440 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
Second order WSGD operators II: A new family of difference schemes for space fractional advection diffusion equation
The second order weighted and shifted Gr\"{u}nwald difference (WSGD)
operators are developed in [Tian et al., arXiv:1201.5949] to solve space
fractional partial differential equations. Along this direction, we further
design a new family of second order WSGD operators; by properly choosing the
weighted parameters, they can be effectively used to discretize space
(Riemann-Liouville) fractional derivatives. Based on the new second order WSGD
operators, we derive a family of difference schemes for the space fractional
advection diffusion equation. By von Neumann stability analysis, it is proved
that the obtained schemes are unconditionally stable. Finally, extensive
numerical experiments are performed to demonstrate the performance of the
schemes and confirm the convergent orders.Comment: 21 pages, 5 figure
High order algorithm for the time-tempered fractional Feynman-Kac equation
We provide and analyze the high order algorithms for the model describing the
functional distributions of particles performing anomalous motion with
power-law jump length and tempered power-law waiting time. The model is derived
in [Wu, Deng, and Barkai, Phys. Rev. E., 84 (2016), 032151], being called the
time-tempered fractional Feynman-Kac equation. The key step of designing the
algorithms is to discretize the time tempered fractional substantial
derivative, being defined as where
and , , and is a real number. The
designed schemes are unconditionally stable and have the global truncation
error , being theoretically proved and numerically
verified in {\em complex} space. Moreover, some simulations for the
distributions of the first passage time are performed, and the second order
convergence is also obtained for solving the `physical' equation (without
artificial source term).Comment: 21 pages, 4 figure
Feynman-Kac Equations for Reaction and Diffusion Processes
This paper provides a theoretical framework of deriving the forward and
backward Feynman-Kac equations for the distribution of functionals of the path
of a particle undergoing both diffusion and chemical reaction. Very general
forms of the equations are obtained. Once given the diffusion type and reaction
rate, a specific forward or backward Feynman-Kac equation can be obtained. The
listed in the paper include the ones for normal/anomalous diffusions and
reactions with linear/nonlinear rates. Using the derived equations, we also
study the occupation time in half-space, the first passage time to a fixed
boundary, and the occupation time in half-space with absorbing or reflecting
boundary conditions.Comment: 15 pages, 4 figure
Convergence proof for the multigrid method of the nonlocal model
Recently, nonlocal models attract the wide interests of scientist. They
mainly come from two applied scientific fields: peridyanmics and anomalous
diffusion. Even though the matrices of the algebraic equation corresponding the
nonlocal models are usually Toeplitz (denote a0 as the principal diagonal
element, a1 as the trailing diagonal element, etc). There are still some
differences for the models in these two fields. For the model of anomalous
diffusion, a0/a1 is uniformly bounded; most of the time, a0/a1 of the model for
peridyanmics is unbounded as the stepsize h tends to zero. Based on the uniform
boundedness of a0/a1, the convergence of the two-grid method is well
established [Chan, Chang, and Sun, SIAM J. Sci. Comput., 19 (1998), pp.
516--529; Pang and Sun, J. Comput. Phys., 231 (2012), pp. 693--703; Chen, Wang,
Cheng, and Deng, BIT, 54 (2014), pp. 623--647]. This paper provides the
detailed proof of the convergence of the two-grid method for the nonlocal model
of peridynamics. Some special cases of the full multigrid and the V-cycle
multigrid are also discussed. The numerical experiments are performed to verify
the convergence.Comment: 21 page
First exit and Dirichlet problem for the nonisotropic tempered -stable processes
This paper discusses the first exit and Dirichlet problems of the
nonisotropic tempered -stable process . The upper bounds of all
moments of the first exit position and the first exit
time are firstly obtained. It is found that the probability density
function of or exponentially decays with the
increase of or , and
,\
. Since is the infinitesimal generator
of the anisotropic tempered stable process, we obtain the Feynman-Kac
representation of the Dirichlet problem with the operator
. Therefore, averaging the generated
trajectories of the stochastic process leads to the solution of the Dirichlet
problem, which is also verified by numerical experiments.Comment: 23 pages, 5 figure
WSLD operators II: the new fourth order difference approximations for space Riemann-Liouville derivative
High order discretization schemes play more important role in fractional
operators than classical ones. This is because usually for classical
derivatives the stencil for high order discretization schemes is wider than low
order ones; but for fractional operators the stencils for high order schemes
and low order ones are the same. Then using high order schemes to solve
fractional equations leads to almost the same computational cost with first
order schemes but the accuracy is greatly improved. Using the fractional linear
multistep methods, Lubich obtains the -th order ()
approximations of the -th derivative () or integral
(\alpha \in(1,2)$ for time dependent problem. By
weighting and shifting Lubich's 2nd order discretization scheme, in [Chen &
Deng, arXiv:1304.7425] we derive a series of effective high order
discretizations for space fractional derivative, called WSLD opeartors there.
