1,872 research outputs found
Extremum principle for the Hadamard derivatives and its application to nonlinear fractional partial differential equations
In this paper we obtain new estimates of the Hadamard fractional derivatives
of a function at its extreme points. The extremum principle is then applied to
show that the initial-boundary-value problem for linear and nonlinear
time-fractional diffusion equations possesses at most one classical solution
and this solution depends continuously on the initial and boundary conditions.
The extremum principle for an elliptic equation with a fractional Hadamard
derivative is also proved
An Analysis of the Rayleigh-Stokes problem for a Generalized Second-Grade Fluid
We study the Rayleigh-Stokes problem for a generalized second-grade fluid
which involves a Riemann-Liouville fractional derivative in time, and present
an analysis of the problem in the continuous, space semidiscrete and fully
discrete formulations. We establish the Sobolev regularity of the homogeneous
problem for both smooth and nonsmooth initial data , including . A space semidiscrete Galerkin scheme using continuous piecewise
linear finite elements is developed, and optimal with respect to initial data
regularity error estimates for the finite element approximations are derived.
Further, two fully discrete schemes based on the backward Euler method and
second-order backward difference method and the related convolution quadrature
are developed, and optimal error estimates are derived for the fully discrete
approximations for both smooth and nonsmooth initial data. Numerical results
for one- and two-dimensional examples with smooth and nonsmooth initial data
are presented to illustrate the efficiency of the method, and to verify the
convergence theory.Comment: 23 pp, 4 figures. The error analysis of the fully discrete scheme is
shortene
A fast implicit difference scheme for solving the generalized time-space fractional diffusion equations with variable coefficients
In this paper, we first propose an unconditionally stable implicit difference
scheme for solving generalized time-space fractional diffusion equations
(GTSFDEs) with variable coefficients. The numerical scheme utilizes the
-type formula for the generalized Caputo fractional derivative in time
discretization and the second-order weighted and shifted Gr\"{u}nwald
difference (WSGD) formula in spatial discretization, respectively. Theoretical
results and numerical tests are conducted to verify the -order
and 2-order of temporal and spatial convergence with the order
of Caputo fractional derivative, respectively. The fast sum-of-exponential
approximation of the generalized Caputo fractional derivative and Toeplitz-like
coefficient matrices are also developed to accelerate the proposed implicit
difference scheme. Numerical experiments show the effectiveness of the proposed
numerical scheme and its good potential for large-scale simulation of GTSFDEs.Comment: 23 pages, 10 tables, 1 figure. Make several corrections again and
have been submitted to a journal at Sept. 20, 2019. Version 2: Make some
necessary corrections and symbols, 13 Jan. 2020. Revised manuscript has been
resubmitted to journa
A unified analysis framework for generalized fractional Moore--Gibson--Thompson equations: Well-posedness and singular limits
In acoustics, higher-order-in-time equations arise when taking into account a
class of thermal relaxation laws in the modeling of sound wave propagation. In
this work, we analyze initial boundary value problems for a family of such
equations and determine the behavior of solutions as the relaxation time
vanishes. In particular, we allow the leading term to be of fractional type.
The studied model can be viewed as a generalization of the well-established
(fractional) Moore--Gibson--Thompson equation with three, in general nonlocal,
convolution terms involving two different kernels. The interplay of these
convolutions will influence the uniform analysis and the limiting procedure. To
unify the theoretical treatment of this class of local and nonlocal
higher-order equations, we relax the classical assumption on the leading-term
kernel and consider it to be a Radon measure. After establishing uniform
well-posedness with respect to the relaxation time of the considered general
model, we connect it, through a delicate singular limit procedure, to
fractional second-order models of linear acoustics
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