Difference Equations: Theory, Applications and Advanced Topics

555 Pages·2018·3.18 MB·english

by  Ronald E. Mickens

Mathematics
MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
Difference Equations: Theory, Applications and Advanced Topics, D
T Difference
Third Edition provides a broad introduction to the mathematics of differ-
h
ence equations and some of their applications. Many worked examples ii
rf
d
illustrate how to calculate both exact and approximate solutions to special f

Ee
classes of difference equations. Along with adding several advanced top- Equations
d
ics, this edition continues to cover general, linear, first-, second-, and n-th ir
t
order difference equations; nonlinear equations that may be reduced to ioe
linear equations; and partial difference equations. nn
c
New to the Third Edition Theory, Applications
e
• New chapter on special topics, including discrete Cauchy–Euler

equations; gamma, beta, and digamma functions; Lambert W-func-
E
and Advanced Topics
tion; Euler polynomials; functional equations; and exact discretiza-
tions of differential equations q
• New chapter on the application of difference equations to complex
u
Third Edition
problems arising in the mathematical modeling of phenomena in
a
engineering and the natural and social sciences
t
• Additional problems in all chapters
i
• Expanded bibliography to include recently published texts related to o
the subject of difference equations
n
This book helps you understand the fundamental concepts and proce- s
dures of difference equations. It uses an informal presentation style, avoid-
ing the minutia of detailed proofs and formal explanations.
Features
• Presents a range of techniques for calculating both exact and ap-
proximate solutions to special classes of difference equations
• Illustrates the use of the techniques through a large number of
worked examples
• Explains how to construct discrete finite difference representations of
differential equations M
• Explores applications in mathematics, physical systems, economics, i
c
biological science, and social science k
e
n Ronald E. Mickens
s
K22684
www.crcpress.com
K22684_cover.indd 1 2/12/15 9:03 AM
MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
Difference
Equations
Theory, Applications
and Advanced Topics
Third Edition
Ronald E. Mickens
Clark Atlanta University
Atlanta, Georgia, USA
K22684_FM.indd 1 2/4/15 12:23 PM
MONOGRAPHS AND RESEARCH NOTES IN MATHEMATICS
Series Editors
John A. Burns
Thomas J. Tucker
Miklos Bona
Michael Ruzhansky
Chi-Kwong Li
Published Titles
Application of Fuzzy Logic to Social Choice Theory, John N. Mordeson, Davender S. Malik
and Terry D. Clark
Blow-up Patterns for Higher-Order: Nonlinear Parabolic, Hyperbolic Dispersion and
Schrödinger Equations, Victor A. Galaktionov, Enzo L. Mitidieri, and Stanislav Pohozaev
Difference Equations: Theory, Applications and Advanced Topics, Third Edition,
Ronald E. Mickens
Dictionary of Inequalities, Second Edition, Peter Bullen
Iterative Optimization in Inverse Problems, Charles L. Byrne
Modeling and Inverse Problems in the Presence of Uncertainty, H. T. Banks, Shuhua Hu,
and W. Clayton Thompson
Set Theoretical Aspects of Real Analysis, Alexander B. Kharazishvili
Signal Processing: A Mathematical Approach, Second Edition, Charles L. Byrne
Sinusoids: Theory and Technological Applications, Prem K. Kythe
Special Integrals of Gradshetyn and Ryzhik: the Proofs – Volume l, Victor H. Moll
Forthcoming Titles
Actions and Invariants of Algebraic Groups, Second Edition, Walter Ferrer Santos
and Alvaro Rittatore
Analytical Methods for Kolmogorov Equations, Second Edition, Luca Lorenzi
Complex Analysis: Conformal Inequalities and the Bierbach Conjecture, Prem K. Kythe
Computational Aspects of Polynomial Identities: Volume l, Kemer’s Theorems, 2nd Edition
Belov Alexey, Yaakov Karasik, Louis Halle Rowen
Cremona Groups and Icosahedron, Ivan Cheltsov and Constantin Shramov
Geometric Modeling and Mesh Generation from Scanned Images, Yongjie Zhang
Groups, Designs, and Linear Algebra, Donald L. Kreher
Handbook of the Tutte Polynomial, Joanna Anthony Ellis-Monaghan and Iain Moffat
Lineability: The Search for Linearity in Mathematics, Juan B. Seoane Sepulveda,
Richard W. Aron, Luis Bernal-Gonzalez, and Daniel M. Pellegrinao
Line Integral Methods and Their Applications, Luigi Brugnano and Felice Iaverno
Microlocal Analysis on Rˆn and on NonCompact Manifolds, Sandro Coriasco
Monomial Algebra, Second Edition, Rafael Villarreal
K22684_FM.indd 2 2/4/15 12:23 PM
Forthcoming Titles (continued)
Nonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory
Under Weak Topology for Nonlinear Operators and Block Operators with Applications
Aref Jeribi and Bilel Krichen
Partial Differential Equations with Variable Exponents: Variational Methods and
Quantitative Analysis, Vicentiu Radulescu and Dusan Repovs
Practical Guide to Geometric Regulation for Distributed Parameter Systems,
Eugenio Aulisa and David S. Gilliam
Reconstructions from the Data of Integrals, Victor Palamodov
Special Integrals of Gradshetyn and Ryzhik: the Proofs – Volume ll, Victor H. Moll
Stochastic Cauchy Problems in Infinite Dimensions: Generalized and Regularized
