Journal ArticleDOI
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
Citations
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On traveling-wave solutions for the scaling-law telegraph equations
TL;DR: In this article, the scaling-law telegraph equations with the Mandelbrot-scaling-law derivative with the traveling-wave solutions with use of the Kohlrausch-Williams-Watts function are considered in detail.
Journal ArticleDOI
Composition of Processes and Related Partial Differential Equations
Mirko D'Ovidio,Enzo Orsingher +1 more
TL;DR: In this paper, various types of compositions involving independent fractional Brownian motions are examined, and the authors show that they can be expressed in terms of independent fractions of Brownians.
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Fractional exponential operators and time-fractional telegraph equation
TL;DR: In this article, the inverse Mellin transform is used for finding an integral representation for a fractional exponential operator, which can be considered as an approach for solving partial fractional differential equations.
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A class of efficient time-stepping methods for multi-term time-fractional reaction-diffusion-wave equations
TL;DR: The truncation error with second-order accuracy is proved under the framework of the shifted convolution quadrature in a family of novel time-stepping methods for the fractional calculus operators with a shifted parameter.
Journal ArticleDOI
Stochastic solutions of a class of Higher order Cauchy problems in $\rd$
TL;DR: In this article, the authors study solutions of a class of higher-order partial differential equations in bounded domains, where the authors express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index 0 < β < 1, or by the absolute value of a symmetric α-stable process with 0 < α ≤ 2.
References
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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).
Journal ArticleDOI
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.