Elliptic Partial Differential Equations of Second Order

Author: David Gilbarg,Neil S. Trudinger

Publisher: Springer

ISBN: 3642617980

Category: Mathematics

Page: 518

View: 6383

From the reviews: "This is a book of interest to any having to work with differential equations, either as a reference or as a book to learn from. The authors have taken trouble to make the treatment self-contained. It (is) suitable required reading for a PhD student." --New Zealand Mathematical Society, 1985
Posted in Mathematics

Elliptic Partial Differential Equations

Author: Qing Han,Fanghua Lin

Publisher: American Mathematical Soc.

ISBN: 0821853139

Category: Mathematics

Page: 147

View: 9273

Elliptic Partial Differential Equations by Qing Han and FangHua Lin is one of the best textbooks I know. It is the perfect introduction to PDE. In 150 pages or so it covers an amazing amount of wonderful and extraordinary useful material. I have used it as a textbook at both graduate and undergraduate levels which is possible since it only requires very little background material yet it covers an enormous amount of material. In my opinion it is a must read for all interested in analysis and geometry, and for all of my own PhD students it is indeed just that. I cannot say enough good things about it--it is a wonderful book. --Tobias Colding This volume is based on PDE courses given by the authors at the Courant Institute and at the University of Notre Dame, Indiana. Presented are basic methods for obtaining various a priori estimates for second-order equations of elliptic type with particular emphasis on maximal principles, Harnack inequalities, and their applications. The equations considered in the book are linear; however, the presented methods also apply to nonlinear problems. This second edition has been thoroughly revised and in a new chapter the authors discuss several methods for proving the existence of solutions of primarily the Dirichlet problem for various types of elliptic equations.
Posted in Mathematics

Partial Differential Equations of Parabolic Type

Author: Avner Friedman

Publisher: Courier Corporation

ISBN: 0486318265

Category: Mathematics

Page: 368

View: 6094

With this book, even readers unfamiliar with the field can acquire sufficient background to understand research literature related to the theory of parabolic and elliptic equations. 1964 edition.
Posted in Mathematics

Stable Solutions of Elliptic Partial Differential Equations

Author: Louis Dupaigne

Publisher: CRC Press

ISBN: 1420066552

Category: Mathematics

Page: 335

View: 8899

Stable solutions are ubiquitous in differential equations. They represent meaningful solutions from a physical point of view and appear in many applications, including mathematical physics (combustion, phase transition theory) and geometry (minimal surfaces). Stable Solutions of Elliptic Partial Differential Equations offers a self-contained presentation of the notion of stability in elliptic partial differential equations (PDEs). The central questions of regularity and classification of stable solutions are treated at length. Specialists will find a summary of the most recent developments of the theory, such as nonlocal and higher-order equations. For beginners, the book walks you through the fine versions of the maximum principle, the standard regularity theory for linear elliptic equations, and the fundamental functional inequalities commonly used in this field. The text also includes two additional topics: the inverse-square potential and some background material on submanifolds of Euclidean space.
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Elliptic Partial Differential Equations

Existence and Regularity of Distributional Solutions

Author: Lucio Boccardo,Gisella Croce

Publisher: Walter de Gruyter

ISBN: 3110315424

Category: Mathematics

Page: 201

View: 1166

Elliptic partial differential equations is one of the main and most active areas in mathematics. In our book we study linear and nonlinear elliptic problems in divergence form, with the aim of providing classical results, as well as more recent developments about distributional solutions. For this reason the book is addressed to master's students, PhD students and anyone who wants to begin research in this mathematical field.
Posted in Mathematics

