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Author: Hideo Kozono
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 430
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Book Description
This volume is the proceedings of the 14th MSJ International Research Institute "Asymptotic Analysis and Singularity", which was held at Sendai, Japan in July 2005. The proceedings contain survey papers and original research papers on nonlinear partial differential equations, dynamical systems, calculus of variations and mathematical physics.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America
Author: Hideo Kozono
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 430
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Book Description
This volume is the proceedings of the 14th MSJ International Research Institute "Asymptotic Analysis and Singularity", which was held at Sendai, Japan in July 2005. The proceedings contain survey papers and original research papers on nonlinear partial differential equations, dynamical systems, calculus of variations and mathematical physics.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America
Author: Hideo Kozono
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 416
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Book Description
This volume is the proceedings of the 14th MSJ International Research Institute "Asymptotic Analysis and Singularity", which was held at Sendai, Japan in July 2005. The proceedings contain survey papers and original research papers on nonlinear partial differential equations, dynamical systems, calculus of variations and mathematical physics.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America
Author: Takashi Suzuki
Publisher: Springer
ISBN: 9462391548
Category : Mathematics
Languages : en
Pages : 450
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Book Description
Mean field approximation has been adopted to describe macroscopic phenomena from microscopic overviews. It is still in progress; fluid mechanics, gauge theory, plasma physics, quantum chemistry, mathematical oncology, non-equilibirum thermodynamics. spite of such a wide range of scientific areas that are concerned with the mean field theory, a unified study of its mathematical structure has not been discussed explicitly in the open literature. The benefit of this point of view on nonlinear problems should have significant impact on future research, as will be seen from the underlying features of self-assembly or bottom-up self-organization which is to be illustrated in a unified way. The aim of this book is to formulate the variational and hierarchical aspects of the equations that arise in the mean field theory from macroscopic profiles to microscopic principles, from dynamics to equilibrium, and from biological models to models that arise from chemistry and physics.
Author: Takashi Suzuki
Publisher: Springer Science & Business Media
ISBN: 9491216228
Category : Mathematics
Languages : en
Pages : 299
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Book Description
A mathematical theory is introduced in this book to unify a large class of nonlinear partial differential equation (PDE) models for better understanding and analysis of the physical and biological phenomena they represent. The so-called mean field approximation approach is adopted to describe the macroscopic phenomena from certain microscopic principles for this unified mathematical formulation. Two key ingredients for this approach are the notions of “duality” according to the PDE weak solutions and “hierarchy” for revealing the details of the otherwise hidden secrets, such as physical mystery hidden between particle density and field concentration, quantized blow up biological mechanism sealed in chemotaxis systems, as well as multi-scale mathematical explanations of the Smoluchowski–Poisson model in non-equilibrium thermodynamics, two-dimensional turbulence theory, self-dual gauge theory, and so forth. This book shows how and why many different nonlinear problems are inter-connected in terms of the properties of duality and scaling, and the way to analyze them mathematically.
Author: Hans G. Kaper
Publisher: CRC Press
ISBN: 1482277069
Category : Mathematics
Languages : en
Pages : 283
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Book Description
Integrates two fields generally held to be incompatible, if not downright antithetical, in 16 lectures from a February 1990 workshop at the Argonne National Laboratory, Illinois. The topics, of interest to industrial and applied mathematicians, analysts, and computer scientists, include singular per
Author: Erich Zauderer
Publisher: John Wiley & Sons
ISBN: 1118031407
Category : Mathematics
Languages : en
Pages : 968
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Book Description
This new edition features the latest tools for modeling, characterizing, and solving partial differential equations The Third Edition of this classic text offers a comprehensive guide to modeling, characterizing, and solving partial differential equations (PDEs). The author provides all the theory and tools necessary to solve problems via exact, approximate, and numerical methods. The Third Edition retains all the hallmarks of its previous editions, including an emphasis on practical applications, clear writing style and logical organization, and extensive use of real-world examples. Among the new and revised material, the book features: * A new section at the end of each original chapter, exhibiting the use of specially constructed Maple procedures that solve PDEs via many of the methods presented in the chapters. The results can be evaluated numerically or displayed graphically. * Two new chapters that present finite difference and finite element methods for the solution of PDEs. Newly constructed Maple procedures are provided and used to carry out each of these methods. All the numerical results can be displayed graphically. * A related FTP site that includes all the Maple code used in the text. * New exercises in each chapter, and answers to many of the exercises are provided via the FTP site. A supplementary Instructor's Solutions Manual is available. The book begins with a demonstration of how the three basic types of equations-parabolic, hyperbolic, and elliptic-can be derived from random walk models. It then covers an exceptionally broad range of topics, including questions of stability, analysis of singularities, transform methods, Green's functions, and perturbation and asymptotic treatments. Approximation methods for simplifying complicated problems and solutions are described, and linear and nonlinear problems not easily solved by standard methods are examined in depth. Examples from the fields of engineering and physical sciences are used liberally throughout the text to help illustrate how theory and techniques are applied to actual problems. With its extensive use of examples and exercises, this text is recommended for advanced undergraduates and graduate students in engineering, science, and applied mathematics, as well as professionals in any of these fields. It is possible to use the text, as in the past, without use of the new Maple material.
