591 research outputs found
Wasserstein Gradient Flow Formulation of the Time-Fractional Fokker-Planck Equation
In this work, we investigate a variational formulation for a time-fractional
Fokker-Planck equation which arises in the study of complex physical systems
involving anomalously slow diffusion. The model involves a fractional-order
Caputo derivative in time, and thus inherently nonlocal. The study follows the
Wasserstein gradient flow approach pioneered by [26]. We propose a JKO type
scheme for discretizing the model, using the L1 scheme for the Caputo
fractional derivative in time, and establish the convergence of the scheme as
the time step size tends to zero. Illustrative numerical results in one- and
two-dimensional problems are also presented to show the approach.Comment: 24 pages, 2 figure
A mixed FEM for a time-fractional Fokker-Planck model
We propose and analyze a mixed finite element method for the spatial
approximation of a time-fractional Fokker--Planck equation in a convex
polyhedral domain, where the given driving force is a function of space. Taking
into account the limited smoothing properties of the model, and considering an
appropriate splitting of the errors, we employed a sequence of clever energy
arguments to show optimal convergence rates with respect to both approximation
properties and regularity results. In particular, error bounds for both primary
and secondary variables are derived in -norm for cases with smooth and
nonsmooth initial data. We further investigate a fully implicit time-stepping
scheme based on a convolution quadrature in time generated by the backward
Euler method. Our main result provides pointwise-in-time optimal -error
estimates for the primary variable. Numerical examples are then presented to
illustrate the theoretical contributions
Stationary states in Langevin dynamics under asymmetric L\'evy noises
Properties of systems driven by white non-Gaussian noises can be very
different from these systems driven by the white Gaussian noise. We investigate
stationary probability densities for systems driven by -stable L\'evy
type noises, which provide natural extension to the Gaussian noise having
however a new property mainly a possibility of being asymmetric. Stationary
probability densities are examined for a particle moving in parabolic, quartic
and in generic double well potential models subjected to the action of
-stable noises. Relevant solutions are constructed by methods of
stochastic dynamics. In situations where analytical results are known they are
compared with numerical results. Furthermore, the problem of estimation of the
parameters of stationary densities is investigated.Comment: 9 pages, 9 figures, 3 table
Generalised fractional diffusion equations for subdiffusion on arbitrarily growing domains
Many physical phenomena occur on domains that grow in time. When the
timescales of the phenomena and domain growth are comparable, models must
include the dynamics of the domain. A widespread intrinsically slow transport
process is subdiffusion. Many models of subdiffusion include a history
dependence. This greatly confounds efforts to incorporate domain growth. Here
we derive the fractional partial differential equations that govern
subdiffusion on a growing domain, based on a Continuous Time Random Walk. This
requires the introduction of a new, comoving, fractional derivative.Comment: 12 pages, 1 figur
Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations
A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach
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