Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

D’Ovidio, Mirko; Orsingher, Enzo; Toaldo, Bruno

doi:10.1080/07362994.2014.962046

Cited by 53 publications

(51 citation statements)

References 21 publications

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“…During the last decade a number of works have appeared in the mathematics literature, particularly those of Orshinger and collaborators [47][48][49], analyzing mathematical and other formal properties of the FTE. However, the fractional equation is set in an ad hoc fashion by replacing the ordinary derivatives that appear in the TE by fractional derivatives which, in addition, may be of various types [49]. Following this way the standard form of the FTE in one dimension reads…”

Section: Introduction and General Scopementioning

confidence: 99%

Fractional telegrapher's equation from fractional persistent random walks

Masoliver

2016

Phys. Rev. E

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We generalize the telegrapher's equation to allow for anomalous transport. We derive the space-time fractional telegrapher's equation using the formalism of the persistent random walk in continuous time. We also obtain the characteristic function of the space-time fractional process and study some particular cases and asymptotic approximations. Similarly to the ordinary telegrapher's equation, the time-fractional equation also presents distinct behaviors for different time scales. Specifically, transitions between different subdiffusive regimes or from superdiffusion to subdiffusion are shown by the fractional equation as time progresses.

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Section: Introduction and General Scopementioning

confidence: 99%

Fractional telegrapher's equation from fractional persistent random walks

Masoliver

2016

Phys. Rev. E

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“…However, for the sake of simplicity and briefness, we have not treated them here, and we will deal with these questions in a future work. In any case, we address the interested reader to the works of Mainardi and collaborators [46,47,62,64] on solutions for fractional diffusion and fractional wave-diffusion equations and to Orsingher and collaborators [65][66][67] on several kinds of solutions to the FTE.…”

Section: Discussionmentioning

confidence: 99%

Three-dimensional telegrapher's equation and its fractional generalization

Masoliver

2017

Phys. Rev. E

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We derive the three-dimensional telegrapher's equation out of a random walk model. The model is a threedimensional version of the multistate random walk where the number of different states form a continuum representing the spatial directions that the walker can take. We set the general equations and solve them for isotropic and uniform walks which finally allows us to obtain the telegrapher's equation in three dimensions. We generalize the isotropic model and the telegrapher's equation to include fractional anomalous transport in three dimensions.

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“…For the technique of the random time change of processes see, for example, [9,20,32,36]. Only for ν > 1, ν represents itself a Lévy process, since it is obtained by subordinating a Lévy process to a stable subordinator.…”

Section: Beghinmentioning

confidence: 99%

“…for some M ∈ N, such that l ν (z, t) is decreasing for z ≥ M The first term in (20) can be bounded as follows: …”

Section: Theoremmentioning

confidence: 99%

Fractional Gamma and Gamma-Subordinated Processes

Beghin

2015

Stochastic Analysis and Applications

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We introduce and study fractional generalizations of the well-known Gamma process, in the following sense: the corresponding densities are proved to satisfy the same differential equation as the usual Gamma process, but with the shift operator replaced by its fractional version of order $\nu >0$. In the case $\nu >1$, the solution corresponds to the density of a Gamma process time-changed by an independent stable subordinator of index $1/\nu $. For $\nu $ less than one an analogous result holds, with the subordinator replaced by the inverse. In this case the fractional Gamma process is proved to be a non-stationary version of the standard one, with power law behavior of the expected value. Hence it can be considered a useful tool in modelling stochastic deterioration in the non-linear cases, a situation which often occurs in real data (see i.e., \cite{VAN} and the references therein).\ud As a consequence of the previous results, the fractional generalizations of some Gamma subordinated processes (i.e. the Variance Gamma, the Geometric Stable and the Negative Binomial) are introduced and the corresponding fractional differential equations are obtained. These processes are particularly relevant for a wide range of financial and technological applications

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Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

Cited by 53 publications

References 21 publications

Fractional telegrapher's equation from fractional persistent random walks

Fractional telegrapher's equation from fractional persistent random walks

Three-dimensional telegrapher's equation and its fractional generalization

Fractional Gamma and Gamma-Subordinated Processes

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