Ferhan Merdivenci Atici

Initial value problems in discrete fractional calculus

This paper is devoted to the study of discrete fractional calculus; the particular goal is to define and solve well-defined discrete fractional difference equations. For this purpose we first carefully develop the commutativity properties of the fractional sum and the fractional difference operators. Then a v-th (0 < v ≤ 1) order fractional difference equation is defined. A nonlinear problem with an initial condition is solved and the corresponding linear problem with constant coefficients is solved as an example. Further, the half-order linear problem with constant coefficients is solved with a method of undetermined coefficients and with a transform method.

An application of time scales to economics

Economics is a discipline in which there appears to be many opportunities for applications of time scales. The time scales approach will not only unify the standard discrete and continuous models in economics, but also, for example, allows for payments which arrive at unequally spaced points in time. We present a dynamic optimization problem from economics, construct a time scales model, and apply calculus of variations to derive a solution. Time scale calculus would allow exploration of a variety of situations in economics.

Two-point boundary value problems for finite fractional difference equations

In this paper, we introduce a two-point boundary value problem for a finite fractional difference equation. We invert the problem and construct and analyse the corresponding Greens function. We then provide an application and obtain sufficient conditions for the existence of positive solutions for a two-point boundary value problem for a nonlinear finite fractional difference equation.

Fractional q-Calculus on a time scale

Abstract The study of fractional q-calculus in this paper serves as a bridge between the fractional q-calculus in the literature and the fractional q-calculus on a time scale , where to ∈ ℝ and 0 < q < 1. By use of time scale calculus notation, we find the proof of many results more straight forward. We shall develop some properties of fractional q-calculus, we shall develop some properties of a q-Laplace transform, and then we shall employ the q-Laplace transform to solve fractional q-difference equations.

Existence and uniqueness results for discrete second-order periodic boundary value problems

Abstract We prove existence and uniqueness results for solutions of second-order nonlinear difference equations on a finite discrete segment with periodic boundary conditions. The results are based on the notion of upper and lower solutions.

Gronwall's inequality on discrete fractional calculus

In this paper, we introduce discrete fractional sum equations and inequalities. We obtain the equivalence of an initial value problem for a discrete fractional equation and a discrete fractional sum equation. Then we give an explicit solution to the linear discrete fractional sum equation. This allows us to state and prove an analogue of Gronwalls inequality on discrete fractional calculus. We employ a nabla, or backward difference; we employ the Riemann-Liouville definition of the fractional difference. As a result, we obtain Gronwalls inequality for discrete calculus with the nabla operator. We illustrate our results with an application that gives continuous dependence of solutions of initial value problems on initial conditions.

A production-inventory model of HMMS on time scales

The aim of this work is to investigate the optimal production and inventory paths of HMMS type models (proposed by Holt, Modigliani, Muth and Simon) on complex time domains. Time scale calculus which is a rapidly growing theory is a main tool for solving and for analyzing the model. This work will enrich management and economics by providing a flexible and capable modelling technique.

Inventory model of deteriorating items on non-periodic discrete-time domains

In this paper, we demonstrate how to model a discrete-time dynamic process on a non-periodic time domain with applications to operations research. We introduce a discrete-time model of inventory with deterioration on domains where time points may be unevenly spaced over a time interval. We formalize the average cost function composed of storage, depreciation and back-ordering costs. The optimal condition is given to locate the optimal point that minimizes the average cost function. Finally, we present simulations to demonstrate how a manager can use this model to make inventory decisions.

First order difference equations with maxima and nonlinear functional boundary value conditions

This paper is devoted to the existence of solutions for a problem of first order difference equations with maxima and with nonlinear functional boundary value conditions. Such boundary conditions include, among others, initial, periodic, antiperiodic and multipoint boundary value conditions, as particular cases.

Convex Functions on Discrete Time Domains

In this paper, we introduce the deûnition of a convex real valued function f deûned on the set of integers, Z. We prove that f is convex on Z if and only if ∆ f ≥ 0 on Z. As a ûrst application of this new concept, we state and prove discrete Hermite-Hadamard inequality using the basics of discrete calculus (i.e. the calculus onZ). Second, we state and prove the discrete fractionalHermite-Hadamard inequality using the basics of discrete fractional calculus. We close the paper by deûning the convexity of a real valued function on any time scale.