Functional Spaces for the Theory of Elliptic Partial Differential Equations

Author: Françoise Demengel,Gilbert Demengel

Publisher: Springer Science & Business Media

ISBN: 1447128079

Category: Mathematics

Page: 465

View: 6815

The theory of elliptic boundary problems is fundamental in analysis and the role of spaces of weakly differentiable functions (also called Sobolev spaces) is essential in this theory as a tool for analysing the regularity of the solutions. This book offers on the one hand a complete theory of Sobolev spaces, which are of fundamental importance for elliptic linear and non-linear differential equations, and explains on the other hand how the abstract methods of convex analysis can be combined with this theory to produce existence results for the solutions of non-linear elliptic boundary problems. The book also considers other kinds of functional spaces which are useful for treating variational problems such as the minimal surface problem. The main purpose of the book is to provide a tool for graduate and postgraduate students interested in partial differential equations, as well as a useful reference for researchers active in the field. Prerequisites include a knowledge of classical analysis, differential calculus, Banach and Hilbert spaces, integration and the related standard functional spaces, as well as the Fourier transformation on the Schwartz space. There are complete and detailed proofs of almost all the results announced and, in some cases, more than one proof is provided in order to highlight different features of the result. Each chapter concludes with a range of exercises of varying levels of difficulty, with hints to solutions provided for many of them.
Posted in Mathematics

An Introduction to Second Order Partial Differential Equations

Classical and Variational Solutions

Author: N.A

Publisher: World Scientific Publishing Company

ISBN: 9813229195

Category: Mathematics

Page: 300

View: 1047

The book extensively introduces classical and variational partial differential equations (PDEs) to graduate and post-graduate students in Mathematics. The topics, even the most delicate, are presented in a detailed way. The book consists of two parts which focus on second order linear PDEs. Part I gives an overview of classical PDEs, that is, equations which admit strong solutions, verifying the equations pointwise. Classical solutions of the Laplace, heat, and wave equations are provided. Part II deals with variational PDEs, where weak (variational) solutions are considered. They are defined by variational formulations of the equations, based on Sobolev spaces. A comprehensive and detailed presentation of these spaces is given. Examples of variational elliptic, parabolic, and hyperbolic problems with different boundary conditions are discussed.
Posted in Mathematics

Partial Differential Equations

Author: Lawrence C. Evans

Publisher: American Mathematical Soc.

ISBN: 0821849743

Category: Mathematics

Page: 749

View: 4789

This is the second edition of the now definitive text on partial differential equations (PDE). It offers a comprehensive survey of modern techniques in the theoretical study of PDE with particular emphasis on nonlinear equations. Its wide scope and clear exposition make it a great text for a graduate course in PDE. For this edition, the author has made numerous changes, including a new chapter on nonlinear wave equations, more than 80 new exercises, several new sections, a significantly expanded bibliography. About the First Edition: I have used this book for both regular PDE and topics courses. It has a wonderful combination of insight and technical detail. ... Evans' book is evidence of his mastering of the field and the clarity of presentation. --Luis Caffarelli, University of Texas It is fun to teach from Evans' book. It explains many of the essential ideas and techniques of partial differential equations ... Every graduate student in analysis should read it. --David Jerison, MIT I use Partial Differential Equations to prepare my students for their Topic exam, which is a requirement before starting working on their dissertation. The book provides an excellent account of PDE's ... I am very happy with the preparation it provides my students. --Carlos Kenig, University of Chicago Evans' book has already attained the status of a classic. It is a clear choice for students just learning the subject, as well as for experts who wish to broaden their knowledge ... An outstanding reference for many aspects of the field. --Rafe Mazzeo, Stanford University
Posted in Mathematics

