Homogenization Of Partial Differential Equations

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Homogenization Of Partial Differential Equations

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Author : Vladimir A. Marchenko
language : en
Publisher: Springer Science & Business Media
Release Date : 2008-12-22

Homogenization Of Partial Differential Equations written by Vladimir A. Marchenko and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-12-22 with Mathematics categories.

A comprehensive study of homogenized problems, focusing on the construction of nonstandard models Details a method for modeling processes in microinhomogeneous media (radiophysics, filtration theory, rheology, elasticity theory, and other domains) Complete proofs of all main results, numerous examples Classroom text or comprehensive reference for graduate students, applied mathematicians, physicists, and engineers

Emerging Problems In The Homogenization Of Partial Differential Equations

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Author : Patrizia Donato
language : en
Publisher: Springer Nature
Release Date : 2021-02-01

Emerging Problems In The Homogenization Of Partial Differential Equations written by Patrizia Donato and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-02-01 with Mathematics categories.

This book contains some of the results presented at the mini-symposium titled Emerging Problems in the Homogenization of Partial Differential Equations, held during the ICIAM2019 conference in Valencia in July 2019. The papers cover a large range of topics, problems with weak regularity data involving renormalized solutions, eigenvalue problems for complicated shapes of the domain, homogenization of partial differential problems with strongly alternating boundary conditions of Robin type with large parameters, multiscale analysis of the potential action along a neuron with a myelinated axon, and multi-scale model of magnetorheological suspensions. The volume is addressed to scientists who deal with complex systems that presents several elements (characteristics, constituents...) of very different scales, very heterogeneous, and search for homogenized models providing an effective (macroscopic) description of their behaviors.

Homogenization Of Differential Operators And Integral Functionals

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Author : V.V. Jikov
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Homogenization Of Differential Operators And Integral Functionals written by V.V. Jikov and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

It was mainly during the last two decades that the theory of homogenization or averaging of partial differential equations took shape as a distinct mathe matical discipline. This theory has a lot of important applications in mechanics of composite and perforated materials, filtration, disperse media, and in many other branches of physics, mechanics and modern technology. There is a vast literature on the subject. The term averaging has been usually associated with the methods of non linear mechanics and ordinary differential equations developed in the works of Poincare, Van Der Pol, Krylov, Bogoliubov, etc. For a long time, after the works of Maxwell and Rayleigh, homogeniza tion problems for· partial differential equations were being mostly considered by specialists in physics and mechanics, and were staying beyond the scope of mathematicians. A great deal of attention was given to the so called disperse media, which, in the simplest case, are two-phase media formed by the main homogeneous material containing small foreign particles (grains, inclusions). Such two-phase bodies, whose size is considerably larger than that of each sep arate inclusion, have been discovered to possess stable physical properties (such as heat transfer, electric conductivity, etc.) which differ from those of the con stituent phases. For this reason, the word homogenized, or effective, is used in relation to these characteristics. An enormous number of results, approximation formulas, and estimates have been obtained in connection with such problems as electromagnetic wave scattering on small particles, effective heat transfer in two-phase media, etc.

Emerging Problems In The Homogenization Of Partial Differential Equations

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Author : Patrizia Donato
language : en
Publisher:
Release Date : 2021

Emerging Problems In The Homogenization Of Partial Differential Equations written by Patrizia Donato and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021 with categories.

This book contains some of the results presented at the mini-symposium titled Emerging Problems in the Homogenization of Partial Differential Equations, held during the ICIAM2019 conference in Valencia in July 2019. The papers cover a large range of topics, problems with weak regularity data involving renormalized solutions, eigenvalue problems for complicated shapes of the domain, homogenization of partial differential problems with strongly alternating boundary conditions of Robin type with large parameters, multiscale analysis of the potential action along a neuron with a myelinated axon, and multi-scale model of magnetorheological suspensions. The volume is addressed to scientists who deal with complex systems that presents several elements (characteristics, constituents...) of very different scales, very heterogeneous, and search for homogenized models providing an effective (macroscopic) description of their behaviors. .

Numerical Homogenization By Localized Decomposition

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Author : Axel Målqvist
language : en
Publisher: SIAM
Release Date : 2020-11-23

Numerical Homogenization By Localized Decomposition written by Axel Målqvist and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-11-23 with Mathematics categories.

