10 research outputs found
Structural derivatives on time scales
We introduce the notion of structural derivative on time scales. The new
operator of differentiation unifies the concepts of fractal and fractional
order derivative and is motivated by lack of classical differentiability of
some self-similar functions. Some properties of the new operator are proved and
illustrated with examples.publishe
Time-Fractional Optimal Control of Initial Value Problems on Time Scales
We investigate Optimal Control Problems (OCP) for fractional systems
involving fractional-time derivatives on time scales. The fractional-time
derivatives and integrals are considered, on time scales, in the
Riemann--Liouville sense. By using the Banach fixed point theorem, sufficient
conditions for existence and uniqueness of solution to initial value problems
described by fractional order differential equations on time scales are known.
Here we consider a fractional OCP with a performance index given as a
delta-integral function of both state and control variables, with time evolving
on an arbitrarily given time scale. Interpreting the Euler--Lagrange first
order optimality condition with an adjoint problem, defined by means of right
Riemann--Liouville fractional delta derivatives, we obtain an optimality system
for the considered fractional OCP. For that, we first prove new fractional
integration by parts formulas on time scales.Comment: This is a preprint of a paper accepted for publication as a book
chapter with Springer International Publishing AG. Submitted 23/Jan/2019;
revised 27-March-2019; accepted 12-April-2019. arXiv admin note: substantial
text overlap with arXiv:1508.0075
Variable order Mittag-Leffler fractional operators on isolated time scales and application to the calculus of variations
We introduce new fractional operators of variable order on isolated time
scales with Mittag-Leffler kernels. This allows a general formulation of a
class of fractional variational problems involving variable-order difference
operators. Main results give fractional integration by parts formulas and
necessary optimality conditions of Euler-Lagrange type.Comment: This is a preprint of a paper whose final and definite form is with
Springe
Existence of solution to a local fractional nonlinear differential equation
We prove existence of solution to a local fractional nonlinear differential equation with initial condition. For that we introduce the notion of tube solution. © 2016 Elsevier B.V
Complex-valued fractional derivatives on time scales
We introduce a notion of fractional (noninteger order) derivative on an arbitrary nonempty closed subset of the real numbers (on a time scale). Main properties of the new operator are proved and several illustrative examples given. © Springer International Publishing Switzerland 2016
Legendre curves on 3-dimensional -Manifolds
Legendre curves play a very important and special role in geometry andtopology of almost contact manifolds.There are certain results known forLegendre curves in 3-dimensional normal almost contact manifolds. The aim ofthis paper is to study Legendre curves of three-dimensional -manifoldswhich are non-normal almost contact manifolds and classifying all biharmonicLegendre curves in these manifolds.Comment: 10 page
A truly conformable calculus on time scales
We introduce the definition of conformable derivative on time scales and develop its calculus. Fundamental properties of the conformable derivative and
integral on time scales are proved. Linear conformable differential equations with constant coefficients are investigated, as well as hyperbolic and trigonometric functions.publishe
The fuzzy Henstock–Kurzweil delta integral on time scales
We investigate properties of the fuzzy Henstock–Kurzweil delta integral (shortly, FHK Δ -integral) on time scales, and obtain two necessary and sufficient conditions for FHK Δ -integrability. The concept of uniformly FHK Δ -integrability is introduced. Under this concept, we obtain a uniformly integrability convergence theorem. Finally, we prove monotone and dominated convergence theorems for the FHK Δ -integral.publishe