Surulescu ◽  
Anna Zhigun
Keyword(s):  
Cell Migration ◽  
General Solution ◽  
Numerical Simulations ◽  
Integral Operators ◽  
Mathematical Framework ◽  
Local Models ◽  
Nonlocal Operators ◽  
Nonlocal Models ◽  
Limit Procedure

AbstractA rigorous limit procedure is presented which links nonlocal models involving adhesion or nonlocal chemotaxis to their local counterparts featuring haptotaxis and classical chemotaxis, respectively. It relies on a novel reformulation of the involved nonlocalities in terms of integral operators applied directly to the gradients of signal-dependent quantities. The proposed approach handles both model types in a unified way and extends the previous mathematical framework to settings that allow for general solution-dependent coefficient functions. The previous forms of nonlocal operators are compared with the new ones introduced in this paper and the advantages of the latter are highlighted by concrete examples. Numerical simulations in 1D provide an illustration of some of the theoretical findings.

scholarly journals Gravity in the infrared and effective nonlocal models

2020 ◽  
Vol 2020 (04) ◽  
pp. 010-010 ◽  
Author(s):  
Enis Belgacem ◽  
Yves Dirian ◽  
Andreas Finke ◽  
Stefano Foffa ◽  
Michele Maggiore
Keyword(s):  
Nonlocal Models
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