235 research outputs found
Brownian subordinators and fractional Cauchy problems
A Brownian time process is a Markov process subordinated to the absolute
value of an independent one-dimensional Brownian motion. Its transition
densities solve an initial value problem involving the square of the generator
of the original Markov process. An apparently unrelated class of processes,
emerging as the scaling limits of continuous time random walks, involve
subordination to the inverse or hitting time process of a classical stable
subordinator. The resulting densities solve fractional Cauchy problems, an
extension that involves fractional derivatives in time. In this paper, we will
show a close and unexpected connection between these two classes of processes,
and consequently, an equivalence between these two families of partial
differential equations.Comment: 18 pages, minor spelling correction
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
Space-time duality for fractional diffusion
Zolotarev proved a duality result that relates stable densities with
different indices. In this paper, we show how Zolotarev duality leads to some
interesting results on fractional diffusion. Fractional diffusion equations
employ fractional derivatives in place of the usual integer order derivatives.
They govern scaling limits of random walk models, with power law jumps leading
to fractional derivatives in space, and power law waiting times between the
jumps leading to fractional derivatives in time. The limit process is a stable
L\'evy motion that models the jumps, subordinated to an inverse stable process
that models the waiting times. Using duality, we relate the density of a
spectrally negative stable process with index to the density of
the hitting time of a stable subordinator with index , and thereby
unify some recent results in the literature. These results also provide a
concrete interpretation of Zolotarev duality in terms of the fractional
diffusion model.Comment: 16 page
Reflected Spectrally Negative Stable Processes and their Governing Equations
This paper explicitly computes the transition densities of a spectrally
negative stable process with index greater than one, reflected at its infimum.
First we derive the forward equation using the theory of sun-dual semigroups.
The resulting forward equation is a boundary value problem on the positive
half-line that involves a negative Riemann-Liouville fractional derivative in
space, and a fractional reflecting boundary condition at the origin. Then we
apply numerical methods to explicitly compute the transition density of this
space-inhomogeneous Markov process, for any starting point, to any desired
degree of accuracy. Finally, we discuss an application to fractional Cauchy
problems, which involve a positive Caputo fractional derivative in time
Space–time fractional derivative operators
Evolution equations for anomalous diffusion employ fractional derivatives in space and time. Linkage between the space-time variables leads to a new type of fractional derivative operator. This paper develops the mathematical foundations of those operators
Subdiffusive transport in intergranular lanes on the Sun. The Leighton model revisited
In this paper we consider a random motion of magnetic bright points (MBP)
associated with magnetic fields at the solar photosphere. The MBP transport in
the short time range [0-20 minutes] has a subdiffusive character as the
magnetic flux tends to accumulate at sinks of the flow field. Such a behavior
can be rigorously described in the framework of a continuous time random walk
leading to the fractional Fokker-Planck dynamics. This formalism, applied for
the analysis of the solar subdiffusion of magnetic fields, generalizes the
Leighton's model.Comment: 7 page
Pennsylvania Folklife Vol. 19, No. 2
• Powwowing in Berks County • Belsnickling in Paxtonville • The Folk Tradition of the Sweetheart Tree • Pigpens and Piglore in Rural Pennsylvania • Gravestones and Ostentation: A Study of Five Delaware County Cemeteries • Notes on Eighteenth-Century Emigration to the British Colonies • A Siegerland Emigrant List of 1738 • Local Place Names: Folk-Cultural Questionnaire No. 14https://digitalcommons.ursinus.edu/pafolklifemag/1038/thumbnail.jp
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