Computer Algebra Recipes for Mathematical Physics - Enns.pdf - Computational Physics - realmodeler

This mathematical physics contribution

to the Computer Algebra Recipes series

is dedicated to my wife Karen,

who lights my path through life.

Richard H. Enns

Computer Algebra Recipes

for Mathematical Physics

Birkhauser

Boston • Basel • Berlin

Richard H. Enns

Simon Fraser University

Department of Physics

Burnaby, B.C. V5A 1S6

Canada

AMS Subject Classiﬁcations (2000): 15A90, 30-XX, 33-XX, 34-XX, 35-XX, 35Qxx, 40-XX,

42-XX, 44-XX, 49-XX, 65-XX, 68-XX, 70-XX, 97U50

ISBN 0-8176-3223-9

Printed on acid-free paper.

2005 Birkhauser Boston

permission of the publisher (Birkhauser Boston, c/o Springer Science + Business Media Inc., Rights

and Permissions, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in con-

nection 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.

Printed in the United States of America. (HP)

987654321 SPIN 10923559

www.birkhauser.com

Preface

This book is a self-contained guide to problem-solving and exploration in math-

ematical physics using the powerful Maple 9.5 computer algebra system (CAS).

With a CAS one cannot only crunch numbers and plot results, but also carry out

the symbolic manipulations which form the backbone of mathematical physics.

The heart of this text consists of over 230 useful and stimulating “classic”

computer algebra worksheets or recipes , which are systematically organized to

cover the major topics presented in the standard Mathematical Physics course

oﬀered to third or fourth year undergraduate physics and engineering students.

The emphasis here is on applications, with only a brief summary of the un-

derlying theoretical ideas being presented. The aim is to show how computer

algebra can not only implement the methods of mathematical physics quickly,

accurately, and e ciently, but can be used to explore more complex examples

which are tedious or di cult or even impossible to implement by hand.

The recipes are grouped into three sections, the introductory Appetizers

dealing with linear ordinary diﬀerential equations (ODEs), series, vectors, and

matrices. The more advanced Entrees cover linear partial diﬀerential equations

(PDEs), scalar and vector ﬁelds, complex variables, integral transforms, and

calculus of variations. Finally, in the Desserts the emphasis is on presenting

some analytic, graphical, and numerical techniques for solving nonlinear ODEs

and PDEs. The numerical methods are also applied to linear ODEs and PDEs.

No prior knowledge of Maple is assumed in this text, the relevant command

structures being introduced on a need-to-know basis. The recipes are thor-

oughly annotated and, on numerous occasions, presented in a “story” format

or in a historical context. Each recipe takes the reader from the analytic formu-

lation or statement of a representative type of mathematical physics problem to

its analytic or numerical solution and to a graphical visualization of the answer,

where relevant. The graphical representations vary from static 2-dimensional

pictures, to contour and vector ﬁeld plots, to 3-dimensional graphs that can

be rotated, to animations in time. For your convenience, all 230 recipes are

included on the accompanying CD.

The range of mathematical physics problems that can be solved with the

enclosed recipes is only limited by your imagination. By altering the parameter

values, or initial conditions, or equation structure, thousands of other problems

can be easily generated and solved. “What if?” questions become answerable.

This should prove extremely useful to instructor and student alike.

Computer Algebra Recipes for Mathematical Physics - Enns.pdf

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