Topics in Engineering Mathematics Mathematics and Its Applications Managing Editor: M. HAZEWINKEL Centre for Mathematics and Computer Science, Amsterdam, The Netherlands Editorial Board: F. CALOGERO, Universita degli Studi di Roma, Italy Yu.1. MANIN, Steklov Institute of Mathematics, Moscow, Russia, C.l.S. M. NIVAT, Universite de Paris VII, Paris, France A. H. G. RINNOOY KAN, Erasmus University, Rotterdam, The Netherlands G.-C. ROTA, M.I.T., Cambridge, Mass., U.SA. Volume 81 Topics in Engineering Mathematics Modeling and Methods edited by Adriaan van der Burgh and Juriaan Simonis Faculty ofTechnical Mathematics and lnformatics, University ofTechnology, Delft, The Netherlands SPRINGER-SCIENCE+BUSINESS MEDIA, B.Y. Library of Congress Cataloging-in-Publication Data Topics In englneerlng mathematlcs : model1ng and methods I edlted by Adriaan van der Burgh and Juriaan Slmonls. p. cm. -- (Mathematlcs and its appl1catlons ; v. 81) ISBN 978-94-010-4800-2 ISBN 978-94-011-1814-9 (eBook) DOI 10.1007/978-94-011-1814-9 1. Englneerlng mathematlcs. 2. Mathematlcal models. 1. Burgh. Adrlaan Her.an Pleter van der. II. Slmonis, Jurlaan, 1943- III. Serles: Mathematics and lts appl1catlons (Kluwer Academic Publ1shers) ; v. 8" TA330.T67 1992 620·.001·51--dc20 92-33608 ISBN 978-94-010-4800-2 Printed on acid-free paper AII Rights Reserved © 1992 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1992 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. SERIES EDITOR'S PREFACE 'Et moi •...• si j'avait su comment en revenir. je One service mathematics bas rendered the n'y serais point a116.' human race. It bas put common sense back Jules Verne where it belongs. on the topmost shelf next to the dusty cauister labelled 'discarded uonsense·. The series is divergent; therefore we may be EricT. Bell able to do something with it. O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlineari ties abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sci ences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One ser vice topology has rendered mathematical physics .. .'; 'One service logic has rendered computer science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d' 8tre of this series. This series, Mathematics and Its Applications, started in 1977. Now that over one hundred volumes have appeared it seems opportune to reexamine its scope. At the time I wrote "Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sci ences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as 'experi mental mathematics', 'CFD', 'completely integrable systems', 'chaos. synergetics and large scale order', which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. " By and large, all this still applies today. It is still true that at first sight mathematics seems rather frag mented and that to find, see, and exploit the deeper underlying interrelations more effort is needed and so are books that can help mathematicians and scientists do so. Accordingly MIA will continue to try to make such books available. If anything, the description I gave in 1977 is now an understatement. To the examples of interaction areas one should add string theory where Riemann surfaces, algebraic geometry, modular functions, knots, quantum field theory, Kac-Moody algebras, monstrous moonshine (and more) all come together. And to the examples of things which can be usefully applied let me add the topic 'finite geometry'; a combination of words which sounds like it might not even exist, let alone be applicable. And yet it is being applied: to statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called pure) mathematics that is not in immediate danger of being applied. And, accordingly, the applied mathematician needs to be aware of much more. Besides analysis and numerics, the traditional workhorses, he may need all kinds of combina torics, algebra, probability, and so on. In addition, the applied scientist needs to cope increasingly with the nonlinear world and the extra vi mathematical sophistication that this requires. For that is where the rewards are. Linear models are honest and a bit sad and depressing: proportional efforts and results. It is in the nonlinear world that infinitesimal inputs may result in macroscopic outputs (or vice versa). To appreciate what I am hinting at: if electronics were linear we would have no fun with transistors and computers; we would have no TV; in fact you would not be reading these lines. There is also no safety in ignoring such outlandish things as nonstandard analysis, superspace and anticommuting integration, p-adic and ultrametric space. All tluee have applications in both electrical engineering and physics. Once, complex numbers were equally outlandish, but they frequently proved the