Advanced Topics In Applied Mathematics - For Engineering And The Physical Sciences

This page intentionally left blank Advanced Topics in Applied Mathematics This book is ideal for engineering, physical science, and applied mathematics students and professionals who want to enhance their mathematical knowledge. Advanced Topics in Applied Mathematics covers four essential applied mathematics topics: Green’s functions, integral equations, Fourier transforms, and Laplace transforms. Also included is a useful discussion of topics such as the Wiener-Hopf method, finite Hilbert transforms, Cagniard–De Hoop method, and the proper orthogonal decomposition. This book reflects Sudhakar Nair’s long classroom experience and includes numerous examples of differential and integral equations from engineering and physics to illustrate the solution procedures. The text includes exercise sets at the end of each chapter and a solutions manual, which is available for instructors. Sudhakar Nair is the Associate Dean for Academic Affairs of the Graduate College, Professor of Mechanical Engineering and Aerospace Engineering, and Professor of Applied Mathematics at the Illinois Institute of Technology in Chicago. He is a Fellow of the ASME, an Associate Fellow of the AIAA, and a member of the American Academy of Mechanics as well as Tau Beta Pi and Sigma Xi. Professor Nair is the author of numerous research articles and Introduction to Continuum Mechanics (2009). A D V A N CED TO P ICS IN APPL I E D MA TH EM AT ICS For Engineering and the Physical Sciences Sudhakar Nair Illinois Institute of Technology C A M B R I D G E U N I V E R S I T Y PRE SS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Tokyo, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9781107006201 © Sudhakar Nair 2011 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Nair, Sudhakar, 1944– author. Advanced Topics in Applied Mathematics: for Engineering and the Physical Sciences/Sudhakar Nair. p. cm Includes index. ISBN 978-1-107-00620-1 (hardback) 1. Differential equations. 2. Engineering mathematics. 3. Mathematical physics. I. Title. TA347.D45N35 2011 620.001 51–dc22 2010052380 ISBN 978-1-107-00620-1 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. Contents Preface page ix 1 Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Heaviside Step Function 1 1.2 Dirac Delta Function 3 1.2.1 Macaulay Brackets 6 1.2.2 Higher Dimensions 7 1.2.3 Test Functions, Linear Functionals, and Distributions 7 1.2.4 Examples: Delta Function 8 1.3 Linear Differential Operators 10 1.3.1 Example: Boundary Conditions 10 1.4 Inner Product and Norm 11 1.5 Green’s Operator and Green’s Function 12 1.5.1 Examples: Direct Integrations 13 1.6 Adjoint Operators 16 1.6.1 Example: Adjoint Operator 17 1.7 Green’s Function and Adjoint Green’s Function 18 1.8 Green’s Function for L 19 1.9 Sturm-Liouville Operator 20 1.9.1 Method of Variable Constants 22 1.9.2 Example: Self-Adjoint Problem 23 1.9.3 Example: Non-Self-Adjoint Problem 24 1.10 Eigenfunctions and Green’s Function 26 1.10.1 Example: Eigenfunctions 28 1.11 Higher-Dimensional Operators 28 1.11.1 Example: Steady-State Heat Conduction in a Plat