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Time-fractional telegraph equations and telegraph processes with brownian time
Enzo Orsingher,Luisa Beghin +1 more
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In this paper, the fundamental solutions to time-fractional telegraph equations of order 2α were studied and the Fourier transform of the solutions for any α and the representation of their inverse, in terms of stable densities, was given.Abstract:
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.read more
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Time-fractional Birth and Death Processes
TL;DR: In this article, the authors provide different representations for a time-fractional birth and death process whose transition probabilities are governed by a time fractional system of differential equations, and provide results for the asymptotic behavior of the process conditioned not to be killed.
Stochastic processes related to the time-fractional diffusion-wave equation
TL;DR: In this paper, the authors considered the question if u(x,t) can be interpreted in a natural way as the sojourn probability density (in point x, evolving in time t) of a randomly wandering particle starting in the origin x=0 at instant t=0.
Structurally damped $\sigma-$evolution equations with power-law memory
Nelson Faustino,Jorge Marques +1 more
TL;DR: In this paper , the authors studied space-time fractional analogues of evolution equations arising in applications toward the theory of structural damping equations of plate/wave type, and they considered an integro-differential counterpart of the σ − evolution equation of the type ∂ 2 t u (t,x ) + µ ( − ∆) σ 2 ∂ t u( t,x) + σ u(x ) = f ( t, x ) with σ > 0 and µ > − 2, that encoded memory of power-law type.
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Nonlocal in-time telegraph equation and telegraph processes with random time
TL;DR: In this paper , the authors studied the properties of a non-markovian version of the telegraph process, whose non-Markovian character comes from a nonlocal in-time evolution equation that is satisfied by its probability density function.
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Book
Fractional Integrals and Derivatives: Theory and Applications
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
Journal ArticleDOI
Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).
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Fractional diffusion and wave equations
W. R. Schneider,W. Wyss +1 more
TL;DR: In this article, the Green's function of fractional diffusion is shown to be a probability density and the corresponding Green's functions are obtained in closed form for arbitrary space dimensions in terms of Fox functions and their properties are exhibited.