Nonlocal Diffusion Problems

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Nonlocal Diffusion Problems

Fuensanta Andreu-Vaillo
Nonlocal Diffusion Problems Book PDF

Author : Fuensanta Andreu-Vaillo
Publisher : American Mathematical Soc.
Page : 274 pages
File Size : 55,5 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821852309

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Nonlocal Diffusion Problems by Fuensanta Andreu-Vaillo Pdf

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 and Applications

Claudia Bucur,Enrico Valdinoci
Nonlocal Diffusion and Applications Book PDF

Author : Claudia Bucur,Enrico Valdinoci
Publisher : Springer
Page : 155 pages
File Size : 50,9 Mb
Release : 2016-04-08
Category : Mathematics
ISBN : 9783319287393

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Nonlocal Diffusion and Applications by Claudia Bucur,Enrico Valdinoci Pdf

Working in the fractional Laplace framework, this book provides models and theorems related to nonlocal diffusion phenomena. In addition to a simple probabilistic interpretation, some applications to water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schrödinger equations are given. Furthermore, an example of an s-harmonic function, its harmonic extension and some insight into a fractional version of a classical conjecture due to De Giorgi are presented. Although the aim is primarily to gather some introductory material concerning applications of the fractional Laplacian, some of the proofs and results are new. The work is entirely self-contained, and readers who wish to pursue related subjects of interest are invited to consult the rich bibliography for guidance.

Nonlocal Diffusion Problems

Anonim
Nonlocal Diffusion Problems Book PDF

Author : Anonim
Publisher : American Mathematical Soc.
Page : 274 pages
File Size : 52,6 Mb
Release : 2010-01-01
Category : Mathematics
ISBN : 9780821875469

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Nonlocal Diffusion Problems by Anonim Pdf

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.

Variational and Diffusion Problems in Random Walk Spaces

José M. Mazón,Marcos Solera-Diana,J. Julián Toledo-Melero
Variational and Diffusion Problems in Random Walk Spaces Book PDF

Author : José M. Mazón,Marcos Solera-Diana,J. Julián Toledo-Melero
Publisher : Springer Nature
Page : 396 pages
File Size : 46,5 Mb
Release : 2023-08-04
Category : Mathematics
ISBN : 9783031335846

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Variational and Diffusion Problems in Random Walk Spaces by José M. Mazón,Marcos Solera-Diana,J. Julián Toledo-Melero Pdf

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.

Nonlocal Modeling, Analysis, and Computation

Qiang Du
Nonlocal Modeling Analysis and Computation Book PDF

Author : Qiang Du
Publisher : SIAM
Page : 181 pages
File Size : 43,6 Mb
Release : 2019-03-20
Category : Science
ISBN : 9781611975611

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Nonlocal Modeling, Analysis, and Computation by Qiang Du Pdf

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.

Fourier Analysis and Nonlinear Partial Differential Equations

Hajer Bahouri,Jean-Yves Chemin,Raphaël Danchin
Fourier Analysis and Nonlinear Partial Differential Equations Book PDF

Author : Hajer Bahouri,Jean-Yves Chemin,Raphaël Danchin
Publisher : Springer Science & Business Media
Page : 530 pages
File Size : 49,7 Mb
Release : 2011-01-03
Category : Mathematics
ISBN : 9783642168307

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Fourier Analysis and Nonlinear Partial Differential Equations by Hajer Bahouri,Jean-Yves Chemin,Raphaël Danchin Pdf

In recent years, the Fourier analysis methods have expereinced a growing interest in the study of partial differential equations. In particular, those techniques based on the Littlewood-Paley decomposition have proved to be very efficient for the study of evolution equations. The present book aims at presenting self-contained, state- of- the- art models of those techniques with applications to different classes of partial differential equations: transport, heat, wave and Schrödinger equations. It also offers more sophisticated models originating from fluid mechanics (in particular the incompressible and compressible Navier-Stokes equations) or general relativity. It is either directed to anyone with a good undergraduate level of knowledge in analysis or useful for experts who are eager to know the benefit that one might gain from Fourier analysis when dealing with nonlinear partial differential equations.

Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions

José Antonio Carrillo,Manuel del Pino,Alessio Figalli,Giuseppe Mingione,Juan Luis Vázquez
Nonlocal and Nonlinear Diffusions and Interactions New Methods and Directions Book PDF

Author : José Antonio Carrillo,Manuel del Pino,Alessio Figalli,Giuseppe Mingione,Juan Luis Vázquez
Publisher : Springer
Page : 280 pages
File Size : 41,8 Mb
Release : 2017-10-03
Category : Mathematics
ISBN : 9783319614946

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Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions by José Antonio Carrillo,Manuel del Pino,Alessio Figalli,Giuseppe Mingione,Juan Luis Vázquez Pdf

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.

Variational Methods for Nonlocal Fractional Problems

Giovanni Molica Bisci,Vicentiu D. Radulescu,Raffaella Servadei
Variational Methods for Nonlocal Fractional Problems Book PDF

Author : Giovanni Molica Bisci,Vicentiu D. Radulescu,Raffaella Servadei
Publisher : Cambridge University Press
Page : 401 pages
File Size : 50,8 Mb
Release : 2016-03-11
Category : Mathematics
ISBN : 9781316571699

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Variational Methods for Nonlocal Fractional Problems by Giovanni Molica Bisci,Vicentiu D. Radulescu,Raffaella Servadei Pdf

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.

A3N2M: Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models

Tadele Mengesha,Abner J. Salgado
A3N2M Approximation Applications and Analysis of Nonlocal Nonlinear Models Book PDF

Author : Tadele Mengesha,Abner J. Salgado
Publisher : Springer Nature
Page : 325 pages
File Size : 40,7 Mb
Release : 2023-09-12
Category : Mathematics
ISBN : 9783031340895

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A3N2M: Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models by Tadele Mengesha,Abner J. Salgado Pdf

This volume collects papers based on plenary and invited talks given at the 50th Barrett Memorial Lectures on Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models that was organized by the University of Tennessee, Knoxville and held virtually in May 2021. The three-day meeting brought together experts from the computational, scientific, engineering, and mathematical communities who work with nonlocal models. These proceedings collect contributions and give a survey of the state of the art in computational practices, mathematical analysis, applications of nonlocal models, and explorations of new application domains. The volume benefits from the mixture of contributions by computational scientists, mathematicians, and application specialists. The content is suitable for graduate students as well as specialists working with nonlocal models and covers topics on fractional PDEs, regularity theory for kinetic equations, approximation theory for fractional diffusion, analysis of nonlocal diffusion model as a bridge between local and fractional PDEs, and more.

Variational Convergence And Stochastic Homogenization Of Nonlinear Reaction-diffusion Problems

Omar Anza Hafsa,Jean-philippe Mandallena,Gerard Michaille
Variational Convergence And Stochastic Homogenization Of Nonlinear Reaction diffusion Problems Book PDF

Author : Omar Anza Hafsa,Jean-philippe Mandallena,Gerard Michaille
Publisher : World Scientific
Page : 321 pages
File Size : 40,9 Mb
Release : 2022-06-21
Category : Mathematics
ISBN : 9789811258503

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Variational Convergence And Stochastic Homogenization Of Nonlinear Reaction-diffusion Problems by Omar Anza Hafsa,Jean-philippe Mandallena,Gerard Michaille Pdf

