297 research outputs found
Stochastic P-bifurcation in a tri-stable Van der Pol system with fractional derivative under Gaussian white noise
In this paper, we study the tri-stable stochastic P-bifurcation problem of a generalized Van der Pol system with fractional derivative under Gaussian white noise excitation. Firstly, using the principle for minimal mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be transformed into an equivalent integer order system. Secondly, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by the stochastic averaging, and using the singularity theory, we find the critical parametric conditions for stochastic P-bifurcation of amplitude of the system, which can make the system switch among the three steady states. Finally, we analyze different types of the stationary PDF curves of the system amplitude qualitatively by choosing parameters corresponding to each region divided by the transition set curves, and the system response can be maintained at the small amplitude near the equilibrium by selecting the appropriate unfolding parameters. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order controller to adjust the response of the system
Stability of solutions of Caputo fractional stochastic differential equations
In this paper, we study the stability of Caputo-type fractional stochastic differential equations. Stochastic stability and stochastic asymptotical stability are shown by stopping time technique. Almost surly exponential stability and pth moment exponentially stability are derived by a new established Itô’s formula of Caputo version. Numerical examples are given to illustrate the main results
Hamiltonian formalism of fractional systems
In this paper we consider a generalized classical mechanics with fractional
derivatives. The generalization is based on the time-clock randomization of
momenta and coordinates taken from the conventional phase space. The fractional
equations of motion are derived using the Hamiltonian formalism. The approach
is illustrated with a simple-fractional oscillator in a free state and under an
external force. Besides the behavior of the coupled fractional oscillators is
analyzed. The natural extension of this approach to continuous systems is
stated. The interpretation of the mechanics is discussed.Comment: 16 pages, 5 figure
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
Fractional and tempered fractional models for Reynolds-averaged Navier-Stokes equations
Turbulence is a non-local phenomenon and has multiple-scales. Non-locality
can be addressed either implicitly or explicitly. Implicitly, by subsequent
resolution of all spatio-temporal scales. However, if directly solved for the
temporal or spatially averaged fields, a closure problem arises on account of
missing information between two points. To solve the closure problem in
Reynolds-averaged Navier-Stokes equations (RANS), an eddy-viscosity hypotheses
has been a popular modelling choice, where it follows either a linear or
non-linear stress-strain relationship. Here, a non-constant diffusivity is
introduced. Such a non-constant diffusivity is also characteristic of
non-Fickian diffusion equation addressing anomalous diffusion process. An
alternative approach, is a fractional derivative based diffusion equations.
Thus, in the paper, we formulate a fractional stress-strain relationship using
variable-order Caputo fractional derivative. This provides new opportunities
for future modelling effort. We pedagogically study of our model construction,
starting from one-sided model and followed by two-sided model. Non-locality at
a point is the amalgamation of all the effects, thus we find the two-sided
model is physically consistent. Further, our construction can also addresses
viscous effects, which is a local process. Thus, our fractional model addresses
the amalgamation of local and non-local process. We also show its validity at
infinite Reynolds number limit. This study is further extended to tempered
fractional calculus, where tempering ensures finite jump lengths, this is an
important remark for unbounded flows. Two tempered definitions are introduced
with a smooth and sharp cutoff, by the exponential term and Heaviside function,
respectively and we also define the horizon of non-local interactions. We
further study the equivalence between the two definitions.Comment: A part of this paper is also available as arXiv preprint
arXiv:2105.03646v1. Tempered F-RANS result first presented at ICTAM 2020+1
held in Italy, 2021 chaired by Prof. A. Quarteroni (postponed by a year due
to pandemic). Results submitted to ICTAM 2020 by Jan. 2020 (refer book of
abstracts, Pages 1235-1236 :
https://iutam.org/wp-content/uploads/2023/06/ABSTRACT_BOOK_ICTAM_2021.pdf
Continuous time random walk and parametric subordination in fractional diffusion
The well-scaled transition to the diffusion limit in the framework of the
theory of continuous-time random walk (CTRW)is presented starting from its
representation as an infinite series that points out the subordinated character
of the CTRW itself. We treat the CTRW as a combination of a random walk on the
axis of physical time with a random walk in space, both walks happening in
discrete operational time. In the continuum limit we obtain a generally
non-Markovian diffusion process governed by a space-time fractional diffusion
equation. The essential assumption is that the probabilities for waiting times
and jump-widths behave asymptotically like powers with negative exponents
related to the orders of the fractional derivatives. By what we call parametric
subordination, applied to a combination of a Markov process with a positively
oriented L\'evy process, we generate and display sample paths for some special
cases.Comment: 28 pages, 18 figures. Workshop 'In Search of a Theory of Complexity'.
Denton, Texas, August 200
A continuous time random walk model of transport in variably saturated heterogeneous porous media
We propose a unified physical framework for transport in variably saturated
porous media. This approach allows fluid flow and solute migration to be
treated as ensemble averages of fluid and solute particles, respectively. We
consider the cases of homogeneous and heterogeneous porous materials. Within a
fractal mobile-immobile (MIM) continuous time random walk framework, the
heterogeneity will be characterized by algebraically decaying particle
retention-times. We derive the corresponding (nonlinear) continuum limit
partial differential equations and we compare their solutions to Monte Carlo
simulation results. The proposed methodology is fairly general and can be used
to track fluid and solutes particles trajectories, for a variety of initial and
boundary conditions.Comment: 12 pages, 9 figure
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