Mathematical Models with Singularities A Zoo of Singular Creatures
The book aims to provide an unifying view of a variety (a 'zoo') of mathematical models with some kind of singular nonlinearity, in the sense that it becomes infinite when the state variable approaches a certain point. Up to 11 different concrete models a
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Pedro J. Torres
Mathematical Models with Singularities A Zoo of Singular Creatures
Atlantis Briefs in Differential Equations Volume 1
Series editors Zuzana Dosla, Brno, Czech Republic Sarka Necasova, Prague 1, Czech Republic Milan Pokorny, Praha 8, Czech Republic
About this Series The aim of the series is rapid dissemination of new results and original methods in the theory of Differential Equations, including topics not yet covered by standard monographs. The series features compact volumes of 75–200 pages, written in a concise, clear way and going directly to the point; the introductory material should be restricted to a minimum or covered by suitable references. For more information on this series and our other book series, please visit our website at: www.atlantis-press.com/publications/books AMSTERDAM—PARIS—BEIJING ATLANTIS PRESS Atlantis Press 29, avenue Laumière 75019 Paris, France
Pedro J. Torres
Mathematical Models with Singularities A Zoo of Singular Creatures
Pedro J. Torres Department of Applied Mathematics University of Granada Granada Spain
ISSN 2405-6405 ISSN 2405-6413 (electronic) Atlantis Briefs in Differential Equations ISBN 978-94-6239-105-5 ISBN 978-94-6239-106-2 (eBook) DOI 10.2991/978-94-6239-106-2 Library of Congress Control Number: 2014958287 © Atlantis Press and the authors 2015 This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher. Printed on acid-free paper
A mi familia
Preface
Talk with M. Hermite. He never evokes a concrete image, yet you soon perceive that the more abstract entities are to him like living creatures. Henri Poincaré, quoted in G. Simmons “Calculus Gems” As time goes on it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen. Paul Adrien Maurice Dirac, in The Relation between Mathematics and Physics
During more than 20 years dedicated to the study of differential equations with singularities, I have crafted what I call my zoo of singular creatures. The inhabitants of this zoo are mathematical models with some kind of singular nonlinearity, in the sense that it becomes infinite when the state variable approaches a certain point. One might say that the zoo is a reversed version of Plato’s cavern, being the mathematical models’ shadows or caricatures of real phenomena in the natural world. Singularities are purely mathematical artifacts, but they are important in the modeling of real-world processes because the main physical forces in nature are singular. Everyday I spend some time in my zoo, contemplating the creatures (some of them are very exotic) and studying their behavior and interconnections. Also, I enjoy hunting new creatures in the related literature. As soon as I find a new model, I add it to my collection. Of course, I am not alone. In this endea
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