Fractional Diffusion Based on Riemann-Liouville Fractional Derivatives †

By Rudolf Hilfer. 2000

250 citations recorded by the source corpus.

A fractional diffusion equation based on Riemann−Liouville fractional derivatives is solved exactly. The initial values are given as fractional integrals. The solution is obtained in terms of H-functions. It differs from the known solution of fractional diffusion equations based on fractional integrals. The solution of fractional diffusion based on a Riemann−Liouville fractional time derivative does not admit a probabilistic interpretation in contrast with fractional diffusion based on fractional integrals. While the fractional initial value problem is well defined and the solution finite at all times, its values for t → 0 are divergent.

Subfields: Mathematical Physics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/ed8db2ee80de06aecefdce30f5bad3231909142c

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Citations in the corpus

Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain (2002, methodology): “Hilfer [ 18 ] presented a closed form solution for Riemann–Liouville based fractional diffusion equation in terms of H -functions.”
Threefold Introduction to Fractional Derivatives (2008, background): “As emphasized in [53] the choice of D0+ in (2.”
OPERATIONAL METHOD FOR THE SOLUTION OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH GENERALIZED RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVES (2009, result): “These findings extended earlier results [19] on the absence of”
A fully discrete difference scheme for a diffusion-wave system (2006, background): “Helfer [9] presented the solution of a fractional diffusion problem in terms of H -functions.”
A new fractional operator of variable order: Application in the description of anomalous diffusion (2016, methodology): “For example, the RiemannLiouville-type model of the AD was considered in [8,9], the Caputo-type model ∗ E-Mails: dyangxiaojun@163.”
Mittag-Leffler Functions and Their Applications (2009, background): “During the last two decades this function has come into prominence after about nine decades of its discovery by a Swedish Mathematician Mittag-Leffler, due to the vast potential of its applications in solving the problems of physical, biological, engineering, and earth sciences, and so forth.”