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Artur M. C. Brito da Cruz

Publications and source records attributed to Artur M. C. Brito da Cruz.

10 recordsLinked to original sources
arXiv2015

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.

math.CA↗
arXiv2014

A Fractional Calculus on Arbitrary Time Scales: Fractional Differentiation and Fractional Integration

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.

math.CA↗
arXiv2013

Symmetric Quantum Calculus

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 $α,β$-symmetric, the $q$-symmetric, and the Hahn symmetric quantum calculus. We introduce the symmetric quantum variational calculus and an Euler-Lagrange type equation for the $q$-symmetric and Hahn's symmetric quantum calculus is proved. We define a symmetric derivative on time scales and derive some of its properties. Finally, we introduce and study the diamond integral, which is a refined version of the diamond-$α$ integral on time scales.

math.CA↗
arXiv2013

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.

math.CA↗
arXiv2012

Symmetric Differentiation on Time Scales

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.

math.CA↗
arXiv2012

Hahn's Symmetric Quantum Variational Calculus

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 type and a sufficient optimality condition for variational problems within the context of Hahn's symmetric calculus. Moreover, we show the effectiveness of Leitmann's direct method when applied to Hahn's symmetric variational calculus. Illustrative examples are provided.

math.OC↗
arXiv2011

A symmetric quantum calculus

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

math.CA↗
arXiv2009

Time Scales and Mathematica

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 time scale calculus are presented. Simultaneously, a package in Mathematica is introduced.

math.CA↗