Nonlocal Diffusion Problems
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Nonlocal Diffusion Problems
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Author : Fuensanta Andreu-Vaillo
language : en
Publisher: American Mathematical Soc.
Release Date : 2010
Nonlocal Diffusion Problems written by Fuensanta Andreu-Vaillo and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010 with Mathematics categories.
Nonlocal diffusion problems arise in a wide variety of applications, including biology, image processing, particle systems, coagulation models, and mathematical finance. These types of problems are also of great interest for their purely mathematical content. This book presents recent results on nonlocal evolution equations with different boundary conditions, starting with the linear theory and moving to nonlinear cases, including two nonlocal models for the evolution of sandpiles. Both existence and uniqueness of solutions are considered, as well as their asymptotic behaviour. Moreover, the authors present results concerning limits of solutions of the nonlocal equations as a rescaling parameter tends to zero. With these limit procedures the most frequently used diffusion models are recovered: the heat equation, the $p$-Laplacian evolution equation, the porous media equation, the total variation flow, a convection-diffusion equation and the local models for the evolution of sandpiles due to Aronsson-Evans-Wu and Prigozhin. Readers are assumed to be familiar with the basic concepts and techniques of functional analysis and partial differential equations. The text is otherwise self-contained, with the exposition emphasizing an intuitive understanding and results given with full proofs. It is suitable for graduate students or researchers. The authors cover a subject that has received a great deal of attention in recent years. The book is intended as a reference tool for a general audience in analysis and PDEs, including mathematicians, engineers, physicists, biologists, and others interested in nonlocal diffusion problems.
Nonlocal Diffusion Problems
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Author :
language : en
Publisher:
Release Date : 2014
Nonlocal Diffusion Problems written by and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with categories.
The Dynamics Of Front Propagation In Nonlocal Reaction Diffusion Equations
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Author : Jean-Michel Roquejoffre
language : en
Publisher: Springer Nature
Release Date : 2024-12-18
The Dynamics Of Front Propagation In Nonlocal Reaction Diffusion Equations written by Jean-Michel Roquejoffre and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-18 with Mathematics categories.
The book provides a self-contained and complete description of the long time evolution of the solutions to a class of one-dimensional reaction–diffusion equations, in which the diffusion is given by an integral operator. The underlying motivation is the mathematical analysis of models for biological invasions. The model under study, while simple looking, is of current use in real-life situations. Interestingly, it arises in totally different contexts, such as the study of branching random walks in probability theory. While the model has attracted a lot of attention, and while many partial results about the time-asymptotic behaviour of its solutions have been proved over the last decades, some basic questions on the sharp asymptotics have remained unanswered. One ambition of this monograph is to close these gaps. In some of the situations that we envisage, the level sets organise themselves into an invasion front that is asymptotically linear in time, up to a correction that converges exponentially in time to a constant. In other situations that constitute the main and newest part of the work, the correction is asymptotically logarithmic in time. Despite these apparently different behaviours, there is an underlying common way of thinking that is underlined. At the end of each chapter, a long set of problems is proposed, many of them rather elaborate and suitable for master's projects or even the first question in a PhD thesis. Open questions are also discussed. The ideas presented in the book apply to more elaborate systems modelling biological invasions or the spatial propagation of epidemics. The models themselves may be multidimensional, but they all have in common a mechanism imposing the propagation in a given direction; examples are presented in the problems that conclude each chapter. These ideas should also be useful in the treatment of further models that we are not able to envisage for the time being. The book is suitable for graduate or PhD students as well as researchers.
Variational And Diffusion Problems In Random Walk Spaces
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Author : José M. Mazón
language : en
Publisher: Springer Nature
Release Date : 2023-08-04
Variational And Diffusion Problems In Random Walk Spaces written by José M. Mazón and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-08-04 with Mathematics categories.
This book presents the latest developments in the theory of gradient flows in random walk spaces. A broad framework is established for a wide variety of partial differential equations on nonlocal models and weighted graphs. Within this framework, specific gradient flows that are studied include the heat flow, the total variational flow, and evolution problems of Leray-Lions type with different types of boundary conditions. With many timely applications, this book will serve as an invaluable addition to the literature in this active area of research. Variational and Diffusion Problems in Random Walk Spaces will be of interest to researchers at the interface between analysis, geometry, and probability, as well as to graduate students interested in exploring these areas.
