Dynamic Equations on Time Scales: An Introduction with Applications

ISBN-13: 9781461266594 / Angielski / Miękka / 2012 / 358 str.

Martin Bohner; Allan Peterson

Opis

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.

Recenzje

"This would be an excellent book to use in a topics course on dynamic equations on time scales at the advanced undergraduate level and/or beginning graduate level."

-Zentralblatt Math

"The monograph under review comes at an excellent time in the rapid development of dynamic equations on time scales. Both authors are authorities in this field of study and they have produced an excellent introduction to it. Much of the material is accessible to upper-level undergraduate mathematics majors, and yet, the results and the techniques are pertinent to active researchers in the area."

-Mathematical Reviews

Spis treści

Preface * The Time Scales Calculus * First Order Linear Equations * Second Order Linear Equations * Self-Adjoint Equations * Linear Systems and Higher Order Equations * Dynamic Inequalities * Linear Symplectic Dynamic Systems * Extensions * Solutions to Selected Problems * Bibliography * Index

O autorze

An introduction to measure chain (time scale) theory with emphasis on
its usefulness in allowing for the simultaneous development of
differential equations and difference equations. Motivating the
subject is the notion that dynamic equations on measure chains can
build bridges between continuous and discrete mathematics. The study
of measure chain theory has also led to applications in the study of
insect population models, neural networks, heat transfer, epidemic
models. This book may stimulate the development of new kinds of
equations with potentially new applications. Includes examples,
exercises, solutions.