Solutions, Irina V. Melnikova and Alexei Filinkov
Symmetry and Quantum Mechanics, Scott Corry
K22684_FM.indd 3 2/4/15 12:23 PM
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Difference
Equations
Theory, Applications
and Advanced Topics
Third Edition
To my wife
Maria,
my son
James Williamson,
my daughter
Leah Marie.
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Contents
List of Figures xvii
List of Tables xix
PREFACE xxi
1 THE DIFFERENCE CALCULUS 1
1.1 GENESIS OF DIFFERENCE EQUATIONS . . . . . . . . . 1
1.1.1 Example A . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Example B . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.3 Example C . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.4 Example D . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.5 Example E . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.6 Example F . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 DEFINITIONS . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Example A . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.2 Example B . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 Example C . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.4 Example D . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 DERIVATION OF DIFFERENCE EQUATIONS . . . . . . 8
1.3.1 Example A . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Example B . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.3 Example C . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.4 Example D . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 EXISTENCE AND UNIQUENESS THEOREM . . . . . . . 10
1.5 OPERATORS ∆ AND E . . . . . . . . . . . . . . . . . . . . 10
1.6 ELEMENTARY DIFFERENCE OPERATORS . . . . . . . 14
1.6.1 Example A . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6.2 Example B . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6.3 Example C . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6.4 Example D . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6.5 Example E . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6.6 Example F . . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 FACTORIAL POLYNOMIALS . . . . . . . . . . . . . . . . 19
1.7.1 Example A . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.2 Example B . . . . . . . . . . . . . . . . . . . . . . . . 24
vii
viii Contents
1.7.3 Example C . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7.4 Example D . . . . . . . . . . . . . . . . . . . . . . . . 26
1.8 OPERATOR ∆ 1 AND THE SUM CALCULUS . . . . . . . 27
−
1.8.1 Example A . . . . . . . . . . . . . . . . . . . . . . . . 33
1.8.2 Example B . . . . . . . . . . . . . . . . . . . . . . . . 34
1.8.3 Example C . . . . . . . . . . . . . . . . . . . . . . . . 35
1.8.4 Example D . . . . . . . . . . . . . . . . . . . . . . . . 35
1.8.5 Example E . . . . . . . . . . . . . . . . . . . . . . . . 36
1.8.6 Example F . . . . . . . . . . . . . . . . . . . . . . . . 36
1.8.7 Example G . . . . . . . . . . . . . . . . . . . . . . . . 37
1.8.8 Example H . . . . . . . . . . . . . . . . . . . . . . . . 37
2 FIRST-ORDER DIFFERENCE EQUATIONS 45
2.1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2 GENERAL LINEAR EQUATION . . . . . . . . . . . . . . . 45
2.2.1 Example A . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.2 Example B . . . . . . . . . . . . . . . . . . . . . . . . 47
2.2.3 Example C . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.4 Example D . . . . . . . . . . . . . . . . . . . . . . . . 48
2.2.5 Example E . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.6 Example F . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.7 Example G . . . . . . . . . . . . . . . . . . . . . . . . 50
n
2.3 y y =(n+1)k . . . . . . . . . . . . . . . . . . . . . . 50
k+1− k
2.3.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4 yk+1 =RkYk . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.4.1 Example A . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4.2 Example B . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4.3 Example C . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4.4 Example D . . . . . . . . . . . . . . . . . . . . . . . . 55
2.5 CONTINUED FRACTIONS . . . . . . . . . . . . . . . . . . 55
2.5.1 Example A . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5.2 Example B . . . . . . . . . . . . . . . . . . . . . . . . 57
2.5.3 Example C . . . . . . . . . . . . . . . . . . . . . . . . 57
2.6 yk+1−kyk =Pk . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.7 AGENERALFIRST-ORDEREQUATION:GEOMETRICAL
METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.7.1 Example A . . . . . . . . . . . . . . . . . . . . . . . . 62
2.7.2 Example B . . . . . . . . . . . . . . . . . . . . . . . . 62
2.7.3 Example C . . . . . . . . . . . . . . . . . . . . . . . . 64
2.8 A GENERAL FIRST-ORDER EQUATION: EXPANSION
TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.8.1 Example A . . . . . . . . . . . . . . . . . . . . . . . . 70
2.8.2 Example B . . . . . . . . . . . . . . . . . . . . . . . . 72
2.8.3 Example C . . . . . . . . . . . . . . . . . . . . . . . . 72
2.8.4 Example D . . . . . . . . . . . . . . . . . . . . . . . . 75