## Eastern Kentucky University

# Derivatives of the Solution to a Dynamic Boundary Value Problem on a Discrete Time Scale

## Institution

Eastern Kentucky University

## Faculty Advisor/ Mentor

Jeffrey Neugebauer

## Abstract

A time scale is an arbitrary nonempty closed subset of the real numbers. The field of time scales calculus was introduced by Stefan Hilger in order to unify discrete and continuous analysis. Some interesting examples of time scales calculus include calculus on the real line, discrete calculus, and q-difference equations. Time scales calculus has numerous applications dealing with biology, engineering, economics, physics, and other areas of science. We began by introducing some basics of time scales, including the forward jump operator, the backwards jump operator, the graininess function, and the delta derivative of a function. We then considered a second order nonlinear dynamic boundary value problem with conjugate boundary conditions on a discrete time scale. A solution of this time scale may be delta differentiated with respect to the boundary points. Here, the delta derivative of the solution solves the boundary value problem consisting of the dynamic analog of the variational equation along with interesting boundary conditions.

Derivatives of the Solution to a Dynamic Boundary Value Problem on a Discrete Time Scale

A time scale is an arbitrary nonempty closed subset of the real numbers. The field of time scales calculus was introduced by Stefan Hilger in order to unify discrete and continuous analysis. Some interesting examples of time scales calculus include calculus on the real line, discrete calculus, and q-difference equations. Time scales calculus has numerous applications dealing with biology, engineering, economics, physics, and other areas of science. We began by introducing some basics of time scales, including the forward jump operator, the backwards jump operator, the graininess function, and the delta derivative of a function. We then considered a second order nonlinear dynamic boundary value problem with conjugate boundary conditions on a discrete time scale. A solution of this time scale may be delta differentiated with respect to the boundary points. Here, the delta derivative of the solution solves the boundary value problem consisting of the dynamic analog of the variational equation along with interesting boundary conditions.