A substantial number of problems in physics, chemical physics, and biology, are modeled through reaction-diffusion equations to describe temperature distribution or chemical substance concentration. For problems arising from ecology, sociology, or population dynamics, they describe the density of some populations or species. In this book the state variable is a concentration, or a density according to the cases. The reaction function may be complex and include time delays terms that model various situations involving maturation periods, resource regeneration times, or incubation periods. The dynamics may occur in heterogeneous media and may depend upon a small or large parameter, as well as the reaction term. From a purely formal perspective, these parameters are indexed by n. Therefore, reaction-diffusion equations give rise to sequences of Cauchy problems.The first part of the book is devoted to the convergence of these sequences in a sense made precise in the book. The second part is dedicated to the specific case when the reaction-diffusion problems depend on a small parameter ∊ₙ intended to tend towards 0. This parameter accounts for the size of small spatial and randomly distributed heterogeneities. The convergence results obtained in the first part, with additionally some probabilistic tools, are applied to this specific situation. The limit problems are illustrated through biological invasion, food-limited or prey-predator models where the interplay between environment heterogeneities in the individual evolution of propagation species plays an essential role. They provide a description in terms of deterministic and homogeneous reaction-diffusion equations, for which numerical schemes are possible.

Nonlocal Continuum Limits of p-Laplacian Problems on Graphs

Imad El Bouchairi,Jalal Fadili,Yosra Hafiene,Abderrahim Elmoataz
Nonlocal Continuum Limits of p Laplacian Problems on Graphs Book PDF

Author : Imad El Bouchairi,Jalal Fadili,Yosra Hafiene,Abderrahim Elmoataz
Publisher : Cambridge University Press
Page : 124 pages
File Size : 45,8 Mb
Release : 2023-04-30
Category : Computers
ISBN : 9781009327879

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Nonlocal Continuum Limits of p-Laplacian Problems on Graphs by Imad El Bouchairi,Jalal Fadili,Yosra Hafiene,Abderrahim Elmoataz Pdf

In this Element, the authors consider fully discretized p-Laplacian problems (evolution, boundary value and variational problems) on graphs. The motivation of nonlocal continuum limits comes from the quest of understanding collective dynamics in large ensembles of interacting particles, which is a fundamental problem in nonlinear science, with applications ranging from biology to physics, chemistry and computer science. Using the theory of graphons, the authors give a unified treatment of all the above problems and establish the continuum limit for each of them together with non-asymptotic convergence rates. They also describe an algorithmic framework based proximal splitting to solve these discrete problems on graphs.

Non-Local Partial Differential Equations for Engineering and Biology

Nikos I. Kavallaris,Takashi Suzuki
Non Local Partial Differential Equations for Engineering and Biology Book PDF

Author : Nikos I. Kavallaris,Takashi Suzuki
Publisher : Springer
Page : 300 pages
File Size : 44,8 Mb
Release : 2017-11-28
Category : Technology & Engineering
ISBN : 9783319679440

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Non-Local Partial Differential Equations for Engineering and Biology by Nikos I. Kavallaris,Takashi Suzuki Pdf

This book presents new developments in non-local mathematical modeling and mathematical analysis on the behavior of solutions with novel technical tools. Theoretical backgrounds in mechanics, thermo-dynamics, game theory, and theoretical biology are examined in details. It starts off with a review and summary of the basic ideas of mathematical modeling frequently used in the sciences and engineering. The authors then employ a number of models in bio-science and material science to demonstrate applications, and provide recent advanced studies, both on deterministic non-local partial differential equations and on some of their stochastic counterparts used in engineering. Mathematical models applied in engineering, chemistry, and biology are subject to conservation laws. For instance, decrease or increase in thermodynamic quantities and non-local partial differential equations, associated with the conserved physical quantities as parameters. These present novel mathematical objects are engaged with rich mathematical structures, in accordance with the interactions between species or individuals, self-organization, pattern formation, hysteresis. These models are based on various laws of physics, such as mechanics of continuum, electro-magnetic theory, and thermodynamics. This is why many areas of mathematics, calculus of variation, dynamical systems, integrable systems, blow-up analysis, and energy methods are indispensable in understanding and analyzing these phenomena. This book aims for researchers and upper grade students in mathematics, engineering, physics, economics, and biology.

