Optimale Steuerung partieller Differentialgleichungen

Theorie, Verfahren und Anwendungen

Author: Fredi Tröltzsch

Publisher: Springer-Verlag

ISBN: 3322968448

Category: Mathematics

Page: 298

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Die mathematische Theorie der optimalen Steuerung hat sich im Zusammenhang mit Berechnungen für die Luft- und Raumfahrt schnell zu einem wichtigen und eigenständigen Gebiet der angewandten Mathematik entwickelt. Die optimale Steuerung durch partielle Differentialgleichungen modellierter Prozesse wird eine numerische Herausforderung der Zukunft sein. Sie erfordert die Analysis nichtlinearer partieller Differentialgleichungen, Optimierung im Funktionenraum, nichtlineare Funktionalanalysis sowie Optimierungsverfahren für extrem große Aufgaben. Im Buch werden entsprechende Grundlagen mit langsam steigendem Schwierigkeitsgrad entwickelt. Grundkenntnisse zu partiellen Differentialgleichungen und der Funktionalanalysis werden jeweils dort gebracht, wo sie konkret nötig sind. Das Buch enthält viele Beispiele und eignet sich als Grundlage für Vorlesungen und Seminare.

Partial Differential Equations

Author: Lawrence C. Evans

Publisher: American Mathematical Soc.

ISBN: 9780821807729

Category: Mathematics

Page: 662

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Lawrence C. Evans presents a comprehensive survey of modern techniques in the theoretical study of partial differential equations, with particular emphasis on nonlinear equations.

Hyperbolic Partial Differential Equations and Geometric Optics

Author: Jeffrey Rauch

Publisher: American Mathematical Soc.

ISBN: 0821872915

Category: Mathematics

Page: 363

View: 7110

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This book introduces graduate students and researchers in mathematics and the sciences to the multifaceted subject of the equations of hyperbolic type, which are used, in particular, to describe propagation of waves at finite speed. Among the topics carefully presented in the book are nonlinear geometric optics, the asymptotic analysis of short wavelength solutions, and nonlinear interaction of such waves. Studied in detail are the damping of waves, resonance, dispersive decay, and solutions to the compressible Euler equations with dense oscillations created by resonant interactions. Many fundamental results are presented for the first time in a textbook format. In addition to dense oscillations, these include the treatment of precise speed of propagation and the existence and stability questions for the three wave interaction equations. One of the strengths of this book is its careful motivation of ideas and proofs, showing how they evolve from related, simpler cases. This makes the book quite useful to both researchers and graduate students interested in hyperbolic partial differential equations. Numerous exercises encourage active participation of the reader. The author is a professor of mathematics at the University of Michigan. A recognized expert in partial differential equations, he has made important contributions to the transformation of three areas of hyperbolic partial differential equations: nonlinear microlocal analysis, the control of waves, and nonlinear geometric optics.

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.

Partielle Differentialgleichungen und numerische Methoden

Author: Stig Larsson,Vidar Thomee

Publisher: Springer-Verlag

ISBN: 3540274227

Category: Mathematics

Page: 272

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Das Buch ist für Studenten der angewandten Mathematik und der Ingenieurwissenschaften auf Vordiplomniveau geeignet. Der Schwerpunkt liegt auf der Verbindung der Theorie linearer partieller Differentialgleichungen mit der Theorie finiter Differenzenverfahren und der Theorie der Methoden finiter Elemente. Für jede Klasse partieller Differentialgleichungen, d.h. elliptische, parabolische und hyperbolische, enthält der Text jeweils ein Kapitel zur mathematischen Theorie der Differentialgleichung gefolgt von einem Kapitel zu finiten Differenzenverfahren sowie einem zu Methoden der finiten Elemente. Den Kapiteln zu elliptischen Gleichungen geht ein Kapitel zum Zweipunkt-Randwertproblem für gewöhnliche Differentialgleichungen voran. Ebenso ist den Kapiteln zu zeitabhängigen Problemen ein Kapitel zum Anfangswertproblem für gewöhnliche Differentialgleichungen vorangestellt. Zudem gibt es ein Kapitel zum elliptischen Eigenwertproblem und zur Entwicklung nach Eigenfunktionen. Die Darstellung setzt keine tiefer gehenden Kenntnisse in Analysis und Funktionalanalysis voraus. Das erforderliche Grundwissen über lineare Funktionalanalysis und Sobolev-Räume wird im Anhang im Überblick besprochen.

