7 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
A linear Galerkin numerical method for a quasilinear subdiffusion equation
We couple the L1 discretization for Caputo derivative in time with spectral
Galerkin method in space to devise a scheme that solves quasilinear
subdiffusion equations. Both the diffusivity and the source are allowed to be
nonlinear functions of the solution. We prove method's stability and
convergence with spectral accuracy in space. The temporal order depends on
solution's regularity in time. Further, we support our results with numerical
simulations that utilize parallelism for spatial discretization. Moreover, as a
side result we find asymptotic exact values of error constants along with their
remainders for discretizations of Caputo derivative and fractional integrals.
These constants are the smallest possible which improves the previously
established results from the literature.Comment: This is the accepted version of the manuscript published in Applied
Numerical Mathematic