Dissipative property for non local evolution equations

2019 ◽  
Vol 26 (1) ◽  
pp. 91-117
Author(s):  
Severino H. da Silva ◽  
Antonio R. G. Garcia ◽  
Bruna E. P. Lucena
Keyword(s):  
Evolution Equations ◽  
Dissipative Property ◽  
Non Local

scholarly journals Using the Split Bregman Algorithm to Solve the Self-repelling Snakes Model

2022 ◽  
Author(s):  
Huizhu Pan ◽  
Jintao Song ◽  
Wanquan Liu ◽  
Ling Li ◽  
Guanglu Zhou ◽  
...  
Keyword(s):  
Evolution Equations ◽  
Active Contour Model ◽  
Operator Splitting ◽  
The Self ◽  
Convergence Time ◽  
Variational Model ◽  
Split Bregman ◽  
Geodesic Active Contour ◽  
Non Local ◽  
Split Bregman Algorithm

AbstractPreserving contour topology during image segmentation is useful in many practical scenarios. By keeping the contours isomorphic, it is possible to prevent over-segmentation and under-segmentation, as well as to adhere to given topologies. The Self-repelling Snakes model (SR) is a variational model that preserves contour topology by combining a non-local repulsion term with the geodesic active contour model. The SR is traditionally solved using the additive operator splitting (AOS) scheme. In our paper, we propose an alternative solution to the SR using the Split Bregman method. Our algorithm breaks the problem down into simpler sub-problems to use lower-order evolution equations and a simple projection scheme rather than re-initialization. The sub-problems can be solved via fast Fourier transform or an approximate soft thresholding formula which maintains stability, shortening the convergence time, and reduces the memory requirement. The Split Bregman and AOS algorithms are compared theoretically and experimentally.

Diffusion Effects Induced by Dislocations in Crystalline Materials Subjected to Large Strains

2015 ◽  
Vol 651-653 ◽  
pp. 89-95
Author(s):  
Raisa Paşcan ◽  
Sanda Cleja-Ţigoiu
Keyword(s):  
Evolution Equations ◽  
Plane Stress State ◽  
Large Strains ◽  
Slip Systems ◽  
State Variables ◽  
Equilibrium Equations ◽  
Crystalline Materials ◽  
Dislocation Densities ◽  
Differential Type ◽  
Non Local

Abstract. We reconsider here the FEM-algorithm for solving the initial and boundary value problems performed within the viscoplastic constitutive framework and proposed in our paper [1]. The problems concerning the deformation of a sheet composed of a single fcc-crystal, generated by different slip systems simultaneously activated, are solved numerically for an in-plane stress state. The variational formulation is associated to the incremental equilibrium equations and is coupled with an update procedure for the state variables, which are described by the differential type equations, as well as for the non-local evolution equations of the dislocation densities. The length scale parameter is introduced into the model through the diffusion-like parameter which enters the evolution equations for dislocation densities. For more accuracy of the simulation, the shape functions have been chosen polynomials with higher than one degree. We do not consider that once a slip system was activated it remains active for the rest of simulation. The activation condition is a key point in the numerical algorithm. As a numerical example, we perform a tensile test of a rectangular and non-rectangular metallic sheet, comparring the results of the simulation when two, respectively eight slip systems are considered.

scholarly journals Existence, uniqueness and stability results for semilinear integrodifferential non-local evolution equations with Random impulse

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6615-6626
Author(s):  
B. Radhakrishnan ◽  
M. Tamilarasi ◽  
P. Anukokila
Keyword(s):  
Hilbert Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Semigroup Theory ◽  
Local Conditions ◽  
Fixed Point Approach ◽  
Non Local ◽  
The Stability ◽  
Random Impulse ◽  
Stability Results

In this paper, authors investigated the existence and uniqueness of random impulsive semilinear integrodifferential evolution equations with non-local conditions in Hilbert spaces. Also the stability results for the same evolution equation has been studied. The results are derived by using the semigroup theory and fixed point approach. An application is provided to illustrate the theory.

scholarly journals Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker–Planck–Kolmogorov equations

2018 ◽  
Vol 21 (5) ◽  
pp. 1203-1237 ◽  
Author(s):  
Yana A. Butko
Keyword(s):  
Evolution Equations ◽  
Approximate Solutions ◽  
Dynamics Simulation ◽  
Fractional Evolution Equations ◽  
Fractional Equations ◽  
Dirichlet Type ◽  
Chernoff Theorem ◽  
Local Operators ◽  
Non Local ◽  
Distributed Order

Abstract We consider operator semigroups generated by Feller processes killed upon leaving a given domain. These semigroups correspond to Cauchy–Dirichlet type initial-exterior value problems in this domain for a class of evolution equations with (possibly non-local) operators. The considered semigroups are approximated by means of the Chernoff theorem. For a class of killed Feller processes, the constructed Chernoff approximation leads to a representation of the solution of the corresponding Cauchy–Dirichlet type problem by a Feynman formula, i.e. by a limit of n-fold iterated integrals of certain functions as n → ∞. Feynman formulae can be used for direct calculations, modelling of underlying dynamics, simulation of underlying stochastic processes. Further, a method to approximate solutions of time-fractional evolution equations is suggested. The method is based on connections between time-fractional and time-non-fractional evolution equations as well as on Chernoff approximations for the latter ones. This method leads to Feynman formulae for solutions of time-fractional evolution equations. A class of distributed order time-fractional equations is considered; Feynman formulae for solutions of the corresponding Cauchy and Cauchy–Dirichlet type problems are obtained.

scholarly journals Mathematical study of a system of multi-dimensional non-local evolution equations describing surfactant-laden two-fluid shear flows

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210307
Author(s):  
Demetrios T. Papageorgiou ◽  
Saleh Tanveer
Keyword(s):  
Global Existence ◽  
Evolution Equations ◽  
Coupled System ◽  
Travelling Wave ◽  
Two Dimensions ◽  
Travelling Wave Solutions ◽  
Rigorous Results ◽  
Wave Solutions ◽  
Non Local ◽  
Two Fluid

This article studies a coupled system of model multi-dimensional partial differential equations (PDEs) that arise in the nonlinear dynamics of two-fluid Couette flow when insoluble surfactants are present on the interface. The equations have been derived previously, but a rigorous study of local and global existence of their solutions, or indeed solutions of analogous systems, has not been considered previously. The evolution PDEs are two-dimensional in space and contain novel pseudo-differential terms that emerge from asymptotic analysis and matching in the multi-scale problem at hand. The one-dimensional surfactant-free case was studied previously, where travelling wave solutions were constructed numerically and their stability investigated; in addition, the travelling wave solutions were justified mathematically. The present study is concerned with some rigorous results of the multi-dimensional surfactant system, including local well posedness and smoothing results when there is full coupling between surfactant dynamics and interfacial motion, and global existence results when such coupling is absent. As far as we know such results are new for non-local thin film equations in either one or two dimensions.

scholarly journals Continuity of global attractors for a class of non local evolution equations

2010 ◽  
Vol 26 (3) ◽  
pp. 1073-1100 ◽  
Author(s):  
Antônio Luiz Pereira ◽  
◽  
Severino Horácio da Silva ◽  
Keyword(s):  
Evolution Equations ◽  
Global Attractors ◽  
Non Local
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