Partielle Differentialgleichungen

Eine Einführung

Author: Walter A. Strauss

Publisher: Springer-Verlag

ISBN: 366312486X

Category: Mathematics

Page: 458

View: 7473

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Dieses Buch ist eine umfassende Einführung in die klassischen Lösungsmethoden partieller Differentialgleichungen. Es wendet sich an Leser mit Kenntnissen aus einem viersemestrigen Grundstudium der Mathematik (und Physik) und legt seinen Schwerpunkt auf die explizite Darstellung der Lösungen. Es ist deshalb besonders auch für Anwender (Physiker, Ingenieure) sowie für Nichtspezialisten, die die Methoden der mathematischen Physik kennenlernen wollen, interessant. Durch die große Anzahl von Beispielen und Übungsaufgaben eignet es sich gut zum Gebrauch neben Vorlesungen sowie zum Selbststudium.

Optimal Control of Partial Differential Equations

Theory, Methods, and Applications

Author: Fredi Tröltzsch

Publisher: American Mathematical Soc.

ISBN: 0821849042

Category: Mathematics

Page: 399

View: 5953

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"Optimal control theory is concerned with finding control functions that minimize cost functions for systems described by differential equations. The methods have found widespread applications in aeronautics, mechanical engineering, the life sciences, and many other disciplines. This book focuses on optimal control problems where the state equation is an elliptic or parabolic partial differential equation. Included are topics such as the existence of optimal solutions, necessary optimality conditions and adjoint equations, second-order sufficient conditions, and main principles of selected numerical techniques. It also contains a survey on the Karush-Kuhn-Tucker theory of nonlinear programming in Banach spaces. The exposition begins with control problems with linear equations, quadratic cost functions and control constraints. To make the book self-contained, basic facts on weak solutions of elliptic and parabolic equations are introduced. Principles of functional analysis are introduced and explained as they are needed. Many simple examples illustrate the theory and its hidden difficulties. This start to the book makes it fairly self-contained and suitable for advanced undergraduates or beginning graduate students. Advanced control problems for nonlinear partial differential equations are also discussed. As prerequisites, results on boundedness and continuity of solutions to semilinear elliptic and parabolic equations are addressed. These topics are not yet readily available in books on PDEs, making the exposition also interesting for researchers. Alongside the main theme of the analysis of problems of optimal control, Tr'oltzsch also discusses numerical techniques. The exposition is confined to brief introductions into the basic ideas in order to give the reader an impression of how the theory can be realized numerically. After reading this book, the reader will be familiar with the main principles of the numerical analysis of PDE-constrained optimization."--Publisher's description.

Introduction to Partial Differential Equations

Author: Peter J. Olver

Publisher: Springer Science & Business Media

ISBN: 3319020994

Category: Mathematics

Page: 636

View: 972

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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.

Partial Differential Equations I

Basic Theory

Author: Michael Eugene Taylor,Eberhard Zeidler

Publisher: Springer Science & Business Media

ISBN: 9780387946535

Category: Mathematics

Page: 563

View: 2328

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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.

Partial Differential Equations

An Introduction

Author: David Colton

Publisher: Courier Corporation

ISBN: 0486138437

Category: Mathematics

Page: 320

View: 1548

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This text offers students in mathematics, engineering, and the applied sciences a solid foundation for advanced studies in mathematics. Features coverage of integral equations and basic scattering theory. Includes exercises, many with answers. 1988 edition.

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.

