Dynamic Equations On Time Scales: An Introduction With Applications
by Allan Peterson /
2012 / English / PDF
6.6 MB Download
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.
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.