Author: L. S. Frank
Publisher:
ISBN:
Category :
Languages : en
Pages : 58
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Book Description
Some classes of Non-linear Second Order Elliptic and Parabolic Partial Differential Operators affected by the presence of a small parameter epsilon are investigated. The reduced problem (epsilon = 0) is characterized by the appearence of a free boundary of the solutions. The Existence, Uniqueness and regularity results are established for both perturbed and reduced problems. Sharp two-sided estimates for the difference of the solutions of the perturbed and reduced problems are proved and some constructive procedures are found out for localizing and computing the free boundary of the reduced problem. The Kinetic Theory of membranes with enzymotic activity is one of the possible fields of applications of the results established, the small parameter being the so-called Michaelis' coefficient. Additional keywords: Netherlands; asymptotics; calculus of variations; convergence; Cauchy problem. (Author).
Author: Alain Bensoussan
Publisher: American Mathematical Soc.
ISBN: 0821853244
Category : Mathematics
Languages : en
Pages : 410
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Book Description
This is a reprinting of a book originally published in 1978. At that time it was the first book on the subject of homogenization, which is the asymptotic analysis of partial differential equations with rapidly oscillating coefficients, and as such it sets the stage for what problems to consider and what methods to use, including probabilistic methods. At the time the book was written the use of asymptotic expansions with multiple scales was new, especially their use as a theoretical tool, combined with energy methods and the construction of test functions for analysis with weak convergence methods. Before this book, multiple scale methods were primarily used for non-linear oscillation problems in the applied mathematics community, not for analyzing spatial oscillations as in homogenization. In the current printing a number of minor corrections have been made, and the bibliography was significantly expanded to include some of the most important recent references. This book gives systematic introduction of multiple scale methods for partial differential equations, including their original use for rigorous mathematical analysis in elliptic, parabolic, and hyperbolic problems, and with the use of probabilistic methods when appropriate. The book continues to be interesting and useful to readers of different backgrounds, both from pure and applied mathematics, because of its informal style of introducing the multiple scale methodology and the detailed proofs.
Author: Joachim Escher
Publisher: Springer
ISBN: 3319125478
Category : Mathematics
Languages : en
Pages : 295
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Book Description
The international workshop on which this proceedings volume is based on brought together leading researchers in the field of elliptic and parabolic equations. Particular emphasis was put on the interaction between well-established scientists and emerging young mathematicians, as well as on exploring new connections between pure and applied mathematics. The volume contains material derived after the workshop taking up the impetus to continue collaboration and to incorporate additional new results and insights.
Author: Oleg Mikitchenko
Publisher:
ISBN:
Category :
Languages : en
Pages : 196
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Book Description
Abstract: The method of resolution of singularities was established in the 17th Century by Newton for finding expansions of solutions of algebraic equations. In this method one uses a polygon in the plane of powers of the variables that appear in the original equation and which is now known as the Newton polygon. Recently, the idea of resolution of singularities was extended by A.D. Bruno into a group of methods, known as Power Geometry, that allows one to compute asymptotics of solutions to ordinary and partial differential equations. In this thesis we show how to solve some problems in asymptotic analysis of differential equations using the resolution of singularities and the power geometry. First, we show how these methods are applied to the linear Airy's equation, doing it in two different ways: by applying the methods directly to the equation and by applying the methods to the autonomous system of ODEs to which the equation can be transformed. This analysis is extended to a larger class of classical second order linear equations with nonautonomous coefficients. Second, we consider a non-linear first order equation for which the origin is an essential singularity, and obtain leading order asymptotic approximations to the solutions in different sectors near the origin, and compare them with numerical solutions. By imposing conditions on the sector boundaries, we obtain approximations of the solutions in the full neighborhood of the origin. Third, we show how the method can be applied to singularly perturbed boundary value problems. We show that the Newton polygon allows us to compute the correct rescaling (or rescalings) of the independent variable as well as to determine the dominant terms of the equation corresponding to this rescaling. We also show how the procedure of matching of inner and outer expansions for such problems can be illustrated by means of the Newton's polygon associated with the equation. We also present a collection of algorithms implemented as a package for the Maple computer algebra system that can be used when one applies power geometry to finding asymptotic expansions of solutions.
