Nonlocal Models with Heterogeneous Localization and Their Application to Seamless Local-Nonlocal Coupling

2019 ◽  
Vol 17 (3) ◽  
pp. 1052-1075 ◽  
Author(s):  
Yunzhe Tao ◽  
Xiaochuan Tian ◽  
Qiang Du
Keyword(s):  
Nonlocal Coupling ◽  
Nonlocal Models

The need of electron microscopy in the study of domains and domain walls in ferroelectrics

1994 ◽  
Vol 52 ◽  
pp. 550-551
Author(s):  
Wenwu Cao
Keyword(s):  
Domain Wall ◽  
Domain Walls ◽  
High Mobility ◽  
Continuum Theory ◽  
Polarization Vector ◽  
Ferroelectric Materials ◽  
Dielectric Constants ◽  
Nonlocal Coupling ◽  
Cubic Symmetry ◽  
Gradient Energy

Domain structures play a key role in determining the physical properties of ferroelectric materials. The formation of these ferroelectric domains and domain walls are determined by the intrinsic nonlinearity and the nonlocal coupling of the polarization. Analogous to soliton excitations, domain walls can have high mobility when the domain wall energy is high. The domain wall can be describes by a continuum theory owning to the long range nature of the dipole-dipole interactions in ferroelectrics. The simplest form for the Landau energy is the so called ϕ model which can be used to describe a second order phase transition from a cubic prototype,where Pi (i =1, 2, 3) are the components of polarization vector, α's are the linear and nonlinear dielectric constants. In order to take into account the nonlocal coupling, a gradient energy should be included, for cubic symmetry the gradient energy is given by,

scholarly journals HOW CRUCIAL IS SMALL WORLD CONNECTIVITY FOR DYNAMICS?

2006 ◽  
Vol 16 (09) ◽  
pp. 2767-2775 ◽  
Author(s):  
PRASHANT M. GADE ◽  
SUDESHNA SINHA
Keyword(s):  
Spectral Properties ◽  
Chaotic Systems ◽  
Signal To Noise Ratio ◽  
Dynamical Behavior ◽  
Small World ◽  
High Strength ◽  
Nonlocal Coupling ◽  
Small Noise ◽  
Optimal Value ◽  
Neuronal Systems

We study the dynamical behavior of the collective field of chaotic systems on small world lattices. Coupled neuronal systems as well as coupled logistic maps are investigated. We observe that significant changes in dynamical properties occur only at a reasonably high strength of nonlocal coupling. Further, spectral features, such as signal-to-noise ratio (SNR), change monotonically with respect to the fraction of random rewiring, i.e. there is no optimal value of the rewiring fraction for which spectral properties are most pronounced. We also observe that for small rewiring, results are similar to those obtained by adding small noise.

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
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