A Basic Course in Partial Differential Equations

Author: Qing Han

Publisher: American Mathematical Soc.

ISBN: 0821852558

Category: Mathematics

Page: 293

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This is a textbook for an introductory graduate course on partial differential equations. Han focuses on linear equations of first and second order. An important feature of his treatment is that the majority of the techniques are applicable more generally. In particular, Han emphasizes a priori estimates throughout the text, even for those equations that can be solved explicitly. Such estimates are indispensable tools for proving the existence and uniqueness of solutions to PDEs, being especially important for nonlinear equations. The estimates are also crucial to establishing properties of the solutions, such as the continuous dependence on parameters. Han's book is suitable for students interested in the mathematical theory of partial differential equations, either as an overview of the subject or as an introduction leading to further study.
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Partial Differential Equations

Author: Lawrence C. Evans

Publisher: American Mathematical Soc.

ISBN: 0821849743

Category: Mathematics

Page: 749

View: 3257

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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
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Partial Differential Equations I

Basic Theory

Author: Michael Eugene Taylor,Eberhard Zeidler

Publisher: Springer Science & Business Media

ISBN: 9780387946535

Category: Mathematics

Page: 563

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This book is intended to be a comprehensive introduction to the subject of partial differential equations. It should be useful to graduate students at all levels beyond that of a basic course in measure theory. It should also be of interest to professional mathematicians in analysis, mathematical physics, and differential geometry. This work will be divided into three volumes, the first of which focuses on the theory of ordinary differential equations and a survey of basic linear PDEs.
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Lectures on Linear Partial Differential Equations

Author: Grigoriĭ Ilʹich Eskin

Publisher: American Mathematical Soc.

ISBN: 0821852841

Category: Mathematics

Page: 410

View: 2033

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This book is a reader-friendly, relatively short introduction to the modern theory of linear partial differential equations. An effort has been made to present complete proofs in an accessible and self-contained form. The first three chapters are on elementary distribution theory and Sobolev spaces with many examples and applications to equations with constant coefficients. The following chapters study the Cauchy problem for parabolic and hyperbolic equations, boundary value problems for elliptic equations, heat trace asymptotics, and scattering theory. The book also covers microlocal analysis, including the theory of pseudodifferential and Fourier integral operators, and the propagation of singularities for operators of real principal type. Among the more advanced topics are the global theory of Fourier integral operators and the geometric optics construction in the large, the Atiyah-Singer index theorem in $\mathbb R^n$, and the oblique derivative problem.
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Introduction to Partial Differential Equations

Author: David Borthwick

Publisher: Springer

ISBN: 3319489364

Category: Mathematics

Page: 283

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This modern take on partial differential equations does not require knowledge beyond vector calculus and linear algebra. The author focuses on the most important classical partial differential equations, including conservation equations and their characteristics, the wave equation, the heat equation, function spaces, and Fourier series, drawing on tools from analysis only as they arise. Within each section the author creates a narrative that answers the five questions: What is the scientific problem we are trying to understand? How do we model that with PDE? What techniques can we use to analyze the PDE? How do those techniques apply to this equation? What information or insight did we obtain by developing and analyzing the PDE? The text stresses the interplay between modeling and mathematical analysis, providing a thorough source of problems and an inspiration for the development of methods.
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Recent Developments in Nonlinear Partial Differential Equations

The Second Symposium on Analysis and PDE's, June 7-10, 2004, Purdue University, West Lafayette, Indiana

Author: Donatella Danielli

Publisher: American Mathematical Soc.

ISBN: 0821837400

Category: Mathematics

Page: 133

View: 4032

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This volume contains research and expository articles based on talks presented at the 2nd Symposium on Analysis and PDEs, held at Purdue University. The Symposium focused on topics related to the theory and applications of nonlinear partial differential equations that are at the forefront of current international research. Papers in this volume provide a comprehensive account of many of the recent developments in the field. The topics featured in this volume include: kinetic formulations of nonlinear PDEs; recent unique continuation results and their applications; concentrations and constrained Hamilton-Jacobi equations; nonlinear Schrodinger equations; quasiminimal sets for Hausdorff measures; Schrodinger flows into Kahler manifolds; and parabolic obstacle problems with applications to finance. The clear and concise presentation in many articles makes this volume suitable for both researchers and graduate students.
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Partial Differential Equations