shortest path between 'real' results. Similarly, the first two topics named have already provided a number of 'wormhole' paths. There is no telling where all this is leading -fortunately. Thus the original scope of the series, which for various (sound) reasons now comprises five subseries: white (Japan), yellow (China), red (USSR), blue (Eastern Europe), and green (everything else), still applies. It has been enlarged a bit to include books treating of the tools from one subdiscipline which are used in others. Thus the series still aims at books dealing with: a central concept which plays an important role in several different mathematical and/or scientific specialization areas; new applications of the results and ideas from one area of scientific endeavour into another; influences which the results, problems and concepts of one field of enquiry have, and have had, on the development of another. The shortest path between two truths in the real Never IeDd books, for 110 ODe ever retums them; domain passes through the complex domain. the only books I bave in my b'bnlly an: books J.Hadamard tbat other folk have lent me. Anatole France La physique De nous donne pas seuIcment I'occasion de resoudre des problancs ... cIIc The function of an expert is DOl to be more rigbt nous fait pressentir Ia solution. than other people, but to be wrong for _ H. Poincare sopbisticated reasons. David Butler Bussum, 14 August 1992 Michie! Hazewinkel To the academic staff of Jurusan Matematika INSTITUT TEKNOWGI SEPULUH NOPEMBER SURABAYA , INDONESIA Contents Series editor's preface Preface ...................................................... xi E. van Groesen A b1eidoscopic emusion into numerical caknlations of differeutial equatious . . . . . . . . . . . . . . . 1 1. Introduction .............................. 2 2. Logistic equation .......................... 6 3. Harmonic oscillator ....................... 10 4. Wave equations .......................... 17 5. Epilogue ............................... 34 Jos van Kan An introduction to the Fmite Element Method . . . . . . 37 1. Introduction............................. 37 2. An elementary example .................... 37 3. The 2-dimensional Poisson equation ........... 48 4. Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 C. Kauffmann Coupling of sound and struc:tural vibrations ........ 61 1. Introduction............................. 61 2. Governing equations and analysis ............. 64 3. Numerical results and discussion . . . . . . . . . . . . . . 73 J. Molenaar Mathematical modeJiag aacI dimCDSioDal aDalysis . . .. 93 1. Introduction............................. 93 2. Mathematical Modeling .................... 94 3. Dimensional Analysis ... . . . . . . . . . . . . . . . . . . . 98 4. Four-wheel steering ...................... 109 x Geert-Jan Olsder About difJerenc:e equations, algebras and discrete evcuts . . .. .. .. .. .. . . . . ... 121 1. Difference equations ..................... 121 2. Changing the algebra ..................... 123 3. Example on production .................. ; 131 4. Some graph theory and the spectral theory of matrices ................. 135 5. A stochastic extension .................... 141 6. Counter versus dater description ............. 144 7. The Z-transform ........................ 145 8. Conclusions ............................ 149 S.W. Rienstra Amustical cIetedion of obstrudioDs in a pipe with a temperature gradient ...•.•..... 151 1. Introduction ............................ 151 2. The problem ........................... 152 3. The model . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 154 4. Solution............................... 160 5. Results ............................... 173 6. Conclusions ............................ 178 C. Roos Interior point approach to linear programming: theory, algorithms and parametric analysis . . . . . . .. 181 1. Introduction............................ 181 2. Central approach to the theory of LP ......... 182 3. A polynomial-time path-following algorithm .... 196 4. Parametric analysis . . . . . . . . . . . . . . . . . . . . . .. 204 F. Twilt Some reflections on Newton's Method . . . . . . . . . .. 217 1. Introduction and motivation ................ 217 2. Rational Newton Flows ................... 219 3. Structural Stability ....................... 222 4. Newton Graphs ......................... 225 5. Characterization of Newton Graphs .......... 229 6. Open problems ......................... 232 AJ. van Zanten Remrrence and induc:tion in Computer Science .... 239 1. Introduction............................ 240 2. Strings of Symbols ....................... 242 3. Gray codes and recursiveness ............... 248 4. Recursive algorithms ..................... 256