Generalized Collocation Methods Solutions to Nonlinear Problems
This book examines various mathematical tools—based on generalized collocation methods—to solve nonlinear problems related to partial differential and integro-differential equations. Covered are specific problems and models related to vehicular traffic fl
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Generalized Collocation Methods Solutions to Nonlinear Problems
Birkh¨auser Boston • Basel • Berlin
Nicola Bellomo Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italia
Bertrand Lods Laboratoire de Math´ematiques Universit´e Blaise Pascal Campus Universitaire des C´ezeaux ` cedex 63177 Aubiere France
Roberto Revelli Dipartimento di Idraulica, Trasporti, ed Infrastrutture Civili Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italia
Luca Ridolfi Dipartimento di Idraulica, Trasporti, ed Infrastrutture Civili Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino Italia
Mathematics Subject Classification: 00A72, 74S25, 74S30
Library of Congress Control Number: 2007932360 ISBN-13: 978-0-8176-4525-0
e-ISBN-13: 978-0-8176-4610-3
Printed on acid-free paper. c 2008 Birkhauser ¨ Boston
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA) and the author, 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. 987654321 www.birkhauser.com
Contents
Preface
. . . . . . . . . . . . . . . . . . . . . ix
Chapter 1. Mathematical Models and Problems in Applied Sciences . . . . . . . . . . 1 1.1
Introduction . . . . . . . . . . . . . . . . . 1
1.2
Some Models in Applied Sciences . . . . . . . . 2
1.3
Dimensional Analysis . . . . . . . . . . . .
16
1.4
Classification of Models and Mathematical Problems . . . . . . . . .
20
Guidelines for the Application of Collocation Methods . . . . . . . . . . .
28
Problems
30
1.5 1.6
. . . . . . . . . . . . . . . . .
Chapter 2. Lagrange and Sinc Collocation Interpolation Methods . . . . . . . 2.1
Introduction . . . . . . . . . . . . . . . .
33 33
v
vi
Contents
2.2
Collocation Methods in One Space Dimension
.
34
2.3
Collocation and Interpolation in Two Space Dimensions . . . . . . . . . . . . .
39
2.4
Examples and Applications
. . . . . . . . .
41
2.5
Critical Analysis . . . . . . . . . . . . . .
50
2.6
Problems
54
. . . . . . . . . . . . . . . . .
Chapter 3. Nonlinear Initial Value Problems in Unbounded Domains . . . . . . .
57
3.1
Introduction . . . . . . . . . . . . . . . .
57
3.2
On the Solution of Initial Value Problems . . .
58
3.3
On the Solution of the Third-Order KdV Model
62
3.4
On the Solution of the Fifth-Order KdV Model
66
3.5
Additional Applications and Discussion . . . .
68
3.6
On the Solution of Ordinary Differential
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