Author: Jürgen Jost

Publisher: Springer Science & Business Media

ISBN: 9780387493183

Category: Mathematics

Page: 356

View: 1124

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This book offers an ideal introduction to the theory of partial differential equations. It focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view towards nonlinear problems. It 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. It also explores connections between elliptic, parabolic, and hyperbolic equations as well as the connection with Brownian motion and semigroups. This second edition features a new chapter on reaction-diffusion equations and systems.
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Introduction to Partial Differential Equations

Author: Peter J. Olver

Publisher: Springer Science & Business Media

ISBN: 3319020994

Category: Mathematics

Page: 636

View: 8364

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This textbook is designed for a one year course covering the fundamentals of partial differential equations, geared towards advanced undergraduates and beginning graduate students in mathematics, science, engineering, and elsewhere. The exposition carefully balances solution techniques, mathematical rigor, and significant applications, all illustrated by numerous examples. Extensive exercise sets appear at the end of almost every subsection, and include straightforward computational problems to develop and reinforce new techniques and results, details on theoretical developments and proofs, challenging projects both computational and conceptual, and supplementary material that motivates the student to delve further into the subject. No previous experience with the subject of partial differential equations or Fourier theory is assumed, the main prerequisites being undergraduate calculus, both one- and multi-variable, ordinary differential equations, and basic linear algebra. While the classical topics of separation of variables, Fourier analysis, boundary value problems, Green's functions, and special functions continue to form the core of an introductory course, the inclusion of nonlinear equations, shock wave dynamics, symmetry and similarity, the Maximum Principle, financial models, dispersion and solutions, Huygens' Principle, quantum mechanical systems, and more make this text well attuned to recent developments and trends in this active field of contemporary research. Numerical approximation schemes are an important component of any introductory course, and the text covers the two most basic approaches: finite differences and finite elements.
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Stochastic Partial Differential Equations

Author: Sergey V. Lototsky,Boris L. Rozovsky

Publisher: Springer

ISBN: 3319586475

Category: Mathematics

Page: 508

View: 9088

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Taking readers with a basic knowledge of probability and real analysis to the frontiers of a very active research discipline, this textbook provides all the necessary background from functional analysis and the theory of PDEs. It covers the main types of equations (elliptic, hyperbolic and parabolic) and discusses different types of random forcing. The objective is to give the reader the necessary tools to understand the proofs of existing theorems about SPDEs (from other sources) and perhaps even to formulate and prove a few new ones. Most of the material could be covered in about 40 hours of lectures, as long as not too much time is spent on the general discussion of stochastic analysis in infinite dimensions. As the subject of SPDEs is currently making the transition from the research level to that of a graduate or even undergraduate course, the book attempts to present enough exercise material to fill potential exams and homework assignments. Exercises appear throughout and are usually directly connected to the material discussed at a particular place in the text. The questions usually ask to verify something, so that the reader already knows the answer and, if pressed for time, can move on. Accordingly, no solutions are provided, but there are often hints on how to proceed. The book will be of interest to everybody working in the area of stochastic analysis, from beginning graduate students to experts in the field.
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Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena

2008-2009 Research Program on Nonlinear Partial Differential Equations, Centre for Advanced Study of the Norwegian Academy of Sciences and Letters, Oslo, Norway

Author: Norske videnskaps-akademi. Research Program on Nonlinear Partial Differential Equations

Publisher: American Mathematical Soc.

ISBN: 082184976X

Category: Mathematics

Page: 389

View: 2891

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This volume presents the state of the art in several directions of research conducted by renowned mathematicians who participated in the research program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo, Norway, during the academic year 2008-09. The main theme of the volume is nonlinear partial differential equations that model a wide variety of wave phenomena. Topics discussed include systems of conservation laws, compressible Navier-Stokes equations, Navier-Stokes-Korteweg type systems in models for phase transitions, nonlinear evolution equations, degenerate/mixed type equations in fluid mechanics and differential geometry, nonlinear dispersive wave equations (Korteweg-de Vries, Camassa-Holm type, etc.), and Poisson interface problems and level set formulations.
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