There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Fractional Differential Equations

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

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

Dynamic Equations on Time Scales: An Introduction with Applications

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

/pdf/dynamic-equations-on-time-scales-3yooyeozaw.pdf

Dynamic Equations on Time Scales

Introduction to the The Time Scales Calculus (Bohner/Guseinov/Peterson) * Some Dynamic Equations (Akin-Bohner/Bohner) * Nabla Dynamic Equations (Anderson/Bullock/Erbe/Peterson/Tran) * Second-Order Self-Adjoint Equations with Mixed Derivatives (Messer) * Riemann and Lebesgue Integration (Bohner/Guseinov) * Lower and Upper Solutions for Two Point Boundary Value Problems (Akin-Bohner/Atici/Kaymakcalan) * Positive Solutions of Boundary Value Problems (Anderson/Avery/Davis/Henderson/Yin) * Disconjugacy and Higher Order Dynamic Equations (Eloe) * Boundary Value Problems on Infinite Intervals: A Topological Approach (Agarwal/Bohner/O'Regan) * Symplectic Dynamic Systems (Dosly/Hilger/Hilscher) * Bibliography * Index

/pdf/advances-in-dynamic-equations-on-time-scales-27mx4rklvv.pdf

Advances in dynamic equations on time scales

https://web.mst.edu/~bohner/papers/deotsas.pdf

Dynamic equations on time scales: a survey

The study of dynamic systems on time scales not only unifies continuous and discrete processes, but also helps in revealing diversities in the corresponding results. In this paper we shall develop basic tools of calculus on time scales such as versions of Taylor’s formula, l’Hospital’s rule, and Kneser’s theorem. Applications of these results in the study of asymptotic and oscillatory behavior of solutions of higher order equations on time scales are addressed. As a further application of Taylor’s formula, Abel-Gontscharoff interpolating polynomial on time scales is constructed and best possible error bounds are offered. We have also included notes at the end of each section which indicate further scope of the calculus developed in this paper.

Martin Bohner

Papers

Dynamic Equations on Time Scales: An Introduction with Applications

Dynamic Equations on Time Scales

Advances in dynamic equations on time scales

Dynamic equations on time scales: a survey

Basic Calculus on Time Scales and some of its Applications