The Space-Fractional Telegraph Equation and the Related Fractional Telegraph Process

Orsingher, Enzo; Zhao, Xuelei

doi:10.1142/s0252959903000050

Cited by 73 publications

(47 citation statements)

References 5 publications

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“…Angulo et al (2000) have studied diffusion equations with space-fractional derivatives (in the sense of Riesz inverse operator). Fractional equations of different type, like the Black and Scholes one (see Wyss (2000)) and the space-fractional telegraph equation (see Orsingher and Zhao (2003)) have recently been considered.…”

Section: Introductionmentioning

confidence: 99%

Time-fractional telegraph equations and telegraph processes with brownian time

Orsingher

Beghin

2003

Probab. Theory Relat. Fields

241

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We study the fundamental solutions to time-fractional telegraph equations of order 2α. We are able to obtain the Fourier transform of the solutions for any α and to give a representation of their inverse, in terms of stable densities. For the special case α = 1/2, we can show that the fundamental solution is the distribution of a telegraph process with Brownian time. In a special case, this becomes the density of the iterated Brownian motion, which is therefore the fundamental solution to a fractional diffusion equation of order 1/2 with respect to time.

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Section: Introductionmentioning

confidence: 99%

Time-fractional telegraph equations and telegraph processes with brownian time

Orsingher

Beghin

2003

Probab. Theory Relat. Fields

241

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“…The characteristic function of the process T 2β (t), t > 0, whose distribution satisfies [14]. Therefore by performing the same steps as in theorem (4.2) we prove that …”

Section: Remark 43mentioning

confidence: 69%

“…The space-fractional telegraph equation (with M. Riesz space derivatives) has been considered in Orsingher and Zhao [14], while the connection between space-fractional equations and asymmetric stable processes has been established in Feller [7]. Fractional telegraph equations from the analytic point of view have been studied by many authors (see Saxena, Mathai and Haubold [18] for equations with n time derivatives).…”

mentioning

confidence: 99%

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

D’Ovidio

Orsingher

Toaldo

2014

Stochastic Analysis and Applications

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Abstract. In this work we construct compositions of vector processes of the form S 2β n c 2 L ν (t) , t > 0, ν ∈ 0, 1 2 , β ∈ (0, 1], n ∈ N, whose distribution is related to space-time fractional n-dimensional telegraph equations. We present within a unifying framework the pde connections of n-dimensional isotropic stable processes S 2β n whose random time is represented by the inverse L ν (t), t > 0, of the superposition of independent positively-skewed stable processes,, independent stable subordinators). As special cases for n = 1, ν = ) and the two-dimensional motion at finite velocity with a random time are investigated. For all these processes we present their counterparts as Brownian motion at delayed stabledistributed time.

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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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The Space-Fractional Telegraph Equation and the Related Fractional Telegraph Process

Cited by 73 publications

References 5 publications

Time-fractional telegraph equations and telegraph processes with brownian time

Time-fractional telegraph equations and telegraph processes with brownian time

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

Three-dimensional telegrapher's equation and its fractional generalization

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