Form Symmetries and Reduction of Order in Difference Equations: 1st Edition (Paperback) book cover

Form Symmetries and Reduction of Order in Difference Equations

1st Edition

By Hassan Sedaghat

CRC Press

325 pages | 31 B/W Illus.

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Form Symmetries and Reduction of Order in Difference Equations presents a new approach to the formulation and analysis of difference equations in which the underlying space is typically an algebraic group. In some problems and applications, an additional algebraic or topological structure is assumed in order to define equations and obtain significant results about them. Reflecting the author’s past research experience, the majority of examples involve equations in finite dimensional Euclidean spaces.

The book first introduces difference equations on groups, building a foundation for later chapters and illustrating the wide variety of possible formulations and interpretations of difference equations that occur in concrete contexts. The author then proposes a systematic method of decomposition for recursive difference equations that uses a semiconjugate relation between maps. Focusing on large classes of difference equations, he shows how to find the semiconjugate relations and accompanying factorizations of two difference equations with strictly lower orders. The final chapter goes beyond semiconjugacy by extending the fundamental ideas based on form symmetries to nonrecursive difference equations.

With numerous examples and exercises, this book is an ideal introduction to an exciting new domain in the area of difference equations. It takes a fresh and all-inclusive look at difference equations and develops a systematic procedure for examining how these equations are constructed and solved.


This book presents a new approach to the formulation and study of difference equations. … The book is well organized. It is addressed to a broad audience in difference equations.

—Vladimir Sh. Burd, Mathematical Reviews, 2012e

Table of Contents


Difference Equations on Groups

Basic definitions

One equation, many interpretations

Examples of difference equations on groups

Semiconjugate Factorization and Reduction of Order

Semiconjugacy and ordering of maps

Form symmetries and SC factorizations

Order-reduction types

SC factorizations as triangular systems

Order-preserving form symmetries

Homogeneous Equations of Degree One

Homogeneous equations on groups

Characteristic form symmetry of HD1 equations

Reductions of order in HD1 equations

Absolute value equation

Type-(k,1) Reductions

Invertible-map criterion

Identity form symmetry

Inversion form symmetry

Discrete Riccati equation of order two

Linear form symmetry

Difference equations with linear arguments

Field-inverse form symmetry

Type-(1,k) Reductions

Linear form symmetry revisited

Separable difference equations

Equations with exponential and power functions

Time-Dependent Form Symmetries

The semiconjugate relation and factorization

Invertible-map criterion revisited

Time-dependent linear form symmetry

SC factorization of linear equations

Nonrecursive Difference Equations

Examples and discussion

Form symmetries, factors, and cofactors

Semi-invertible map criterion

Quadratic difference equations

An order-preserving form symmetry

Appendix: Asymptotic Stability on the Real Line



Notes and Problems appear at the end of each chapter.

About the Author

Hassan Sedaghat is a professor of mathematics at Virginia Commonwealth University. His research interests include difference equations and discrete dynamical systems and their applications in mathematics, economics, biology, and medicine.

About the Series

Advances in Discrete Mathematics and Applications

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Differential Equations
SCIENCE / Mathematical Physics