Variational Methods For Nonlocal Fractional Problems
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Author : Giovanni Molica Bisci
language : en
Publisher: Cambridge University Press
Release Date : 2016-03-11
Variational Methods For Nonlocal Fractional Problems written by Giovanni Molica Bisci and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-03-11 with Mathematics categories.
This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.
Nonlocal Modeling Analysis And Computation
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Author : Qiang Du
language : en
Publisher: SIAM
Release Date : 2019-03-20
Nonlocal Modeling Analysis And Computation written by Qiang Du and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-03-20 with Science categories.
Studies of complexity, singularity, and anomaly using nonlocal continuum models are steadily gaining popularity. This monograph provides an introduction to basic analytical, computational, and modeling issues and to some of the latest developments in these areas. Nonlocal Modeling, Analysis, and Computation includes motivational examples of nonlocal models, basic building blocks of nonlocal vector calculus, elements of theory for well-posedness and nonlocal spaces, connections to and coupling with local models, convergence and compatibility of numerical approximations, and various applications, such as nonlocal dynamics of anomalous diffusion and nonlocal peridynamic models of elasticity and fracture mechanics. A particular focus is on nonlocal systems with a finite range of interaction to illustrate their connection to local partial differential equations and fractional PDEs. These models are designed to represent nonlocal interactions explicitly and to remain valid for complex systems involving possible singular solutions and they have the potential to be alternatives for as well as bridges to existing models. The author discusses ongoing studies of nonlocal models to encourage the discovery of new mathematical theory for nonlocal continuum models and offer new perspectives on traditional models, analytical techniques, and algorithms.
Fractional Diffusion Equations And Anomalous Diffusion
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Author : Luiz Roberto Evangelista
language : en
Publisher: Cambridge University Press
Release Date : 2018-01-25
Fractional Diffusion Equations And Anomalous Diffusion written by Luiz Roberto Evangelista and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-01-25 with Mathematics categories.
Presents a unified treatment of anomalous diffusion problems using fractional calculus in a wide range of applications across scientific and technological disciplines.
Nonlinear Diffusion Equations
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Author : Zhuoqun Wu
language : en
Publisher: World Scientific
Release Date : 2001
Nonlinear Diffusion Equations written by Zhuoqun Wu and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.
Nonlinear diffusion equations, an important class of parabolic equations, come from a variety of diffusion phenomena which appear widely in nature. They are suggested as mathematical models of physical problems in many fields, such as filtration, phase transition, biochemistry and dynamics of biological groups. In many cases, the equations possess degeneracy or singularity. The appearance of degeneracy or singularity makes the study more involved and challenging. Many new ideas and methods have been developed to overcome the special difficulties caused by the degeneracy and singularity, which enrich the theory of partial differential equations.This book provides a comprehensive presentation of the basic problems, main results and typical methods for nonlinear diffusion equations with degeneracy. Some results for equations with singularity are touched upon.
Reaction Diffusion Waves
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Author : Arnaud Ducrot
language : en
Publisher: Editions Publibook
Release Date : 2009
Reaction Diffusion Waves written by Arnaud Ducrot and has been published by Editions Publibook this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009 with Differential operators categories.
Nonlocal And Nonlinear Diffusions And Interactions New Methods And Directions
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Author : José Antonio Carrillo
language : en
Publisher: Springer
Release Date : 2017-10-03
Nonlocal And Nonlinear Diffusions And Interactions New Methods And Directions written by José Antonio Carrillo and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-10-03 with Mathematics categories.
Presenting a selection of topics in the area of nonlocal and nonlinear diffusions, this book places a particular emphasis on new emerging subjects such as nonlocal operators in stationary and evolutionary problems and their applications, swarming models and applications to biology and mathematical physics, and nonlocal variational problems. The authors are some of the most well-known mathematicians in this innovative field, which is presently undergoing rapid development. The intended audience includes experts in elliptic and parabolic equations who are interested in extending their expertise to the nonlinear setting, as well as Ph.D. or postdoctoral students who want to enter into the most promising research topics in the field.