arXiv subjects

Caibin Zeng

Publications and source records attributed to Caibin Zeng.

7 recordsLinked to original sources

arXiv2023

Smooth center-stable/unstable manifolds and foliations of stochastic evolution equations with non-dense domain

The current paper is devoted to the asymptotic behavior of a class of stochastic PDE. More precisely, with the help of the theory of integrated semigroups and a crucial estimate of the random Stieltjes convolution, we study the existence and smoothness of center-unstable invariant manifolds and center-stable foliations for a class of stochastic PDE with non-dense domain through the Lyapunov-Perron method. Finally, we give two examples about a stochastic age-structured model and a stochastic parabolic equation to illustrate our results.

math.DS↗

arXiv2023

Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise

Little seems to be known about the invariant manifolds for stochastic partial differential equations (SPDEs) driven by nonlinear multiplicative noise. Here we contribute to this aspect and analyze the Lu-Schmalfuß conjecture [Garrido-Atienza, et al., J. Differential Equations, 248(7):1637--1667, 2010] on the existence of stable manifolds for a class of parabolic SPDEs driven by nonlinear mutiplicative fractional noise. We emphasize that stable manifolds for SPDEs are infinite-dimensional objects, and the classical Lyapunov-Perron method cannot be applied, since the Lyapunov-Perron operator does not give any information about the backward orbit. However, by means of interpolation theory, we construct a suitable function space in which the discretized Lyapunov-Perron-type operator has a unique fixed point. Based on this we further prove the existence and smoothness of local stable manifolds for such SPDEs.

math.PR↗

arXiv2021

Mean-square invariant manifolds for ill-posed stochastic evolution equations driven by nonlinear noise

This paper discerns the invariant manifold of a class of ill-posed stochastic evolution equations driven by a nonlinear multiplicative noise. To be more precise, we establish the existence of mean-square random unstable invariant manifold and only mean-square stable invariant set. Due to the lack of the Hille-Yosida condition, we construct a modified variation of constants formula by the resolvent operator. With the price of imposing an unusual condition involving a non-decreasing map, we set up the Lyapunov-Perron method and derive the required estimates. We also emphasize that the Lyapunov-Perron map in the forward time loses the invariant due to the adaptedness, we alternatively establish the existence of mean-square random stable sets.

math.DS↗

arXiv2020

Stochastic differential equations with noise perturbations and Wong-Zakai approximation of fractional Brownian motion

In this article we study effects that small perturbations in the noise have to the solution of differential equations driven by Hölder continuous functions of order $H>\frac12$. As an application, we consider stochastic differential equations driven by a fractional Brownian motion. We introduce a Wong--Zakai type stationary approximation to the fractional Brownian motions and prove that it converges in a suitable space. Moreover, we provide sharp results on the rate of convergence in the $p$-norm. Our stationary approximation is suitable for all values of $H\in (0,1)$.

math.PR↗

arXiv2015

Global Padé approximations of the generalized Mittag-Leffler function and its inverse

This paper proposes a global Padé approximation of the generalized Mittag-Leffler function $E_{α,β}(-x)$ with $x\in[0,+\infty)$. This uniform approximation can account for both the Taylor series for small arguments and asymptotic series for large arguments. Based on the complete monotonicity of the function $E_{α,β}(-x)$, we work out the global Padé approximation [1/2] for the particular cases $\{0<α<1, β>α\}$, $\{0<α=β<1\}$, and $\{α=1, β>1\}$, respectively. Moreover, these approximations are inverted to yield a global Padé approximation of the inverse generalized Mittag-Leffler function $-L_{α,β}(x)$ with $x\in(0,1/Γ(β)]$. We also provide several examples with selected values $α$ and $β$ to compute the relative error from the approximations. Finally, we point out the possible applications using our established approximations in the ordinary and partial time-fractional differential equations in the sense of Riemann-Liouville.

math.CA↗

arXiv2013

Optimal random search, fractional dynamics and fractional calculus

What is the most efficient search strategy for the random located target sites subject to the physical and biological constraints? Previous results suggested the Lévy flight is the best option to characterize this optimal problem, however, which ignores the understanding and learning abilities of the searcher agents. In the paper we propose the Continuous Time Random Walk (CTRW) optimal search framework and find the optimum for both of search length's and waiting time's distributions. Based on fractional calculus technique, we further derive its master equation to show the mechanism of such complex fractional dynamics. Numerous simulations are provided to illustrate the non-destructive and destructive cases.

math.OC↗

arXiv2010

Homotopy perturbation method for fractional-order Burgers-Poisson equation

In this paper, the fractional-order Burgers-Poisson equation is introduced by replacing the first-order time derivative by fractional derivative of order $α$. Both exact and approximate explicit solutions are obtained by employing homotopy perturbation method. The numerical results reveal that the proposed method is very effective and simple for handling fractional-order differential equations.

nlin.PS↗