Book Description
Since the publication of "Spectral Methods in Fluid Dynamics", spectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. While retaining the tight integration between the theoretical and practical aspects of spectral methods that was the hallmark of the earlier book, Canuto et al. now incorporate the many improvements in the algorithms and the theory of spectral methods that have been made since 1988. The initial treatment Fundamentals in Single Domains discusses the fundamentals of the approximation of solutions to ordinary and partial differential equations on single domains by expansions in smooth, global basis functions. The first half of the book provides the algorithmic details of orthogonal expansions, transform methods, spectral discretization of differential equations plus their boundary conditions, and solution of the discretized equations by direct and iterative methods. The second half furnishes a comprehensive discussion of the mathematical theory of spectral methods on single domains, including approximation theory, stability and convergence, and illustrative applications of the theory to model boundary-value problems. Both the algorithmic and theoretical discussions cover spectral methods on tensor-product domains, triangles and tetrahedra. All chapters are enhanced with material on the Galerkin with numerical integration version of spectral methods. The discussion of direct and iterative solution methods is greatly expanded as are the set of numerical examples that illustrate the key properties of the various types of spectral approximations and the solution algorithms.
A companion book "Evolution to Complex Geometries and Applications to Fluid Dynamics" contains an extensive survey of the essential algorithmic and theoretical aspects of spectral methods for complex geometries and provides detailed discussions of spectral algorithms for fluid dynamics in simple and complex geometries.
Book Description
The theory of distributions is an extension of classical analysis, an area of particular importance in the field of linear partial differential equations. Underlying it is the theory of topological vector spaces, but it is possible to give a systematic presentation without a knowledge of this. The material in this book, based on graduate lectures given over a number of years requires few prerequisites but the treatment is rigorous throughout. From the outset, the theory is developed in several variables. It is taken as far as such important topics as Schwartz kernels, the Paley-Wiener-Schwartz theorem and Sobolev spaces. In this second edition, the notion of the wavefront set of a distribution is introduced. It allows many operations on distributions to be extended to larger classes and gives much more precise understanding of the nature of the singularities of a distribution. This is done in an elementary fashion without using any involved theories. This account will be useful to graduate students and research workers who are interested in the applications of analysis in mathematics and mathematical physics.
Customer Reviews:
An Excellent Introduction.......2000-11-02
Every physicist and mathematician uses distributions (sometimes called generalized functions), albeit often unknowingly. From its origins in the Dirac delta `function', distribution theory continues to influence many research areas from quantum mechanics to partial differential equations, but has also grown into an important field in its own right. For anyone interested in learning about the field, this is clearly the first port of call. It presents a balanced introduction to the subject on a level suitable for anyone with a basic grounding in analysis (no knowledge of functional analysis is required).
The book begins by defining the two building blocks of the theory---test functions and distributions. It then quickly expands, filling in the important details of differentiation, multiplication, tensor products and convolution. All of this is written with sufficient mathematical rigor, but never too much that it interferes with the basic understanding of the subject, and is supported throughout by useful exercises. The book then builds up the theory of Fourier and Laplace transforms of distributions, which has important applications in the study of linear partial differential equations. The second edition contains an indispensable new chapter on the calculus of wavefront sets, which, among its uses, allows the propagation of singularities of solutions to partial differential equations to be properly treated. All in all, while the book is not for the common man, and does require a certain level of mathematical maturity, it does present an excellent introduction to an important, and often poorly understood, area of mathematics.
Book Description
One of the most fundamental and active areas in mathematics, the theory of partial differential equations (PDEs) is essential in the modeling of natural phenomena. PDEs have a wide range of interesting and important applications in every branch of applied mathematics, physics, and engineering, including fluid dynamics, elasticity, and optics.
This significantly expanded fourth edition is designed as an introduction to the theory and applications of linear PDEs. The authors provide fundamental concepts, underlying principles, a wide range of applications, and various methods of solutions to PDEs. In addition to essential standard material on the subject, the book contains new material that is not usually covered in similar texts and reference books, including conservation laws, the spherical wave equation, the cylindrical wave equation, higher-dimensional boundary-value problems, the finite element method, fractional partial differential equations, and nonlinear partial differential equations with applications.
Key features include:
* Applications to a wide variety of physical problems in numerous interdisciplinary areas
* Over 900 worked examples and exercises dealing with problems in fluid mechanics, gas dynamics, optics, plasma physics, elasticity, biology, and chemistry
* Historical comments on partial differential equations
* Solutions and hints to selected exercises
* A comprehensive bibliography—comprised of many standard texts and reference books, as well as a set of selected classic and recent papers—for readers interested in learning more about the modern treatment of the subject
Linear Partial Differential Equations for Scientists and Engineers, Fourth Edition will primarily serve as a textbook for the first two courses in PDEs, or in a course on advanced engineering mathematics. The book may also be used as a reference for graduate students, researchers, and professionals in modern applied mathematics, mathematical physics, and engineering. Readers will gain a solid mathematical background in PDEs, sufficient to start interdisciplinary collaborative research in a variety of fields.