This book presents the first survey of the Localized Orthogonal Decomposition (LOD) method, a pioneering approach for the numerical homogenization of partial differential equations with multiscale data beyond periodicity and scale separation. The authors provide a careful error analysis, including previously unpublished results, and a complete implementation of the method in MATLAB. They also reveal how the LOD method relates to classical homogenization and domain decomposition. Illustrated with numerical experiments that demonstrate the significance of the method, the book is enhanced by a survey of applications including eigenvalue problems and evolution problems. Numerical Homogenization by Localized Orthogonal Decomposition is appropriate for graduate students in applied mathematics, numerical analysis, and scientific computing. Researchers in the field of computational partial differential equations will find this self-contained book of interest, as will applied scientists and engineers interested in multiscale simulation.

G Convergence And Homogenization Of Nonlinear Partial Differential Operators

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Author : A. A. Pankov
language : en
Publisher:
Release Date : 2014-01-15

G Convergence And Homogenization Of Nonlinear Partial Differential Operators written by A. A. Pankov and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with categories.

The General Theory Of Homogenization

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Author : Luc Tartar
language : en
Publisher: Springer Science & Business Media
Release Date : 2009-12-03

The General Theory Of Homogenization written by Luc Tartar and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-12-03 with Science categories.

Homogenization is not about periodicity, or Gamma-convergence, but about understanding which effective equations to use at macroscopic level, knowing which partial differential equations govern mesoscopic levels, without using probabilities (which destroy physical reality); instead, one uses various topologies of weak type, the G-convergence of Sergio Spagnolo, the H-convergence of François Murat and the author, and some responsible for the appearance of nonlocal effects, which many theories in continuum mechanics or physics guessed wrongly. For a better understanding of 20th century science, new mathematical tools must be introduced, like the author’s H-measures, variants by Patrick Gérard, and others yet to be discovered.

Mathematical Problems In Elasticity And Homogenization

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Author : O.A. Oleinik
language : en
Publisher: Elsevier
Release Date : 2009-06-15

Mathematical Problems In Elasticity And Homogenization written by O.A. Oleinik and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-06-15 with Mathematics categories.

This monograph is based on research undertaken by the authors during the last ten years. The main part of the work deals with homogenization problems in elasticity as well as some mathematical problems related to composite and perforated elastic materials. This study of processes in strongly non-homogeneous media brings forth a large number of purely mathematical problems which are very important for applications. Although the methods suggested deal with stationary problems, some of them can be extended to non-stationary equations. With the exception of some well-known facts from functional analysis and the theory of partial differential equations, all results in this book are given detailed mathematical proof. It is expected that the results and methods presented in this book will promote further investigation of mathematical models for processes in composite and perforated media, heat-transfer, energy transfer by radiation, processes of diffusion and filtration in porous media, and that they will stimulate research in other problems of mathematical physics and the theory of partial differential equations.

Periodic Homogenization Of Elliptic Systems

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Author : Zhongwei Shen
language : en
Publisher: Springer
Release Date : 2018-09-04

Periodic Homogenization Of Elliptic Systems written by Zhongwei Shen and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-04 with Mathematics categories.

This monograph surveys the theory of quantitative homogenization for second-order linear elliptic systems in divergence form with rapidly oscillating periodic coefficients in a bounded domain. It begins with a review of the classical qualitative homogenization theory, and addresses the problem of convergence rates of solutions. The main body of the monograph investigates various interior and boundary regularity estimates that are uniform in the small parameter e>0. Additional topics include convergence rates for Dirichlet eigenvalues and asymptotic expansions of fundamental solutions, Green functions, and Neumann functions. The monograph is intended for advanced graduate students and researchers in the general areas of analysis and partial differential equations. It provides the reader with a clear and concise exposition of an important and currently active area of quantitative homogenization.

Homogenization Theory For Multiscale Problems

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Author : Xavier Blanc
language : en
Publisher: Springer Nature
Release Date : 2023-04-29

Homogenization Theory For Multiscale Problems written by Xavier Blanc and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-04-29 with Mathematics categories.

The book provides a pedagogic and comprehensive introduction to homogenization theory with a special focus on problems set for non-periodic media. The presentation encompasses both deterministic and probabilistic settings. It also mixes the most abstract aspects with some more practical aspects regarding the numerical approaches necessary to simulate such multiscale problems. Based on lecture courses of the authors, the book is suitable for graduate students of mathematics and engineering.