Linear systems of fractional nabla difference equations

In this paper we shall consider a linear system of fractional nabla difference equations with constant coefficients. We shall construct the fundamental matrix for the homogeneous system and the causal Green’s function for the nonhomogeneous system. We employ transform methods and series methods, and we illustrate analogies with classical first-order differential or difference equations. We shall close the paper with an asymptotic result that follows from the analysis of a half-order nabla difference equation

A transform method in discrete fractional calculus

We begin with an introduction to a calculus of fractional finite differences. We extend the discrete Laplace transform to develop a discrete transform method. We define a family of finite fractional difference equations and employ the transform method to obtain solutions

Discrete fractional calculus with the nabla operator

Properties of discrete fractional calculus in the sense of a backward difference are introduced and developed. Exponential laws and a product rule are developed and relations to the forward fractional calculus are explored. Properties of the Laplace transform for the nabla derivative on the time scale of integers are developed and a fractional finite difference equation is solved with a transform method. As a corollary, two new identities for the gamma function are exhibited

On the Definitions of Nabla Fractional Operators

We show that two recent definitions of discrete nabla fractional sum operators are related. Obtaining such a relation between two operators allows one to prove basic properties of the one operator by using the known properties of the other. We illustrate this idea with proving power rule and commutative property of discrete fractional sum operators. We also introduce and prove summation by parts formulas for the right and left fractional sum and difference operators, where we employ the Riemann-Liouville definition of the fractional difference. We formalize initial value problems for nonlinear fractional difference equations as an application of our findings. An alternative definition for the nabla right fractional difference operator is also introduced

Gronwall\u27s inequality on discrete fractional calculus

AbstractIn 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 Gronwall’s 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 Gronwall’s 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

The quasilinearization method for boundary value problems on time scales

AbstractIn this paper, we apply the method of quasilinearization to a family of boundary value problems for second order dynamic equations −yΔ∇+q(t)y=H(t,y) on time scales. The results include a variety of possible cases when H is either convex or a splitting of convex and concave parts and whether lower and upper solutions are of natural form or of natural coupled form