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This book summarizes the qualitative theory of differential
equations with or without delays, collecting recent oscillation
studies important to applications and further developments in
mathematics, physics, engineering, and biology. The authors address
oscillatory and nonoscillatory properties of first-order delay and
neutral delay differential equations, second-order delay and
ordinary differential equations, higher-order delay differential
equations, and systems of nonlinear differential equations. The
final chapter explores key aspects of the oscillation of dynamic
equations on time scales-a new and innovative theory that
accomodates differential and difference equations simultaneously.
This book comprises selected papers of the 25th International
Conference on Difference Equations and Applications, ICDEA 2019,
held at UCL, London, UK, in June 2019. The volume details the
latest research on difference equations and discrete dynamical
systems, and their application to areas such as biology, economics,
and the social sciences. Some chapters have a tutorial style and
cover the history and more recent developments for a particular
topic, such as chaos, bifurcation theory, monotone dynamics, and
global stability. Other chapters cover the latest personal research
contributions of the author(s) in their particular area of
expertise and range from the more technical articles on abstract
systems to those that discuss the application of difference
equations to real-world problems. The book is of interest to both
Ph.D. students and researchers alike who wish to keep abreast of
the latest developments in difference equations and discrete
dynamical systems.
This book provides an extensive survey on Lyapunov-type
inequalities. It summarizes and puts order into a vast literature
available on the subject, and sketches recent developments in this
topic. In an elegant and didactic way, this work presents the
concepts underlying Lyapunov-type inequalities, covering how they
developed and what kind of problems they address. This survey
starts by introducing basic applications of Lyapunov's
inequalities. It then advances towards even-order, odd-order, and
higher-order boundary value problems; Lyapunov and Hartman-type
inequalities; systems of linear, nonlinear, and quasi-linear
differential equations; recent developments in Lyapunov-type
inequalities; partial differential equations; linear difference
equations; and Lyapunov-type inequalities for linear, half-linear,
and nonlinear dynamic equations on time scales, as well as linear
Hamiltonian dynamic systems. Senior undergraduate students and
graduate students of mathematics, engineering, and science will
benefit most from this book, as well as researchers in the areas of
ordinary differential equations, partial differential equations,
difference equations, and dynamic equations. Some background in
calculus, ordinary and partial differential equations, and
difference equations is recommended for full enjoyment of the
content.
This book offers the reader an overview of recent developments of
multivariable dynamic calculus on time scales, taking readers
beyond the traditional calculus texts. Covering topics from
parameter-dependent integrals to partial differentiation on time
scales, the book's nine pedagogically oriented chapters provide a
pathway to this active area of research that will appeal to
students and researchers in mathematics and the physical sciences.
The authors present a clear and well-organized treatment of the
concept behind the mathematics and solution techniques, including
many practical examples and exercises.
This book offers the reader an overview of recent developments of
multivariable dynamic calculus on time scales, taking readers
beyond the traditional calculus texts. Covering topics from
parameter-dependent integrals to partial differentiation on time
scales, the book's nine pedagogically oriented chapters provide a
pathway to this active area of research that will appeal to
students and researchers in mathematics and the physical sciences.
The authors present a clear and well-organized treatment of the
concept behind the mathematics and solution techniques, including
many practical examples and exercises.
These proceedings of the 20th International Conference on
Difference Equations and Applications cover the areas of difference
equations, discrete dynamical systems, fractal geometry, difference
equations and biomedical models, and discrete models in the natural
sciences, social sciences and engineering. The conference was held
at the Wuhan Institute of Physics and Mathematics, Chinese Academy
of Sciences (Hubei, China), under the auspices of the International
Society of Difference Equations (ISDE) in July 2014. Its purpose
was to bring together renowned researchers working actively in the
respective fields, to discuss the latest developments, and to
promote international cooperation on the theory and applications of
difference equations. This book will appeal to researchers and
scientists working in the fields of difference equations, discrete
dynamical systems and their applications.
These proceedings of the 20th International Conference on
Difference Equations and Applications cover the areas of difference
equations, discrete dynamical systems, fractal geometry, difference
equations and biomedical models, and discrete models in the natural
sciences, social sciences and engineering. The conference was held
at the Wuhan Institute of Physics and Mathematics, Chinese Academy
of Sciences (Hubei, China), under the auspices of the International
Society of Difference Equations (ISDE) in July 2014. Its purpose
was to bring together renowned researchers working actively in the
respective fields, to discuss the latest developments, and to
promote international cooperation on the theory and applications of
difference equations. This book will appeal to researchers and
scientists working in the fields of difference equations, discrete
dynamical systems and their applications.
