Martin Bohner Discusses the Study of Dynamic Equations
Emerging Research FRonts Commentary, February 2011
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Article: Oscillation of second order nonlinear dynamic equations on time scales
Authors: Bohner, M;Saker, SH |
Martin Bohner talks with ScienceWatch.com and answers a few questions about this month's Emerging Research Front paper in the field of Mathematics.
Why do you think your paper is highly
cited?
The concept of time scales analysis is a fairly new idea. In 1988, it was introduced by the German mathematician Stefan Hilger in his Ph.D. thesis and supervised by Bernd Aulbach. It combines the traditional areas of continuous and discrete analysis into one theory. After the publication of two textbooks in this area (by Bohner and Peterson, 2001, 2003), more and more researchers were getting involved in this fast-growing field of mathematics.
Our paper is one of the first papers in nonlinear oscillation theory on time scales, and the reason why it is highly cited is that numerous mathematicians worldwide have started to work on oscillation theory, in particular, nonlinear oscillation theory, on time scales.
Does it describe a new discovery, methodology, or
synthesis of knowledge?
Coauthor, Samir Saker
The study of dynamic equations brings together the traditional research areas of (ordinary and partial) differential and difference equations. It allows one to handle these two research areas at the same time, hence shedding light on the reasons for their seeming discrepancies. In fact, many new results for the continuous and discrete cases have been obtained by studying the more general time scales case.
Would you summarize the significance of your paper
in layman's terms?
In his book Men of Mathematics (Simon and Schuster, New York, pp. 13/14, 1937), Eric Temple Bell wrote: "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." Time scales analysis accomplishes exactly this. But not only can this theory unify continuous and discrete analysis, it is also able to extend the study of differential (continuous) and difference (discrete) equations to a more general class of dynamic equations, which includes, e.g., quantum-difference equations.
How did you become involved in this research, and
how would you describe the particular challenges, setbacks, and
successes that you've encountered along the way?
Both authors have a background in oscillation theory for differential and difference equations (both of them wrote books on this topic) and both authors also have a background in the unified theory, dynamic equations on time scales (both of them wrote books on this topic). Hence it was fairly natural to combine all of this knowledge by writing this paper.
Where do you see your research leading in the
future?
Research in the area of dynamic equations on time scales is still just at
the beginning. Currently there are about 400 researchers worldwide who have
published about 600 research articles in the area of time scales. The
number of these researchers is steadily growing and this research area is
wide open. This calculus has potential applications in such areas as
engineering, biology, economics and finance, physics, chemistry, social
sciences, medical sciences, mathematics education, and
others.
Dr. Martin Bohner
Missouri S&T
Department of Mathematics
Department of Economics
Rolla, MO, USA
Current Address:
University of Ulm
Faculty of Mathematics and Economics
Institute of Mathematical Finance
Ulm, Germany
Dr. Samir H. Saker
Mansoura University
Faculty of Science
Department of Mathematics
Mansoura, Egypt
Current Address:
King Saud University
College of Science
Department of Mathematics
Riyadh, Saudi Arabia
KEYWORDS: OSCILLATION; SECOND ORDER NONLINEAR DYNAMIC EQUATION; TIME SCALE; RICCATI TRANSFORMATION TECHNIQUE; POSITIVE SOLUTION; MEASURE CHAIN; DISCONJUGACY.