Difference Equations For Scientists And Engineering: Interdisciplinary Difference Equations

330 Pages·2019·4.613 MB·English

by  Michael A. Radin

DIFFERENCE
EQUATIONS
FOR
SCIENTISTS
AND
ENGINEERING

Interdisciplinary Difference Equations
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DIFFERENCE
EQUATIONS

FOR
SCIENTISTS
AND
ENGINEERING

Interdisciplinary Difference Equations
Michael A Radin
Rochester Institute of Technology, USA

World Scientific
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Library of Congress Cataloging-in-Publication Data
Names: Radin, Michael A. (Michael Alexander), author.
Title: Difference equations for scientists and engineering : interdisciplinary difference equations /
Michael A. Radin (Rochester Institute of Technology, USA).
Description: New Jersey : World Scientific, 2019. | Includes bibliographical references.
Identifiers: LCCN 2019008183| ISBN 9789811203855 (hard cover : alk. paper) |
ISBN 9789811202964 (pbk : alk. paper)
Subjects: LCSH: Difference equations. | Differential equations, Linear. | Nonlinear difference equations.
Classification: LCC QA431 .R3255 2019 | DDC 515/.625--dc23
LC record available at https://lccn.loc.gov/2019008183
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
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PREFACE
Learning about structures and their details is essential for development
and enhancement of our intuitions, analytical skills and deducting reason-
ing skills. For instance, it is pertinent to decipher specific patterns when
studying weather systems, when learning to play a musical instrument,
when studying a foreign language, when learning computer program-
ming, when studying transportation routes and schedules, when analyz-
ing engineering structures, when studying behaviors and other similar
applications.
The aim of this interdisciplinary book is to capture and recognize the
emergence of patterns when obtaining an explicit solution to determine the
monotonic properties and periodic patterns. We will progressively expand
our instincts with repetitive style examples that will inductively guide
us to developing theorems and deeper mastery of convergence nature of

solutions, oscillatory character of solutions and periodic traits of solutions
by addressing the following questions:
• Is the solution monotonically increasing or monotonically decreasing?
• Does the solution approach the limit from above or from below?
• Is the solution oscillatory?
• What is the period of the given cycle? An even ordered or an odd ordered
period?
• The existence and uniqueness of periodic cycles? Are periodic cycles
unique or is every solution periodic?
• Are all the terms of the cycle positive or negative, and do they alternate?
• When do eventually periodic solutions arise and why?
I invite you to the discovery voyage in deciphering monotonic, oscillatory,
periodic and chaotic behavior of solutions of first order, second order and
v
vi Difference Equations for Scientists and Engineering
higher order difference equations. We will study the monotonic properties
andperiodictraitsoflineardifferenceequations,rationaldifferenceequations
in exponential form, piecewise difference equations and max-type difference
equations, and discuss various applications in population dynamics, bio-
logical sciences, signal processing, economics and neural networking. Our
plan is to develop inductive intuition that will help us recognize specific
structuresofpatternsofmonotonicproperties,periodiccyclesandeventually
periodic cycles and to develop theorems after numerous repetitive types
of examples. The intents are to widen our inductive reasoning skills and
to develop techniques such as proof by induction, proof by contradiction
and how additional fundamentals such as Number Theory, Combinatorics,
Abstract Algebra and analyzing computer observations will blend in as
pieces of the puzzle to address deeper research questions and welcome the
interdisciplinary research atmosphere. The aim is to enhance efficiency in
fast speed computing when we write computer programs that decipher the
patterns of the periodic cycles and the transient terms inductively that will
lead to new unanswered questions to pioneers.
AUTHOR INTRODUCTION
Michael A. Radin earned his Ph.D. at the University of Rhode Island in
2001 andis currently anassociate professor of mathematics at theRochester
Institute of Technology. Michael started his journey analyzing difference
equationswithperiodicandeventuallyperiodicsolutionsaspartofhisPh.D.
thesis and has several publications on boundedness and periodic nature of
solutions of rational difference equations, max-type difference equations and
piecewise difference equations. Michael published several papers together
with his Master’s students and undergraduate students at RIT and has
publications with students and colleagues from Riga Technical University,
University of Latvia and several other European Universities.
Michael also has publications in applied mathematics related topics
such as Neural Networking, Modelling Extinct Civilizations and Modelling
Human Emotions. Michael organized numerous sessions on difference equa-

tions and applications at the annual American Mathematical Society
meetings. Recently Michael published four manuscripts on international
pedagogy and has been invited as a keynote speaker at several international
and interdisciplinary conferences. Michael taught courses and conducted
seminars on these related topics during his spring 2009 sabbatical at the
Aegean University in Greece and during his spring 2016 sabbatical at Riga
Technical University and at the University of Latvia.
During his spare time Michael spends time outdoors and is an avid
landscape photographer. In addition, Michael is an active poet and has
several published poemsin the Le Mot Juste. Spendingtime outdoors and
active landscapephotography widensandexpandsMichael’s understandings
of nature’s patterns and cadences.
vii
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CONTENTS
Preface v
Author Introduction vii
1. Introduction 1
1.1 Recursive Sequences . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Order of a Difference Equation and Explicit Solution . . . 6
1.3 Equilibrium Points . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Convergent Sequences (Solutions) . . . . . . . . . . . . . . 15
1.5 Periodic Sequences (Solutions) . . . . . . . . . . . . . . . . 19
1.6 Complex Numbers and Periodic Cycles . . . . . . . . . . . 22
1.7 Specific Patterns of Periodic Cycles . . . . . . . . . . . . . 24
1.8 Eventually Constant Sequences (Solutions) . . . . . . . . . 26

1.9 Eventually Periodic Sequences (Solutions) . . . . . . . . . . 27
1.10 Additional Examples of Periodic and Eventually
Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . 29
1.11 Divergent (Unbounded) Sequences (Solutions) . . . . . . . 31
1.12 Chapter 1 Exercises . . . . . . . . . . . . . . . . . . . . . . 33
2. First Order Linear Difference Equations 41
2.1 Homogeneous First Order Linear Difference Equations . . . 42
2.2 Nonhomogeneous First Order Linear Difference
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 Non-autonomous First Order Linear Difference
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
ix