Artur M. C. Brito da Cruz

A fractional calculus on arbitrary time scales

We introduce a general notion of fractional (noninteger) derivative for functions defined on arbitrary time scales. The basic tools for the time-scale fractional calculus (fractional differentiation and fractional integration) are then developed. As particular cases, one obtains the usual time-scale Hilger derivative when the order of differentiation is one, and a local approach to fractional calculus when the time scale is chosen to be the set of real numbers. HighlightsWe introduce a general fractional calculus on an arbitrary time scale.The basic tools for the time-scale fractional calculus are rigorously developed.The time-scale Hilger derivative is obtained when the order of differentiation is one.Kolwankar-Gangal approach is obtained when the time scale is the set of real numbers.

Higher-order Hahn’s quantum variational calculus

Abstract We prove a necessary optimality condition of Euler–Lagrange type for quantum variational problems involving Hahn’s derivatives of higher-order.

Symmetric differentiation on time scales

Abstract We define a symmetric derivative on an arbitrary nonempty closed subset of the real numbers and derive some of its properties. It is shown that real-valued functions defined on time scales that are neither delta nor nabla differentiable can be symmetric differentiable.

Nonsymmetric and symmetric fractional calculi on arbitrary nonempty closed sets

We introduce a nabla, a delta, and a symmetric fractional calculus on arbitrary nonempty closed subsets of the real numbers. These fractional calculi provide a study of differentiation and integration of noninteger order on discrete, continuous, and hybrid settings. Main properties of the new fractional operators are investigated, and some fundamental results presented, illustrating the interplay between discrete and continuous behaviors.

A Symmetric Quantum Calculus

We introduce the α,β-symmetric difference derivative and the α,β-symmetric Norlund sum. The associated symmetric quantum calculus is developed, which can be seen as a generalization of the forward and backward h-calculus.

The Diamond Integral on Time Scales

We define a more general type of integral on time scales. The new diamond integral is a refined version of the diamond-alpha integral introduced in 2006 by Sheng et al. A mean value theorem for the diamond integral is proved, as well as versions of Holder’s, Cauchy–Schwarz’s, and Minkowski’s inequalities.

The q-symmetric variational calculus

We bring a new approach to the study of quantum calculus and introduce the q-symmetric variational calculus. We prove a necessary optimality condition of Euler-Lagrange type and a sufficient optimality condition for symmetric quantum variational problems. The results are illustrated with an example.

A Symmetric Nörlund Sum with Application to Inequalities

Properties of an α,β-symmetric Norlund sum are studied. Inspired in the work by Agarwal et al., α,β-symmetric quantum versions of Holder’s, Cauchy–Schwarz’s and Minkowski’s inequalities are obtained.