418 research outputs found
Time-fractional Caputo derivative versus other integro-differential operators in generalized Fokker-Planck and generalized Langevin equations
Fractional diffusion and Fokker-Planck equations are widely used tools to
describe anomalous diffusion in a large variety of complex systems. The
equivalent formulations in terms of Caputo or Riemann-Liouville fractional
derivatives can be derived as continuum limits of continuous time random walks
and are associated with the Mittag-Leffler relaxation of Fourier modes,
interpolating between a short-time stretched exponential and a long-time
inverse power-law scaling. More recently, a number of other
integro-differential operators have been proposed, including the
Caputo-Fabrizio and Atangana-Baleanu forms. Moreover, the conformable
derivative has been introduced. We here study the dynamics of the associated
generalized Fokker-Planck equations from the perspective of the moments, the
time averaged mean squared displacements, and the autocovariance functions. We
also study generalized Langevin equations based on these generalized operators.
The differences between the Fokker-Planck and Langevin equations with different
integro-differential operators are discussed and compared with the dynamic
behavior of established models of scaled Brownian motion and fractional
Brownian motion. We demonstrate that the integro-differential operators with
exponential and Mittag-Leffler kernels are not suitable to be introduced to
Fokker-Planck and Langevin equations for the physically relevant diffusion
scenarios discussed in our paper. The conformable and Caputo Langevin equations
are unveiled to share similar properties with scaled and fractional Brownian
motion, respectively.Comment: 26 pages, 7 figures, RevTe
Fractional Calculus: Historical Notes Apuntes Históricos del Cálculo Fraccionario
En este trabajo, presentamos algunos apuntes históricos del cálculo fraccionario Local, y destacamos algunas propiedades y aplicaciones de estas nuevas herramientas matemicasIn this paper, we present some historical notes to Generalized Calculus, sometimes called Local Fractional Calculus, and highlight some properties and applications of these new mathematical tool
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
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