Also by L. Debnath:
Nonlinear Partial Differential Equations for Scientists and Engineers, Second Edition, ISBN 0-8176-4323-0.
Customer Reviews:
Aimed at the Person who needs to do Real Work.......2007-04-05
Partial Differential equations are really the first time that the science or entineering student begins to see that math can really represent real life situations that are more complex than the almost trivial examples he has studied up until now.
This book, now in its fourth edition reflects comments made by users: students, faculty and researchers as well as covering the major new discoveries of methods for the solution of partial differential equations in the years since the last edition. The book is aimed at the advanced undergraduate or beginning graduate student, and it should be useful as a research reference for professionals in mathematics, science, engineering and the other applied sciences. In addition other fields of study from the social sciented to financial analysis have begin to use more advanced mathematics in their calculations.
The intent in this book is to provide the working physicist/engineer or whatever with the tools he needs to perform his work. In a few places this has meant a slight slacking of the rigor normally required by the pure mathematician in favor of producing solutions that give the user the answers he needs.
Book Description
This book is intended for students who wish to get an introduction to the theory of partial differential equations. The author focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view towards nonlinear problems. These are maximum principle methods (particularly important for numerical analysis schemes), parabolic equations, variational methods, and continuity methods. This book also develops the main methods for obtaining estimates for solutions of elliptic equations: Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. Connections between elliptic, parabolic, and hyperbolic equations are explored, as well as the connection with Brownian motion and semigroups. This book can be utilized for a one-year course on partial differential equations.
For the new edition the author has added a new chapter on reaction-diffusion equations and systems. There is also new material on Neumann boundary value problems, Poincaré inequalities, expansions, as well as a new proof of the Hölder regularity of solutions of the Poisson equation.
Jürgen Jost is Co-Director of the Max Planck Institute for Mathematics in the Sciences and Professor of Mathematics at the University of Leipzig. He is the author of a number of Springer books, including Dynamical Systems (2005), Postmodern Analysis (3rd ed. 2005, also translated into Japanese), Compact Riemann Surfaces (3rd ed. 2006) and Riemannian Geometry and Geometric Analysis (4th ed., 2005). The present book is an expanded translation of the original German version, Partielle Differentialgleichungen (1998).
Customer Reviews:
Elliptic PDE's done right.......2007-06-16
I'm no specialist in PDE's but I boldly would like to review the present book.
This book is intended(in my personal view) as a in-depth-but-not-pedantic introduction to Elliptic equations (which, if one considers the original title in german "Partielle Differentialgleichungen - Elliptische Gleichungen" makes complete sense).
As any other descent book, it doesn't aim at list down results in a encyclopedic way, but rather tour-guide the reader in the beautiful subject of Elliptic PDE's.
Motivated by Dirichlet's principle, the author introduces variational methods for Elliptic PDE's and a good deal of the regularity program for that class of equations(including non-linear equations). Most of what follows the first chapter, is a successful attempt to generalize to other elliptic equations the techniques useful in the qualitative theory of the Dirichlet problem for the Poisson equation. Some of the methods, as far as I understand them, can be generalized to Schroedinger's equation too.
The interested reader can consult also the book on analysis by Elliot Lieb and the textbook on variational methods by Prof. S. Hildebrandt(and references therein).
Average customer rating:
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Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (Ergebnisse der Mathematik und ihrer Grenzgebiete. ... / A Series of Modern Surveys in Mathematics)
Michael Struwe
Manufacturer: Springer
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Calculus of Variations (Cambridge Studies in Advanced Mathematics)
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Functional Analysis (Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs and Tracts)
ASIN: 3540664793 |
Book Description
Hilbert's talk at the second International Congress of 1900 in Paris marked the beginning of a new era in the calculus of variations. A development began which, within a few decades, brought tremendous success, highlighted by the 1929 theorem of Ljusternik and Schnirelman on the existence of three distinct prime closed geodesics on any compact surface of genus zero, and the 1930/31 solution of Plateau's problem by Douglas and Radó. The book gives a concise introduction to variational methods and presents an overview of areas of current research in the field. The third edition gives a survey on new developments in the field. References have been updated and a small number of mistakes have been rectified.
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Vortices in the Magnetic Ginzburg-Landau Model (Progress in Nonlinear Differential Equations and Their Applications)
Etienne Sandier , and
Sylvia Serfaty
Manufacturer: Birkhäuser Boston
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ASIN: 0817643168 |
Book Description
With the discovery of type-II superconductivity by Abrikosov, the prediction of vortex lattices, and their experimental observation, quantized vortices have become a central object of study in superconductivity, superfluidity, and Bose--Einstein condensation. This book presents the mathematics of superconducting vortices in the framework of the acclaimed two-dimensional Ginzburg-Landau model, with or without magnetic field, and in the limit of a large Ginzburg-Landau parameter, kappa.
