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Inna Shingareva Carlos Lizárraga-Celaya

Solving Nonlinear Partial Differential Equations with Maple and Mathematica

SpringerWienNewYork

Prof. Dr. Inna Shingareva

Department of Mathematics, University of Sonora, Sonora, Mexico [email protected]

Dr. Carlos Lizárraga-Celaya

Department of Physics, University of Sonora, Sonora, Mexico

[email protected]

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machines or similar means, and storage in data banks. Product Liability: The publisher can give no guarantee for all the information contained in this book. This does also refer to information about drug dosage and application thereof. In every individual case the respective user must check its accuracy by consulting other pharmaceutical literature. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

© 2011 Springer-Verlag / Wien SpringerWienNewYork is part of Springer Science + Business Media springer.at Cover Design: WMX Design, 69126 Heidelberg, Germany Typesetting: Camera ready by the authors With 20 Figures Printed on acid-free and chlorine-free bleached paper SPIN: 80021221 Library of Congress Control Number: 2011929420

ISBN 978-3-7091-0516-0 e-ISBN 978-3-7091-0517-7 DOI 10.1007/978-3-7091-0517-7 SpringerWienNewYork

Preface

The study of partial differential equations (PDEs) goes back to the 18th century, as a result of analytical investigations of a large set of physical models (works by Euler, Cauchy, d’Alembert, Hamilton, Jacobi, Lagrange, Laplace, Monge, and many others). Since the mid 19th century (works by Riemann, Poincar`e, Hilbert, and others), PDEs became an essential tool for studying other branches of mathematics. The most important results in determining explicit solutions of nonlinear partial differential equations have been obtained by S. Lie [91]. Many analytical methods rely on the Lie symmetries (or symmetry continuous transformation groups). Nowadays these transformations can be performed using computer algebra systems (e.g., Maple and Mathematica). Currently PDE theory plays a central role within the general advancement of mathematics, since they help us to describe the evolution of many phenomena in various fields of science, engineering, and numerous other applications. Since the 20th century, the investigation of nonlinear PDEs has become an independent field expanding in many research directions. One of these directions is, symbolic and numerical computations of solutions of nonlinear PDEs, which is considered in this book. It should be noted that the main ideas on practical computations of solutions of PDEs were first indicated by H. Poincar`e in 1890 [121]. However the solution techniques of such proble