Ewa Girejko

Overview of fractional h-difference operators

Fractional difference operators and their properties are discussed. We give a characterization of three operators that we call Grunwald-Letnikov, Riemann-Liouville and Caputo like difference operators. We show relations among them. In the paper, linear fractional h-difference equations are described. We give formulas of solutions to initial value problems. Crucial formulas are gathered in the tables presented in the last section of the paper.

The contingent epiderivative and the calculus of variations on time scales

The calculus of variations on time scales is considered. We propose a new approach to the subject that consists of applying a differentiation tool called the contingent epiderivative. It is shown that the contingent epiderivative applied to the calculus of variations on time scales is very useful: it allows to unify the delta and nabla approaches previously considered in the literature. Generalized versions of the Euler–Lagrange necessary optimality conditions are obtained, both for the basic problem of the calculus of variations and isoperimetric problems. As particular cases one gets the recent delta and nabla results.

Comparison of h-Difference Fractional Operators

We compare three different types of h-difference fractional operators: Grunwald-Letnikov, Caputo, Riemann-Liouville types of operators. There is introduced the formula for fundamental matrix of solutions for linear systems of h-difference fractional equations with Grunwald-Letnikov type operator while the one with Caputo type or Riemann-Liouville type is well known. We present new formulas for linear control systems with the mentioned operators.

Fractional nonlinear systems with sequential operators

In the paper possible approximation of solutions to initial value problems stated for fractional nonlinear equations with sequential derivatives of Caputo type is presented. We proved that values of Caputo derivatives in continuous case can be approximated by corresponding values of h-difference operators with h being small enough. Numerical examples are presented.

Stability of nonlinear

In the paper we study the subject of stability of systems with

h

h

-difference systems with

-differences of Caputo-, Riemann-Liouville- and Grunwald-Letnikov-type with

n

n

fractional orders

fractional orders. The equivalent descriptions of fractional

A unified approach to the calculus of variations on time scales

h