59,457 research outputs found
Boundary Conditions for Fractional Diffusion
This paper derives physically meaningful boundary conditions for fractional
diffusion equations, using a mass balance approach. Numerical solutions are
presented, and theoretical properties are reviewed, including well-posedness
and steady state solutions. Absorbing and reflecting boundary conditions are
considered, and illustrated through several examples. Reflecting boundary
conditions involve fractional derivatives. The Caputo fractional derivative is
shown to be unsuitable for modeling fractional diffusion, since the resulting
boundary value problem is not positivity preserving
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
On stable solutions of boundary reaction-diffusion equations and applications to nonlocal problems with Neumann data
We study reaction-diffusion equations in cylinders with possibly nonlinear
diffusion and possibly nonlinear Neumann boundary conditions. We provide a
geometric Poincar\'e-type inequality and classification results for stable
solutions, and we apply them to the study of an associated nonlocal problem. We
also establish a counterexample in the corresponding framework for the
fractional Laplacian
Distributed-order fractional Cauchy problems on bounded domains
In a fractional Cauchy problem, the usual first order time derivative is
replaced by a fractional derivative. The fractional derivative models time
delays in a diffusion process. The order of the fractional derivative can be
distributed over the unit interval, to model a mixture of delay sources. In
this paper, we provide explicit strong solutions and stochastic analogues for
distributed-order fractional Cauchy problems on bounded domains with Dirichlet
boundary conditions. Stochastic solutions are constructed using a non-Markovian
time change of a killed Markov process generated by a uniformly elliptic second
order space derivative operator.Comment: 29 page
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