Mathematical Physics with Partial Differential Equations

Author: James Kirkwood

Publisher: Academic Press

ISBN: 0128147601

Category: Mathematics

Page: 492

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Mathematical Physics with Partial Differential Equations, Second Edition, is designed for upper division undergraduate and beginning graduate students taking mathematical physics taught out by math departments. The new edition is based on the success of the first, with a continuing focus on clear presentation, detailed examples, mathematical rigor and a careful selection of topics. It presents the familiar classical topics and methods of mathematical physics with more extensive coverage of the three most important partial differential equations in the field of mathematical physics—the heat equation, the wave equation and Laplace’s equation. The book presents the most common techniques of solving these equations, and their derivations are developed in detail for a deeper understanding of mathematical applications. Unlike many physics-leaning mathematical physics books on the market, this work is heavily rooted in math, making the book more appealing for students wanting to progress in mathematical physics, with particularly deep coverage of Green’s functions, the Fourier transform, and the Laplace transform. A salient characteristic is the focus on fewer topics but at a far more rigorous level of detail than comparable undergraduate-facing textbooks. The depth of some of these topics, such as the Dirac-delta distribution, is not matched elsewhere. New features in this edition include: novel and illustrative examples from physics including the 1-dimensional quantum mechanical oscillator, the hydrogen atom and the rigid rotor model; chapter-length discussion of relevant functions, including the Hermite polynomials, Legendre polynomials, Laguerre polynomials and Bessel functions; and all-new focus on complex examples only solvable by multiple methods. Introduces and evaluates numerous physical and engineering concepts in a rigorous mathematical framework Provides extremely detailed mathematical derivations and solutions with extensive proofs and weighting for application potential Explores an array of detailed examples from physics that give direct application to rigorous mathematics Offers instructors useful resources for teaching, including an illustrated instructor's manual, PowerPoint presentations in each chapter and a solutions manual

Nonlinear Partial Differential Equations in Differential Geometry

Author: Robert Hardt

Publisher: American Mathematical Soc.

ISBN: 9780821804315

Category: Mathematics

Page: 339

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What distinguishes differential geometry in the last half of the twentieth century from its earlier history is the use of nonlinear partial differential equations in the study of curved manifolds, submanifolds, mapping problems, and function theory on manifolds, among other topics. The differential equations appear as tools and as objects of study, with analytic and geometric advances fueling each other in the current explosion of progress in this area of geometry in the last twenty years. This book contains lecture notes of minicourses at the Regional Geometry Institute at Park City, Utah, in July 1992. Presented here are surveys of breaking developments in a number of areas of nonlinear partial differential equations in differential geometry. The authors of the articles are not only excellent expositors, but are also leaders in this field of research. All of the articles provide in-depth treatment of the topics and require few prerequisites and less background than current research articles.

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

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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.

Partial Differential Equations

Author: Jürgen Jost

Publisher: Springer Science & Business Media

ISBN: 9780387954288

Category: Mathematics

Page: 325

View: 3191

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Modern and systematic treatment of main approaches; Several additions have been made to the German edition, most notably coverage of eigenvalues and expansions; Emphasis on methods relevant for both linear and nonlinear equations; Contains chapter summaries, detailed illustrations and numerous exercises

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: 5793

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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.

Stochastic Partial Differential Equations

Author: Sergey V. Lototsky,Boris L. Rozovsky

Publisher: Springer

ISBN: 3319586475

Category: Mathematics

Page: 508

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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.

Partial Differential Equations in Mechanics 1

Fundamentals, Laplace's Equation, Diffusion Equation, Wave Equation

Author: A.P.S. Selvadurai

Publisher: Springer Science & Business Media

ISBN: 3662040069

Category: Technology & Engineering

Page: 596

View: 3546

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This two-volume work focuses on partial differential equations (PDEs) with important applications in mechanical and civil engineering, emphasizing mathematical correctness, analysis, and verification of solutions. The presentation involves a discussion of relevant PDE applications, its derivation, and the formulation of consistent boundary conditions.

Handbook of Differential Equations: Stationary Partial Differential Equations

Author: Michel Chipot

Publisher: Elsevier

ISBN: 9780080521831

Category: Mathematics

Page: 626

View: 7587

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A collection of self contained state-of-the art surveys. The authors have made an effort to achieve readability for mathematicians and scientists from other fields, for this series of handbooks to be a new reference for research, learning and teaching. - written by well-known experts in the field - self contained volume in series covering one of the most rapid developing topics in mathematics