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Abstract and Applied Analysis
Volume 2007, Article ID 58373, 24 pages
http://dx.doi.org/10.1155/2007/58373
Research Article

The Convolution on Time Scales

1Department of Mathematics and Statistics, University of Missouri-Rolla, Rolla, MO 65401, USA
2Department of Mathematics, Atilim University, Incek, Ankara 06836, Turkey

Received 19 March 2007; Accepted 24 May 2007

Academic Editor: Allan C. Peterson

Copyright © 2007 Martin Bohner and Gusein Sh. Guseinov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main theme in this paper is an initial value problem containing a dynamic version of the transport equation. Via this problem, the delay (or shift) of a function defined on a time scale is introduced, and the delay in turn is used to introduce the convolution of two functions defined on the time scale. In this paper, we give some elementary properties of the delay and of the convolution and we also prove the convolution theorem. Our investigation contains a study of the initial value problem under consideration as well as some results about power series on time scales. As an extensive example, we consider the q-difference equations case.