Partial Differential Equations of Applied Mathematics

Author: Erich Zauderer

Publisher: Wiley-Interscience

ISBN: 9780471315162

Category: Mathematics

Page: 894

View: 5388

The only comprehensive guide to modeling, characterizing, and solving partial differential equations This classic text by Erich Zauderer provides a comprehensive account of partial differential equations and their applications. Dr. Zauderer develops mathematical models that give rise to partial differential equations and describes classical and modern solution techniques. With an emphasis on practical applications, he makes liberal use of real-world examples, explores both linear and nonlinear problems, and provides approximate as well as exact solutions. He also describes approximation methods for simplifying complicated solutions and for solving linear and nonlinear problems not readily solved by standard methods. 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 continues in a less statistical vein to cover an exceptionally broad range of topics, including stabilities, singularities, transform methods, the use of Green's functions, and perturbation and asymptotic treatments. Features that set Partial Differential Equations of Applied Mathematics, Second Edition above all other texts in the field include: Coverage of random walk problems, discontinuous and singular solutions, and perturbation and asymptotic methods More than 800 practice exercises, many of which are fully worked out Numerous up-to-date examples from engineering and the physical sciences Partial Differential Equations of Applied Mathematics, Second Edition is a superior advanced-undergraduate to graduate-level text for students in engineering, the sciences, and applied mathematics. The title is also a valuable working resource for professionals in these fields. Dr. Zauderer received his doctorate in mathematics from the New York University-Courant Institute. Prior to joining the staff of Polytechnic University, he was a Senior Weitzmann Fellow of the Weitzmann Institute of Science in Rehovot, Israel.
Posted in Mathematics

Finite Difference Methods for Ordinary and Partial Differential Equations

Steady-State and Time-Dependent Problems

Author: Randall J. LeVeque

Publisher: SIAM

ISBN: 9780898717839

Category: Differential equations

Page: 339

View: 860

This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.
Posted in Differential equations

Partial Differential Equations

Author: Jürgen Jost

Publisher: Springer Science & Business Media

ISBN: 1461448093

Category: Mathematics

Page: 410

View: 832

This book offers an ideal graduate-level introduction to the theory of partial differential equations. The first part of the book describes the basic mathematical problems and structures associated with elliptic, parabolic, and hyperbolic partial differential equations, and explores the connections between these fundamental types. Aspects of Brownian motion or pattern formation processes are also presented. The second part focuses on existence schemes and develops estimates for solutions of elliptic equations, such as Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. In particular, the reader will learn the basic techniques underlying current research in elliptic partial differential equations. This revised and expanded third edition is enhanced with many additional examples that will help motivate the reader. New features include a reorganized and extended chapter on hyperbolic equations, as well as a new chapter on the relations between different types of partial differential equations, including first-order hyperbolic systems, Langevin and Fokker-Planck equations, viscosity solutions for elliptic PDEs, and much more. Also, the new edition contains additional material on systems of elliptic partial differential equations, and it explains in more detail how the Harnack inequality can be used for the regularity of solutions.
Posted in Mathematics

Multiple Integrals in the Calculus of Variations

Author: Charles Bradfield Morrey Jr.

Publisher: Springer Science & Business Media

ISBN: 9783540699521

Category: Mathematics

Page: 506

View: 2895

From the reviews: "...the book contains a wealth of material essential to the researcher concerned with multiple integral variational problems and with elliptic partial differential equations. The book not only reports the researches of the author but also the contributions of his contemporaries in the same and related fields. The book undoubtedly will become a standard reference for researchers in these areas. ...The book is addressed mainly to mature mathematical analysts. However, any student of analysis will be greatly rewarded by a careful study of this book." M. R. Hestenes in Journal of Optimization Theory and Applications "The work intertwines in masterly fashion results of classical analysis, topology, and the theory of manifolds and thus presents a comprehensive treatise of the theory of multiple integral variational problems." L. Schmetterer in Monatshefte für Mathematik "The book is very clearly exposed and contains the last modern theory in this domain. A comprehensive bibliography ends the book." M. Coroi-Nedeleu in Revue Roumaine de Mathématiques Pures et Appliquées
Posted in Mathematics

Partial Differential Equations of Mathematical Physics

Author: S. L. Sobolev

Publisher: Courier Corporation

ISBN: 9780486659640

Category: Science

Page: 427

View: 4788

This volume presents an unusually accessible introduction to equations fundamental to the investigation of waves, heat conduction, hydrodynamics, and other physical problems. Topics include derivation of fundamental equations, Riemann method, equation of heat conduction, theory of integral equations, Green's function, and much more. The only prerequisite is a familiarity with elementary analysis. 1964 edition.
Posted in Science

