《Dynamic equations on time scales a survey》.pdf

Journal of Computational and Applied Mathematics 141 (2002) 1–26 /locate/cam Dynamic equations on time scales: a survey a b ; ∗ c d Ravi Agarwal , Martin Bohner , Donal O’Regan , Allan Peterson a National University of Singapore, Department of Mathematics, Singapore 119260 b University of Missouri-Rolla, Department of Mathematics an dStatistics, Rolla, MO 65409-0020, USA cNational University of Ireland, Department of Mathematics, Galway, Irelan d University of Nebraska-Lincoln, Department of Mathematics an dStatistics, Lincoln, NE 68588-0323, USA Received 20 August 2000; received in revised form 26 February 2001 Abstract The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area of mathematics that has recently received a lot of attention. It has been created in order to unify the study of dierential and dierence equations. In this paper we give an introduction to the time scales calculus. We also present various properties of the exponential function on an arbitrary time scale, and use it to solve linear dynamic equations of rst order. Several examples and applications, among them an insect population model, are considered. We then use the exponential function to dene hyperbolic and trigonometric functions and use those to solve linear dynamic equations of second order with constant coecients. Finally, we consider self-adjoint eq