Principles of Partial Differential Equations
This concise book covers the classical tools of PDE theory used in today's science and engineering: characteristics, the wave propagation, the Fourier method, distributions, Sobolev spaces, fundamental solutions, and Green's functions. The approach
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Alexander Komech • Andrew Komech
Principles of Partial Differential Equations
Alexander Komech Faculty of Mathematics Vienna University 1090 Vienna Austria [email protected]
Andrew Komech Department of Mathematics Texas A&M University College Station, TX 77843 USA [email protected]
Series Editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA [email protected]
ISBN 978-1-4419-1095-0 e-ISBN 978-1-4419-1096-7 DOI 10.2007/978-1-4419-1096-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009932894
Mathematics Subject Classification (2000): 32-XX, 35-00, 35-01 © Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
Cover art: Olga Rozmakhova, [email protected] Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
This book is intended to give the reader an opportunity to master solving PDE problems. Our main goal was to have a concise text that would cover the classical tools of PDE theory that are used in today’s science and engineering, such as characteristics, the wave propagation, the Fourier method, distributions, Sobolev spaces, fundamental solutions, and Green’s functions. While introductory Fourier method – based PDE books do not give an adequate description of these areas, the more advanced PDE books are quite theoretical and require a high level of mathematical background from a reader. This book was written specifically to fill this gap, satisfying the demand of the wide range of end users who need the knowledge of how to solve the PDE problems and at the same time are not going to specialize in this area of mathematics. Arguably, this is the shortest PDE course, which stretches far beyond common, Fourier method – based PDE texts. For example, [Hab03], which is a common thorough textbook on partial differential equations, teaches a similar set of tools while being about five times longer. The book is problem-oriented. The theoretical part is rigorous yet short. Sometimes we refer the reader to textbooks that give wider coverage of the theory. Yet, important theoretical details are presented with care, while the hints give the reader an opportunity to restore the argume
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