98 citations · 229 across the 9 of their papers we have counts for
9 papers
Nonsymmetric and Symmetric Fractional Calculi on Arbitrary Nonempty Closed Sets
Nadia Benkhettou, Artur M. C. Brito da Cruz, Delfim F. M. Torres
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 differentia…
A Fractional Calculus on Arbitrary Time Scales: Fractional Differentiation and Fractional Integration
Nadia Benkhettou, Artur M. C. Brito da Cruz, Delfim F. M. Torres
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 (fraction…
Symmetric Quantum Calculus
Artur M. C. Brito da Cruz
We generalize the Hahn variational calculus by studying problems of the calculus of variations with higher-order derivatives. The symmetric quantum calculus is studied, namely the…
Symmetric Differentiation on Time Scales
Artur M. C. Brito da Cruz, Natalia Martins, Delfim F. M. Torres
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 tim…
Hahn's Symmetric Quantum Variational Calculus
Artur M. C. Brito da Cruz, Natalia Martins, Delfim F. M. Torres
We introduce and develop the Hahn symmetric quantum calculus with applications to the calculus of variations. Namely, we obtain a necessary optimality condition of Euler-Lagrange t…
A symmetric Norlund sum with application to inequalities
Artur M. C. Brito da Cruz, Natalia Martins, Delfim F. M. Torres
Properties of an -symmetric Norlund sum are studied. Inspired in the work by Agarwal et al., -symmetric quantum versions of Holder, Cauchy-Schwarz and Minkowski inequalit…
A symmetric quantum calculus
Artur M. C. Brito da Cruz, Natalia Martins, Delfim F. M. Torres
We introduce the -symmetric difference derivative and the -symmetric Nörlund sum. The associated symmetric quantum calculus is developed, which can be seen as a generaliz…
Higher-order Hahn's quantum variational calculus
Artur M. C. Brito da Cruz, Natalia Martins, Delfim F. M. Torres
We prove a necessary optimality condition of Euler-Lagrange type for quantum variational problems involving Hahn's derivatives of higher-order.
Time Scales and Mathematica
Artur M. C. Brito da Cruz, Helena Sofia Rodrigues, Delfim F. M. Torres
Time scales are a model of time, where the continuous and the discrete time cases are considered and merged into the same framework. In this paper some basic definitions of the tim…