3,623 research outputs found
Aging Feynman-Kac Equation
Aging, the process of growing old or maturing, is one of the most widely seen
natural phenomena in the world. For the stochastic processes, sometimes the
influence of aging can not be ignored. For example, in this paper, by analyzing
the functional distribution of the trajectories of aging particles performing
anomalous diffusion, we reveal that for the fraction of the occupation time
of strong aging particles,
with coefficient , having no relation with the aging time
and and being completely different from the case of weak (none) aging.
In fact, we first build the models governing the corresponding functional
distributions, i.e., the aging forward and backward Feynman-Kac equations; the
above result is one of the applications of the models. Another application of
the models is to solve the asymptotic behaviors of the distribution of the
first passage time, . The striking discovery is that for weakly aging
systems, , while
for strongly aging systems, behaves as .Comment: 13 pages, 7 figure
Liouville's theorem for the generalized harmonic function
In this paper, we give a more physical proof of Liouville's theorem for a
class generalized harmonic functions by the method of parabolic equation
The Tur\'an problem for a family of tight linear forests
Let be a family of -graphs. The Tur\'an number
is defined to be the maximum number of edges in an
-graph of order that is -free. The famous Erd\H{o}s
Matching Conjecture shows that where
represents the -graph consisting of disjoint edges.
Motivated by this conjecture, we consider the Tur\'an problem for tight linear
forests. A tight linear forest is an -graph whose connected components are
all tight paths or isolated vertices. Let be the
family of all tight linear forests of order with edges in -graphs.
In this paper, we prove that for sufficiently large ,
where and if and with , if and
with . The proof is based on the weak regularity lemma for
hypergraphs. We also conjecture that for arbitrary satisfying $k \equiv 1\
(mod\ r)d$ in the above result equals 0. We prove that the
proposed conjecture implies the Erd\H{o}s Matching Conjecture directly
The Tur\'{a}n Number for Spanning Linear Forests
For a set of graphs , the extremal number is
the maximum number of edges in a graph of order not containing any subgraph
isomorphic to some graph in . If contains a graph on
vertices, then we often call the problem a spanning Tur\'{a}n problem. A
linear forest is a graph whose connected components are all paths and isolated
vertices. In this paper, we let be the set of all linear
forests of order with at least edges. We prove that when
and , Clearly,
the result is interesting when
Discontinuous Galerkin methods and their adaptivity for the tempered fractional (convection) diffusion equations
This paper focuses on the adaptive discontinuous Galerkin (DG) methods for
the tempered fractional (convection) diffusion equations. The DG schemes with
interior penalty for the diffusion term and numerical flux for the convection
term are used to solve the equations, and the detailed stability and
convergence analyses are provided. Based on the derived posteriori error
estimates, the local error indicator is designed. The theoretical results and
the effectiveness of the adaptive DG methods are respectively verified and
displayed by the extensive numerical experiments. The strategy of designing
adaptive schemes presented in this paper works for the general PDEs with
fractional operators.Comment: 31 pages, 5 figure
Galerkin Finite Element Approximations for Stochastic Space-Time Fractional Wave Equations
The traditional wave equation models wave propagation in an ideal conducting
medium. For characterizing the wave propagation in inhomogeneous media with
frequency dependent power-law attenuation, the space-time fractional wave
equation appears; further incorporating the additive white Gaussian noise
coming from many natural sources leads to the stochastic space-time fractional
wave equation. This paper discusses the Galerkin finite element approximations
for the stochastic space-time fractional wave equation forced by an additive
space-time white noise. We firstly discretize the space-time additive noise,
which introduces a modeling error and results in a regularized stochastic
space-time fractional wave equation; then the regularity of the regularized
equation is analyzed. For the discretization in space, the finite element
approximation is used and the definition of the discrete fractional Laplacian
is introduced. We derive the mean-squared -norm priori estimates for the
modeling error and for the approximation error to the solution of the
regularized problem; and the numerical experiments are performed to confirm the
estimates. For the time-stepping, we calculate the analytically obtained
Mittag-Leffler type function.Comment: 28 page
Localization and ballistic diffusion for the tempered fractional Brownian-Langevin motion
This paper further discusses the tempered fractional Brownian motion, its
ergodicity, and the derivation of the corresponding Fokker-Planck equation.