This text presents complete and mathematically rigorous versions of both results either already known by physicists or applied mathematicians, or entirely new. It begins by introducing mathematical tools such as the vortex balls construction and Jacobian estimates. Among the applications presented are: the determination of the vortex densities and vortex locations for energy minimizers in a wide range of regimes of applied fields, the precise expansion of the so-called first critical field in a bounded domain, the existence of branches of solutions with given numbers of vortices, and the derivation of a criticality condition for vortex densities of non-minimizing solutions. Thus, this book retraces in an almost entirely self-contained way many results that are scattered in series of articles, while containing a number of previously unpublished results as well.
The book also provides a list of open problems and a guide to the increasingly diverse mathematical literature on Ginzburg--Landau related topics. It will benefit both pure and applied mathematicians, physicists, and graduate students having either an introductory or an advanced knowledge of the subject.
Book Description
In modern theoretical physics, gauge field theories are of great importance since they keep internal symmetries and account for phenomena such as spontaneous symmetry breaking, the quantum Hall effect, charge fractionalization, superconductivity and supergravity. This monograph discusses specific examples of self-dual gauge field structures, including the Chern--Simons model, the abelian--Higgs model, and Yang--Mills gauge field theory.
The author builds a foundation for gauge theory and self-dual vortices by introducing the basic mathematical language of gauge theory and formulating examples of Chern-Simons-Higgs theories (in both abelian and non-abelian settings). Thereafter, the Electroweak theory and self-gravitating Electroweak strings are examined. The final chapters treat elliptic problems involving Chern—Simmons models, concentration-compactness principles, and Maxwell—Chern—Simons vortices.
Many open questions still remain in the field and are examined in this work in connection with Liouville-type equations and systems. The goal of this text is to form an understanding of self-dual solutions arising in a variety of physical contexts and thus is ideal for graduate students and researchers interested in partial differential equations and mathematical physics.
Book Description
This book is the second part of the text Differential Equations: A Dynamical Systems Approach written by John Hubbard and Beverly West. It is a continuation of the subject matter discussed in the first book, with an emphasis on systems of ordinary differential equations. This book will be most appropriate for upper level undergraduate and graduate students in the fields of mathematics, engineering, applied mathematics, as well as in the life sciences, physics, and economics. This book opens with an introduction, and follows with chapters on systems of differential equations, systems of linear differential equations, and systems of nonlinear differential equations. The book continues with structural stability, bifurcations, and an appendix on linear algebra. The authors also include an appendix containing important theorems from parts I and II, as well as answers to selected problems.
Customer Reviews:
Excellent book!.......1999-10-30
Very clear and intuitive upper-undergrad/graduate level text. I highly recomend it.
Book Description
Written for students of mathematics and the physical sciences, this superb treatment offers modern mathematical techniques for setting up and analyzing problems. Topics include elementary modeling, partial differential equations of the 1st order, potential theory, parabolic equations, much more. Prerequisites are a course in advanced calculus and basic knowledge of matrix methods.
Customer Reviews:
a classic text in applied mathematics.......2007-09-01
This is one of the finest textbooks on applied mathematics. It is appropriate for a wide range of levels and disciplines. A moderately paced graduate class can finish most of the chapters in a full year. The first two chapters cover material that is difficult to find elsewhere and nowhere is it better handled. The real boon to this book is the constant integration of physical reasoning with solid mathematical theory. To be sure, many of the more complicated ideas are only mentioned, but the later chapters pick these up and go much further. It covers basic heat and wave applications, but also Brownian motion and gas dynamics. A solid calculus background is needed, which includes vector calculus and convergence issues. Eigenvalues and some functional analysis are needed. Many books on elementary partial differential equations spend all their time on the big three: heat, wave and Laplace. This is too low a level for any graduate class and doesn't create the excitement other applications provide. On the other hand, this book is accessible to many people. Often, graduate texts are thoroughly awash with Sobolev spaces and weak convergence and the physical problem, i.e., the actual model, is all but ignored. This classic text is the bridge between elementary and advanced applied mathematics. In some respects, it is self-contained. Guenther and Lee have written an indispensable book.
Partial Differential Equations of Mathematical Physics & Integral Equations.......2007-01-16
Book came in excellent condition and was shipped quickly. Highly recommend this seller.
I agree with the reader from CAL.......2003-03-26
The book skip too many steps and most of the time, the equation is presented without explanation...Don't waste money.
Book that dosn't make sense.......1999-09-07
This book is one of the worse I ever see representing the mathematics for physics, the mathematics and the physics both was not explained as it should, the authors are not familiar at all with the concept of physics make their material in the book related to this subject unclear and vague. The book has too many gaps as I see it has too many jumps when a physical phenomena represented using mathematics. Physical concept was most of the times not even mention, and I believe that the author should stick only with mathematics since they try to explain physics they don't really familiar with.
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