The Finite Element Method for Elliptic Problems

Author: Philippe G. Ciarlet

Publisher: SIAM

ISBN: 0898715148

Category: Mathematics

Page: 530

View: 5079

This is the only book available that fully analyzes the mathematical foundations of the finite element method. Not only is it valuable reference and introduction to current research, it is also a working textbook for graduate courses in numerical analysis, including useful figures and exercises of varying difficulty.
Posted in Mathematics

Handbook of Linear Partial Differential Equations for Engineers and Scientists, Second Edition

Author: Andrei D. Polyanin,Vladimir E. Nazaikinskii

Publisher: CRC Press

ISBN: 1466581492

Category: Mathematics

Page: 1609

View: 2984

Includes nearly 4,000 linear partial differential equations (PDEs) with solutions Presents solutions of numerous problems relevant to heat and mass transfer, wave theory, hydrodynamics, aerodynamics, elasticity, acoustics, electrodynamics, diffraction theory, quantum mechanics, chemical engineering sciences, electrical engineering, and other fields Outlines basic methods for solving various problems in science and engineering Contains much more linear equations, problems, and solutions than any other book currently available Provides a database of test problems for numerical and approximate analytical methods for solving linear PDEs and systems of coupled PDEs New to the Second Edition More than 700 pages with 1,500+ new first-, second-, third-, fourth-, and higher-order linear equations with solutions Systems of coupled PDEs with solutions Some analytical methods, including decomposition methods and their applications Symbolic and numerical methods for solving linear PDEs with Maple, Mathematica, and MATLAB® Many new problems, illustrative examples, tables, and figures To accommodate different mathematical backgrounds, the authors avoid wherever possible the use of special terminology, outline some of the methods in a schematic, simplified manner, and arrange the material in increasing order of complexity.
Posted in Mathematics

Partial Differential Equations in Action

From Modelling to Theory

Author: Sandro Salsa

Publisher: Springer

ISBN: 3319150936

Category: Mathematics

Page: 701

View: 3600

The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering. It has evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background in numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In turn the second part, chapters 6 to 11, concentrates on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear boundary and initial-boundary value problems.
Posted in Mathematics

A First Course in Partial Differential Equations

with Complex Variables and Transform Methods

Author: H. F. Weinberger

Publisher: Courier Corporation

ISBN: 0486132048

Category: Mathematics

Page: 480

View: 2753

Suitable for advanced undergraduate and graduate students, this text presents the general properties of partial differential equations, including the elementary theory of complex variables. Solutions. 1965 edition.
Posted in Mathematics

From Classical to Modern Analysis

Author: Rinaldo B. Schinazi

Publisher: Springer

ISBN: 3319945831

Category:

Page: N.A

View: 8061

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Numerical Solution of Partial Differential Equations by the Finite Element Method

Author: Claes Johnson

Publisher: Courier Corporation

ISBN: 0486131599

Category: Mathematics

Page: 288

View: 8357

An accessible introduction to the finite element method for solving numeric problems, this volume offers the keys to an important technique in computational mathematics. Suitable for advanced undergraduate and graduate courses, it outlines clear connections with applications and considers numerous examples from a variety of science- and engineering-related specialties.This text encompasses all varieties of the basic linear partial differential equations, including elliptic, parabolic and hyperbolic problems, as well as stationary and time-dependent problems. Additional topics include finite element methods for integral equations, an introduction to nonlinear problems, and considerations of unique developments of finite element techniques related to parabolic problems, including methods for automatic time step control. The relevant mathematics are expressed in non-technical terms whenever possible, in the interests of keeping the treatment accessible to a majority of students.
Posted in Mathematics

Lectures on Nonlinear Hyperbolic Differential Equations

Author: Lars Hörmander

Publisher: Springer Science & Business Media

ISBN: 9783540629214

Category: Mathematics

Page: 289

View: 4363

In this introductory textbook, a revised and extended version of well-known lectures by L. Hörmander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail.
Posted in Mathematics