As the sequel of the previous work, we further provide new high order schemes
for space fractional derivatives by weighting and shifting Lubich's 3rd and 4th
order discretizations. In particular, we prove that the obtained 4th order
approximations are effective for space fractional derivatives. And the
corresponding schemes are used to solve the space fractional diffusion equation
with variable coefficients.Comment: 22 pages, 2 figure
Multiresolution Galerkin method for solving the functional distribution of anomalous diffusion described by time-space fractional diffusion equation
The functional distributions of particle trajectories have wide applications,
including the occupation time in half-space, the first passage time, and the
maximal displacement, etc. The models discussed in this paper are for
characterizing the distribution of the functionals of the paths of anomalous
diffusion described by time-space fractional diffusion equation. This paper
focuses on providing effective computation methods for the models. Two kinds of
time stepping schemes are proposed for the fractional substantial derivative.
The multiresolution Galerkin method with wavelet B-spline is used for space
approximation. Compared with the finite element or spectral polynomial bases,
the wavelet B-spline bases have the advantage of keeping the Toeplitz structure
of the stiffness matrix, and being easy to generate the matrix elements and to
perform preconditioning. The unconditional stability and convergence of the
provided schemes are theoretically proved and numerically verified. Finally, we
also discuss the efficient implementations and some extensions of the schemes,
such as the wavelet preconditioning and the non-uniform time discretization.Comment: 31 pages, 1 figur
High order schemes for the tempered fractional diffusion equations
L\'{e}vy flight models whose jumps have infinite moments are mathematically
used to describe the superdiffusion in complex systems. Exponentially tempering
the Levy measure of L\'{e}vy flights leads to the tempered stable L\'{e}vy
processes which combine both the -stable and Gaussian trends; and the
very large jumps are unlikely and all their moments exist. The probability
density functions of the tempered stable L\'{e}vy processes solve the tempered
fractional diffusion equation. This paper focuses on designing the high order
difference schemes for the tempered fractional diffusion equation on bounded
domain. The high order difference approximations, called the tempered and
weighted and shifted Gr\"{u}nwald difference (tempered-WSGD) operators, in
space are obtained by using the properties of the tempered fractional calculus
and weighting and shifting their first order Gr\"{u}nwald type difference
approximations. And the Crank-Nicolson discretization is used in the time
direction. The stability and convergence of the presented numerical schemes are
established; and the numerical experiments are performed to confirm the
theoretical results and testify the effectiveness of the schemes.Comment: 23 pages, 3 figure
Applications of Wavelet Bases to The Numerical Solutions of Fractional PDEs
For describing the probability distribution of the positions and times of
particles performing anomalous motion, fractional PDEs are derived from the
continuous time random walk models with waiting time distribution having
divergent first order moment and/or jump length distribution which has
divergent second order moment. It can be noted that the fractional PDEs are
essentially dealing with the multiscale issues. Generally the regularity of the
solutions for fractional PDEs is weak at the areas close to boundary and
initial time. This paper focuses on developing the applications of wavelet
bases to numerically solving fractional PDEs and digging out the potential
benefits of wavelet methods comparing with other numerical methods, especially
in the aspects of realizing preconditioning, adaptivity, and keeping the
Toeplitz structure. More specifically, the contributions of this paper are as
follows: 1. the techniques of efficiently generating stiffness matrix with
computational cost are provided for first, second, and any
order bases; 2. theoretically and numerically discuss the effective
preconditioner for time-independent equation and multigrid method for
time-dependent equation, respectively; 3. the wavelet adaptivity is detailedly
discussed and numerically applied to solving the time-dependent (independent)
equations. In fact, having reliable, simple, and local regularity indicators is
the striking benefit of the wavelet in adaptively solving fractional PDEs (it
seems hard to give a local posteriori error estimate for the adaptive finite
element method because of the global property of the operator).Comment: 38 page
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