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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
Distributed Order Calculus and Equations of Ultraslow Diffusion
We consider diffusion type equations with a distributed order derivative in
the time variable. This derivative is defined as the integral in of
the Caputo-Dzhrbashian fractional derivative of order with a
certain positive weight function. Such equations are used in physical
literature for modeling diffusion with a logarithmic growth of the mean square
displacement. In this work we develop a mathematical theory of such equations,
study the derivatives and integrals of distributed order.Comment: 39 pages. To appear in J. Math. Anal. App
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