This article is concerned with the practical stability performance of nonlinear impulsive stochastic functional differential systems driven by G-Brownian motion (G-ISFDSs). Comparing with traditional ...Lyapunov stability theory, practical stability can portray qualitative behaviour and quantitative properties of suggested systems. By employing G-Itô formula, Lyapunov-Razumikhin approach and stochastic analysis theory, some novel conditions for pth moment practical exponential stability and quasi-sure global practical uniform exponential stability of G-ISFDSs are established. The obtained results show that impulses may influence dynamic behaviour of the addressed system. Two numerical examples are given to verify the validity of our developed results.
Hydrothermal behavior of nanofluid fluid between two parallel plates is studied. One of the plates is externally heated, and the other plate, through which coolant fluid is injected, expands or ...contracts with time. The effective thermal conductivity and viscosity of nanofluid are calculated by KKL correlation. The effects of the nanoparticle volume fraction, Reynolds number, Expansion ratio and power law index on Hydrothermal behavior are investigated. Results show that heat transfer enhancement has direct relationship with Reynolds number when power law index is equals to zero but opposite trend is observed for other values of power law index.
•Nanofluid Hydrothermal behavior in a porous channel is considered.•Nusselt number is an increasing function of ϕ, R and m.•Nusselt number is a decreasing function of α.
Abstract
We study the diffusive motion of a particle in a subharmonic potential of the form
U
(
x
) = |
x
|
c
(0 <
c
< 2) driven by long-range correlated, stationary fractional Gaussian noise
ξ
α
(
t
...) with 0 <
α
⩽ 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent
α
. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation
c
> 2(1 − 1/
α
) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Lévy flights in shallow external potentials.
Stochastic differential equations (SDEs) are very useful in modeling many problems in biology, economic data, turbulence, and medicine. Fractional Brownian motion (fBm) and variable‐order fractional ...Brownian motion (vofBm) are suitable alternatives to standard Brownian motion (sBm) for describing and modeling many phenomena, since the increments of these processes are dependent of the past and for
H>12$$ \mathcal{H}>\frac{1}{2} $$ these increments have the property of long‐term dependence. Classical mathematical techniques such as Ito's calculus do not work for stochastic computations on fBm and vofBm due to they are not semi‐Martingale for
H(ξ)≠12$$ \mathcal{H}\left(\xi \right)\ne \frac{1}{2} $$. Therefore, solving these equations is much more difficult than solving SDEs with sBm. On the other hand, these equations do not have an analytical solution, so we have to use numerical methods to find their solution. In this paper, a computational approach based on hybrid of block‐pulse and parabolic functions (HBPFs) has been introduced for simulating vofBm and solving a modern class of SDEs. The mechanism of this approach is based on stochastic and fractional integration operational matrices, which transform the intended problem to a nonlinear system of algebraic equations. Thus, the complexity of solving the mentioned problem is reduced significantly. Also, convergence analysis of the expressed method has been theoretically examined. Finally, the accuracy and efficiency of the proposed algorithm have been experimentally investigated through some test problems and comparison of obtained results with results of previous papers. High accurate numerical results are obtained by using a small number of basic functions. Therefore, this method deals with smaller matrices and vectors, which is one of the most important advantage of our suggested method. Also, presenting an applicable procedure to construct vofBm is another innovation of this work. To gain this aim, at first, discretized sBm is generated via fundamental features of this process, and afterward, block‐pulse functions (BPFs) and HBPFs are utilized for simulating discretized vofBm. Finally, spline interpolation method has been employed to provide a continuous path of vofBm.
The diffusion motion of spin-bearing molecules is considerably affected in confined environments. In Nuclear Magnetic Resonance (NMR) experiments, under the presence of magnetic field gradients, this ...movement can be encoded in spin phase and the NMR signal attenuation due to diffusion can be evaluated. This paper considered this effect in both normal and anomalous diffusion by means of the Langevin stochastic model within the Gaussian phase approximation (GPA) for a constant gradient temporal profile. The phenomenological approach illustrates the emergence of fractional order exponential decay in the NMR signal and retrieves the classical result from Hanh echo.
•The generalized Langevin equation is introduced for diffusion-based NMR experiments.•The NMR signal due to diffusion is expressed within the Gaussian phase approximation.•The constant gradient profile is considered.•The exact analytic solution for a memory kernel that decays as a power law is given.•The approach extends and retrieves the classic result from Hahn echo.
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