Nonlocal Diffusion Problems

Author: Fuensanta Andreu-Vaillo,Jose M. Mazon,Julio D. Rossi,J. Julian Toledo-Melero

Publisher: American Mathematical Soc.

ISBN: 0821852302

Category: Mathematics

Page: 256

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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.

Variational Methods for Nonlocal Fractional Problems

Author: Giovanni Molica Bisci,Vicentiu D. Radulescu,Raffaella Servadei

Publisher: Cambridge University Press

ISBN: 1107111943

Category: Mathematics

Page: 400

View: 3569

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A thorough graduate-level introduction to the variational analysis of nonlinear problems described by nonlocal operators.

Nonlocal Diffusion and Applications

Author: Claudia Bucur,Enrico Valdinoci

Publisher: Springer

ISBN: 3319287397

Category: Mathematics

Page: 155

View: 2077

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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.

Meshfree Methods for Partial Differential Equations VII

Author: Michael Griebel,Marc Alexander Schweitzer

Publisher: Springer

ISBN: 3319068989

Category: Mathematics

Page: 324

View: 7628

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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.

Handbook of Peridynamic Modeling

Author: Florin Bobaru,John T. Foster,Philippe H Geubelle,Stewart A. Silling

Publisher: CRC Press

ISBN: 1315355949

Category: Mathematics

Page: 548

View: 1625

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This handbook covers the peridynamic modeling of failure and damage. Peridynamics is a reformulation of continuum mechanics based on integration of interactions rather than spatial differentiation of displacements. The book extends the classical theory of continuum mechanics to allow unguided modeling of crack propagation/fracture in brittle, quasi-brittle, and ductile materials; autonomous transition from continuous damage/fragmentation to fracture; modeling of long-range forces within a continuous body; and multiscale coupling in a consistent mathematical framework.

Elliptic Partial Differential Equations

Volume 2: Reaction-Diffusion Equations

Author: Vitaly Volpert

Publisher: Springer

ISBN: 3034808135

Category: Mathematics

Page: 784

View: 6616

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If we had to formulate in one sentence what this book is about, it might be "How partial differential equations can help to understand heat explosion, tumor growth or evolution of biological species". These and many other applications are described by reaction-diffusion equations. The theory of reaction-diffusion equations appeared in the first half of the last century. In the present time, it is widely used in population dynamics, chemical physics, biomedical modelling. The purpose of this book is to present the mathematical theory of reaction-diffusion equations in the context of their numerous applications. We will go from the general mathematical theory to specific equations and then to their applications. Existence, stability and bifurcations of solutions will be studied for bounded domains and in the case of travelling waves. The classical theory of reaction-diffusion equations and new topics such as nonlocal equations and multi-scale models in biology will be considered.

The Octagonal PETs

Author: Richard Evan Schwartz

Publisher: American Mathematical Soc.

ISBN: 1470415224

Category: Mathematics

Page: 212

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A polytope exchange transformation is a (discontinuous) map from a polytope to itself that is a translation wherever it is defined. The 1-dimensional examples, interval exchange transformations, have been studied fruitfully for many years and have deep connections to other areas of mathematics, such as Teichmüller theory. This book introduces a general method for constructing polytope exchange transformations in higher dimensions and then studies the simplest example of the construction in detail. The simplest case is a 1-parameter family of polygon exchange transformations that turns out to be closely related to outer billiards on semi-regular octagons. The 1-parameter family admits a complete renormalization scheme, and this structure allows for a fairly complete analysis both of the system and of outer billiards on semi-regular octagons. The material in this book was discovered through computer experimentation. On the other hand, the proofs are traditional, except for a few rigorous computer-assisted calculations.

Functional Inequalities: New Perspectives and New Applications

New Perspectives and New Applications

Author: Nassif Ghoussoub,Amir Moradifam

Publisher: American Mathematical Soc.

ISBN: 0821891529

Category: Mathematics

Page: 299

View: 9035

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"The book describes how functional inequalities are often manifestations of natural mathematical structures and physical phenomena, and how a few general principles validate large classes of analytic/geometric inequalities, old and new. This point of view leads to "systematic" approaches for proving the most basic inequalities, but also for improving them, and for devising new ones--sometimes at will and often on demand. These general principles also offer novel ways for estimating best constants and for deciding whether these are attained in appropriate function spaces. As such, improvements of Hardy and Hardy-Rellich type inequalities involving radially symmetric weights are variational manifestations of Sturm's theory on the oscillatory behavior of certain ordinary differential equations. On the other hand, most geometric inequalities, including those of Sobolev and Log-Sobolev type, are simply expressions of the convexity of certain free energy functionals along the geodesics on the Wasserstein manifold of probability measures equipped with the optimal mass transport metric. Caffarelli-Kohn-Nirenberg and Hardy-Rellich-Sobolev type inequalities are then obtained by interpolating the above two classes of inequalities via the classical ones of Hölder. The subtle Moser-Onofri-Aubin inequalities on the two-dimensional sphere are connected to Liouville type theorems for planar mean field equations."--Publisher's website.

Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems

Author: Gershon Kresin,V. G. Maz_i_a_

Publisher: American Mathematical Soc.

ISBN: 0821889818

Category: Mathematics

Page: 317

View: 3907

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The main goal of this book is to present results pertaining to various versions of the maximum principle for elliptic and parabolic systems of arbitrary order. In particular, the authors present necessary and sufficient conditions for validity of the classical maximum modulus principles for systems of second order and obtain sharp constants in inequalities of Miranda-Agmon type and in many other inequalities of a similar nature. Somewhat related to this topic are explicit formulas for the norms and the essential norms of boundary integral operators. The proofs are based on a unified approach using, on one hand, representations of the norms of matrix-valued integral operators whose target spaces are linear and finite dimensional, and, on the other hand, on solving certain finite dimensional optimization problems. This book reflects results obtained by the authors, and can be useful to research mathematicians and graduate students interested in partial differential equations.

Nonlinear Wave Methods for Charge Transport

Author: Luis L. Bonilla,Stephen W. Teitsworth

Publisher: John Wiley & Sons

ISBN: 9783527628681

Category: Science

Page: 280

View: 7859

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The present book introduces and develops mathematical techniques for the treatment of nonlinear waves and singular perturbation methods at a level that is suitable for graduate students, researchers and faculty throughout the natural sciences and engineering. The practice of implementing these techniques and their value are largely realized by showing their application to problems of nonlinear wave phenomena in electronic transport in solid state materials, especially bulk semiconductors and semiconductor superlattices. The authors are recognized leaders in this field, with more than 30 combined years of contributions.

Air Pollution Modeling and Its Application VI (Nato Challenges of Modern Society, Vol 11)

Author: Han Van Dop

Publisher: Springer

ISBN: 9780306428142

Category: Science

Page: 702

View: 3862

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Proceedings held Sept. 1988. The gradually changing concentration of trace gases in the global troposphere due to man's activity is becoming a matter of serious concern. The topics treated in this volume include: emission inventories for source and treatment in air pollution dispersion models; modelling of accidental releases; regional and global scale dispersion, including boundary layer-free troposphere exchange processes and subgrid scale parameterisations; model verification and policy implications; new developments in dispersion modelling and theory. Annotation copyrighted by Book News, Inc., Portland, OR