Search results for "Nonlocal"

Optimal control in models with conductive‐radiative heat transfer

2003

In this paper an optimal control problem for the elliptic boundary value problem with nonlocal boundary conditions is considered. It is shown that the weak solutions of the boundary value problem depend smoothly on the control parameter and that the cost functional of the optimal control problem is Frechet differentiable with respect to the control parameter. Optimalus valdymas modeliuose su laidžiu-radioaktyviu šilumos pernešimu Santrauka Darbe nagrinejamas nelokalaus kraštinio uždavinio optimalaus valdymo uždavinys. Parodyta, kad silpnasis kraštinio uždavinio sprendinys tolydžiai priklauso nuo valdomojo parametro, taigi, optimalaus valdymo tikslo funkcija yra diferencijuojama Freše prasme…

Mixed Finite Elements for multilayered smart plates nonlocal analysis

2022

A mixed finite element formulation for the Eringen’s nonlocal analysis of smart, magneto-electro-elastic, multilayered plates is presented. Finite elements for different refined higher order plate layerwise theories are systematically developed. They ensure interface continuity and allow associating different values of the nonlocal parameter to the laminate layers. Standard 9-node and 16-node isoparametric, quadrilateral finite elements have been implemented and tested, showing the characteristics and limitations of the proposed approach.

Refined layer-wise models for nonlocal analysis of magneto-electro-elastic plates

Size-dependent theories of continuum mechanics are an important tool for structural and material modeling in engineering applications, with particular regard to those involving micro- and nano-scales. Among various approaches proposed in the literature to account for the effect of the microstructure via continuum models, the Eringen’s nonlocal elasticity model incorporates important features of material behavior via a differential stress-strain relationship involving a scale coefficient, or characteristic length, depending on the material microstructure [1]. In the framework of Eringen’s nonlocal elasticity, plate theories have been reformulated for homogeneous and multilayered configuratio…

Quantitative Approximation Properties for the Fractional Heat Equation

2017

In this note we analyse \emph{quantitative} approximation properties of a certain class of \emph{nonlocal} equations: Viewing the fractional heat equation as a model problem, which involves both \emph{local} and \emph{nonlocal} pseudodifferential operators, we study quantitative approximation properties of solutions to it. First, relying on Runge type arguments, we give an alternative proof of certain \emph{qualitative} approximation results from \cite{DSV16}. Using propagation of smallness arguments, we then provide bounds on the \emph{cost} of approximate controllability and thus quantify the approximation properties of solutions to the fractional heat equation. Finally, we discuss genera…

The Calderón problem for the fractional wave equation: Uniqueness and optimal stability

2021

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial di…