Then we introduce the generalized Langevin equation with the tempered
fractional Gaussian noise for a free particle, called tempered fractional
Langevin equation (tfLe). While the tempered fractional Brownian motion
displays localization diffusion for the long time limit and for the short time
its mean squared displacement has the asymptotic form , we show that
the asymptotic form of the mean squared displacement of the tfLe transits from
(ballistic diffusion for short time) to , and then to
(again ballistic diffusion for long time). On the other hand, the overdamped
tfLe has the transition of the diffusion type from to
(ballistic diffusion). The tfLe with harmonic potential is also considered.Comment: 19 pages, 9 figure
Langevin dynamics for L\'evy walk with memory
Memory effects, sometimes, can not be neglected. In the framework of
continuous time random walk, memory effect is modeled by the correlated waiting
times. In this paper, we derive the two-point probability distribution of the
stochastic process with correlated increments as well as the one of its inverse
process, and present the Langevin description of L\'evy walk with memory, i.e.,
correlated waiting times. Based on the built Langevin picture, the properties
of aging and nonstationary are discussed. The Langevin system exhibits
sub-ballistic superdiffusion if the friction force is involved, while it
displays super-ballistic diffusion or hyperdiffusion if there is no friction.
It is discovered that the correlation of waiting times suppresses the diffusion
behavior whether there is friction or not, and the stronger the correlation of
waiting times becomes, the slower the diffusion is. In particular, the
correlation function, correlation coefficient, ergodicity, and scaling property
of the corresponding stochastic process are also investigated.Comment: 11 pages, 4 figure
L\'{e}vy-walk-like Langevin dynamics
Continuous time random walks and Langevin equations are two classes of
stochastic models for describing the dynamics of particles in the natural
world. While some of the processes can be conveniently characterized by both of
them, more often one model has significant advantages (or has to be used)
compared with the other one. In this paper, we consider the weakly damped
Langevin system coupled with a new subordinator|-dependent subordinator
with . We pay attention to the diffusion behaviour of the
stochastic process described by this coupled Langevin system, and find the
super-ballistic diffusion phenomena for the system with an unconfined potential
on velocity but sub-ballistic superdiffusion phenomenon with a confined
potential, which is like L\'{e}vy walk for long times. One can further note
that the two-point distribution of inverse subordinator affects mean square
displacement of this coupled weakly damped Langevin system in essential.Comment: 24 pages, 4 figure
Turing instability and Turing-Hopf bifurcation in diffusive Schnakenberg systems with gene expression time delay
For delayed reaction-diffusion Schnakenberg systems with Neumann boundary
conditions, critical conditions for Turing instability are derived, which are
necessary and sufficient. And existence conditions for Turing, Hopf and
Turing-Hopf bifurcations are established. Normal forms truncated to order 3 at
Turing-Hopf singularity of codimension 2, are derived. By investigating
Turing-Hopf bifurcation, the parameter regions for the stability of a periodic
solution, a pair of spatially inhomogeneous steady states and a pair of
spatially inhomogeneous periodic solutions, are derived in
parameter plane ( for time delay, for diffusion rate). It
is revealed that joint effects of diffusion and delay can lead to the
occurrence of mixed spatial and temporal patterns. Moreover, it is also
demonstrated that various spatially inhomogeneous patterns with different
spatial frequencies can be achieved via changing the diffusion rate. And, the
phenomenon that time delay may induce a failure of Turing instability observed
by Gaffney and Monk (2006) are theoretically explained
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