662 research outputs found
Review of Some Promising Fractional Physical Models
Fractional dynamics is a field of study in physics and mechanics
investigating the behavior of objects and systems that are characterized by
power-law non-locality, power-law long-term memory or fractal properties by
using integrations and differentiation of non-integer orders, i.e., by methods
of the fractional calculus. This paper is a review of physical models that look
very promising for future development of fractional dynamics. We suggest a
short introduction to fractional calculus as a theory of integration and
differentiation of non-integer order. Some applications of
integro-differentiations of fractional orders in physics are discussed. Models
of discrete systems with memory, lattice with long-range inter-particle
interaction, dynamics of fractal media are presented. Quantum analogs of
fractional derivatives and model of open nano-system systems with memory are
also discussed.Comment: 38 pages, LaTe
A survey on fuzzy fractional differential and optimal control nonlocal evolution equations
We survey some representative results on fuzzy fractional differential
equations, controllability, approximate controllability, optimal control, and
optimal feedback control for several different kinds of fractional evolution
equations. Optimality and relaxation of multiple control problems, described by
nonlinear fractional differential equations with nonlocal control conditions in
Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Computational and Applied Mathematics', ISSN: 0377-0427.
Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication
20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515
Enlarged Controllability of Riemann-Liouville Fractional Differential Equations
We investigate exact enlarged controllability for time fractional diffusion
systems of Riemann-Liouville type. The Hilbert uniqueness method is used to
prove exact enlarged controllability for both cases of zone and pointwise
actuators. A penalization method is given and the minimum energy control is
characterized.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Computational and Nonlinear Dynamics', ISSN 1555-1415, eISSN
1555-1423, CODEN JCNDDM, available at
[http://computationalnonlinear.asmedigitalcollection.asme.org]. Submitted
10-Aug-2017; Revised 28-Sept-2017 and 24-Oct-2017; Accepted 05-Nov-201
A formulation of the fractional Noether-type theorem for multidimensional Lagrangians
This paper presents the Euler-Lagrange equations for fractional variational
problems with multiple integrals. The fractional Noether-type theorem for
conservative and nonconservative generalized physical systems is proved. Our
approach uses well-known notion of the Riemann-Liouville fractional derivative.Comment: Submitted 26-SEP-2011; accepted 3-MAR-2012; for publication in
Applied Mathematics Letter
Stationarity-conservation laws for certain linear fractional differential equations
The Leibniz rule for fractional Riemann-Liouville derivative is studied in
algebra of functions defined by Laplace convolution. This algebra and the
derived Leibniz rule are used in construction of explicit form of
stationary-conserved currents for linear fractional differential equations. The
examples of the fractional diffusion in 1+1 and the fractional diffusion in d+1
dimensions are discussed in detail. The results are generalized to the mixed
fractional-differential and mixed sequential fractional-differential systems
for which the stationarity-conservation laws are obtained. The derived currents
are used in construction of stationary nonlocal charges.Comment: 28 page
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
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