Excellent introductory material on the calculus of time scales and
dynamic equations.; Numerous examples and exercises illustrate the
diverse application of dynamic equations on time scales.; Unified
and systematic exposition of the topics allows good transitions
from chapter to chapter.; Contributors include Anderson, M. Bohner,
Davis, Dosly, Eloe, Erbe, Guseinov, Henderson, Hilger, Hilscher,
Kaymakcalan, Lakshmikantham, Mathsen, and A. Peterson, founders and
leaders of this field of study.; Useful as a comprehensive resource
of time scales and dynamic equations for pure and applied
mathematicians.; Comprehensive bibliography and index complete this
text.
On becoming familiar with difference equations and their close re
lation to differential equations, I was in hopes that the theory of
difference equations could be brought completely abreast with that
for ordinary differential equations. [HUGH L. TURRITTIN, My
Mathematical Expectations, Springer Lecture Notes 312 (page 10),
1973] A major task of mathematics today is to harmonize the
continuous and the discrete, to include them in one comprehensive
mathematics, and to eliminate obscurity from both. [E. T. BELL, Men
of Mathematics, Simon and Schuster, New York (page 13/14), 1937]
The theory of time scales, which has recently received a lot of
attention, was introduced by Stefan Hilger in his PhD thesis [159]
in 1988 (supervised by Bernd Aulbach) in order to unify continuous
and discrete analysis. This book is an intro duction to the study
of dynamic equations on time scales. Many results concerning
differential equations carryover quite easily to corresponding
results for difference equations, while other results seem to be
completely different in nature from their continuous counterparts.
The study of dynamic equations on time scales reveals such
discrepancies, and helps avoid proving results twice, once for
differential equa tions and once for difference equations. The
general idea is to prove a result for a dynamic equation where the
domain of the unknown function is a so-called time scale, which is
an arbitrary nonempty closed subset of the reals.
Excellent introductory material on the calculus of time scales and
dynamic equations.; Numerous examples and exercises illustrate the
diverse application of dynamic equations on time scales.; Unified
and systematic exposition of the topics allows good transitions
from chapter to chapter.; Contributors include Anderson, M. Bohner,
Davis, Dosly, Eloe, Erbe, Guseinov, Henderson, Hilger, Hilscher,
Kaymakcalan, Lakshmikantham, Mathsen, and A. Peterson, founders and
leaders of this field of study.; Useful as a comprehensive resource
of time scales and dynamic equations for pure and applied
mathematicians.; Comprehensive bibliography and index complete this
text.
The study of dynamic equations on a measure chain (time scale) goes
back to its founder S. Hilger (1988), and is a new area of still
fairly theoretical exploration in mathematics. Motivating the
subject is the notion that dynamic equations on measure chains can
build bridges between continuous and discrete mathematics. Further,
the study of measure chain theory has led to several important
applications, e.g., in the study of insect population models,
neural networks, heat transfer, and epidemic models. Key features
of the book: * Introduction to measure chain theory; discussion of
its usefulness in allowing for the simultaneous development of
differential equations and difference equations without having to
repeat analogous proofs * Many classical formulas or procedures for
differential and difference equations cast in a new light * New
analogues of many of the "special functions" studied * Examination
of the properties of the "exponential function" on time scales,
which can be defined and investigated using a particularly simple
linear equation * Additional topics covered: self-adjoint
equations, linear systems, higher order equations, dynamic
inequalities, and symplectic dynamic systems * Clear, motivated
exposition, beginning with preliminaries and progressing to more
sophisticated text * Ample examples and exercises throughout the
book * Solutions to selected problems Requiring only a first
semester of calculus and linear algebra, Dynamic Equations on Time
Scales may be considered as an interesting approach to differential
equations via exposure to continuous and discrete analysis. This
approach provides an early encounter with many applications in such
areas as biology, physics, and engineering. Parts of the book may
be used in a special topics seminar at the senior undergraduate or
beginning graduate levels. Finally, the work may serve as a
reference to stimulate the development of new kinds of equations
with potentially new applications.
This book comprises selected papers of the 25th International
Conference on Difference Equations and Applications, ICDEA 2019,
held at UCL, London, UK, in June 2019. The volume details the
latest research on difference equations and discrete dynamical
systems, and their application to areas such as biology, economics,
and the social sciences. Some chapters have a tutorial style and
cover the history and more recent developments for a particular
topic, such as chaos, bifurcation theory, monotone dynamics, and
global stability. Other chapters cover the latest personal research
contributions of the author(s) in their particular area of
expertise and range from the more technical articles on abstract
systems to those that discuss the application of difference
equations to real-world problems. The book is of interest to both
Ph.D. students and researchers alike who wish to keep abreast of
the latest developments in difference equations and discrete
dynamical systems.
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