Operator and Norm Inequalities and Related Topics

Richard M. Aron,Mohammad Sal Moslehian,Ilya M. Spitkovsky,Hugo J. Woerdeman
Operator and Norm Inequalities and Related Topics Book PDF

Author : Richard M. Aron,Mohammad Sal Moslehian,Ilya M. Spitkovsky,Hugo J. Woerdeman
Publisher : Springer Nature
Page : 822 pages
File Size : 53,7 Mb
Release : 2022-08-10
Category : Mathematics
ISBN : 9783031021046

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Operator and Norm Inequalities and Related Topics by Richard M. Aron,Mohammad Sal Moslehian,Ilya M. Spitkovsky,Hugo J. Woerdeman Pdf

Inequalities play a central role in mathematics with various applications in other disciplines. The main goal of this contributed volume is to present several important matrix, operator, and norm inequalities in a systematic and self-contained fashion. Some powerful methods are used to provide significant mathematical inequalities in functional analysis, operator theory and numerous fields in recent decades. Some chapters are devoted to giving a series of new characterizations of operator monotone functions and some others explore inequalities connected to log-majorization, relative operator entropy, and the Ando-Hiai inequality. Several chapters are focused on Birkhoff–James orthogonality and approximate orthogonality in Banach spaces and operator algebras such as C*-algebras from historical perspectives to current development. A comprehensive account of the boundedness, compactness, and restrictions of Toeplitz operators can be found in the book. Furthermore, an overview of the Bishop-Phelps-Bollobás theorem is provided. The state-of-the-art of Hardy-Littlewood inequalities in sequence spaces is given. The chapters are written in a reader-friendly style and can be read independently. Each chapter contains a rich bibliography. This book is intended for use by both researchers and graduate students of mathematics, physics, and engineering.

Fractional Diffusion Equations and Anomalous Diffusion

Luiz Roberto Evangelista,Ervin Kaminski Lenzi
Fractional Diffusion Equations and Anomalous Diffusion Book PDF

Author : Luiz Roberto Evangelista,Ervin Kaminski Lenzi
Publisher : Cambridge University Press
Page : 361 pages
File Size : 45,6 Mb
Release : 2018-01-25
Category : Mathematics
ISBN : 9781107143555

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Fractional Diffusion Equations and Anomalous Diffusion by Luiz Roberto Evangelista,Ervin Kaminski Lenzi Pdf

Presents a unified treatment of anomalous diffusion problems using fractional calculus in a wide range of applications across scientific and technological disciplines.

Meshfree Methods for Partial Differential Equations VII

Michael Griebel,Marc Alexander Schweitzer
Meshfree Methods for Partial Differential Equations VII Book PDF

Author : Michael Griebel,Marc Alexander Schweitzer
Publisher : Springer
Page : 323 pages
File Size : 53,9 Mb
Release : 2014-12-02
Category : Mathematics
ISBN : 9783319068985

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Meshfree Methods for Partial Differential Equations VII by Michael Griebel,Marc Alexander Schweitzer Pdf

Meshfree methods, particle methods, and generalized finite element methods have witnessed substantial development since the mid 1990s. The growing interest in these methods is due in part to the fact that they are extremely flexible numerical tools and can be interpreted in a number of ways. For instance, meshfree methods can be viewed as a natural extension of classical finite element and finite difference methods to scattered node configurations with no fixed connectivity. Furthermore, meshfree methods offer a number of advantageous features which are especially attractive when dealing with multiscale phenomena: a priori knowledge about particular local behavior of the solution can easily be introduced in the meshfree approximation space, and coarse-scale approximations can be seamlessly refined with fine-scale information. This volume collects selected papers presented at the Seventh International Workshop on Meshfree Methods, held in Bonn, Germany in September 2013. They address various aspects of this highly dynamic research field and cover topics from applied mathematics, physics and engineering.