1st Edition

Form Symmetries and Reduction of Order in Difference Equations

By Hassan Sedaghat Copyright 2011
    325 Pages 31 B/W Illustrations
    by CRC Press

    328 Pages 31 B/W Illustrations
    by CRC Press

    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.

    Introduction

    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

    References

    Index

    Notes and Problems appear at the end of each chapter.

    Biography

    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.

    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