Time-fractional telegraph equations and telegraph processes with brownian time
TL;DR: 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.
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TL;DR: In this survey paper, nearly all types of Mittag-Leffler type functions existing in the literature are presented and an attempt is made to present nearly an exhaustive list of references to make the reader familiar with the present trend of research in Mittag, Leffler, and type functions and their applications.
Abstract: Motivated essentially by the success of the applications of the Mittag-Leffler functions in many areas
of science and engineering, the authors present, in a unified manner, a detailed account or rather a brief
survey of the Mittag-Leffler function, generalized Mittag-Leffler functions, Mittag-Leffler type functions,
and their interesting and useful properties. Applications of G. M. Mittag-Leffler functions in certain areas of
physical and applied sciences are also demonstrated. 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. In this survey paper, nearly all types of Mittag-Leffler type functions existing in the
literature are presented. An attempt is made to present nearly an exhaustive list of references concerning
the Mittag-Leffler functions to make the reader familiar with the present trend of research in Mittag-Leffler
type functions and their applications.
661 citations
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TL;DR: A detailed survey of Mittag-Leffler type functions can be found in this article, where the authors present a detailed account or rather a brief survey of the Mittag Leffler function, generalized Mittag leffler functions and their interesting and useful properties.
Abstract: Motivated essentially by the success of the applications of the Mittag-Leffler functions in many areas of science and engineering, the authors present in a unified manner, a detailed account or rather a brief survey of the Mittag- Leffler function, generalized Mittag-Leffler functions, Mittag-Leffler type functions, and their interesting and useful properties. Applications of Mittag-Leffler functions in certain areas of physical and applied sciences are also demonstrated. During the last two decades this function has come into prominence after about nine decades of its discovery by a Swedish mathematician G.M. Mittag-Leffler, due its vast potential of its applications in solving the problems of physical, biological, engineering and earth sciences etc. In this survey paper, nearly all types of Mittag-Leffler type functions existing in the literature are presented. An attempt is made to present nearly an exhaustive list of references concerning the Mittag-Leffler functions to make the reader familiar with the present trend of research in Mittag-Leffler type functions and their applications.
528 citations
TL;DR: In this paper, the authors considered a time fractional diffusion equation on a finite domain and proposed a computationally effective implicit difference approximation (IDA) method to solve the problem, and proved that the IDA is unconditionally stable and convergent with O(tau+h^2) time and space steps.
Abstract: In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.
236 citations
Cites background from "Time-fractional telegraph equations..."
...Enzo et al. [ 3 ] and Luisa et al. [13] considered and proved the solutions to the Cauchy problem of the fractional telegraph equation can be expressed as the distribution of a suitable composition of dierent processes....
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TL;DR: In this article, three different fractional versions of the standard Poisson process and some related results concerning the distribution of order statistics and the compound poisson process are presented, and a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by fractional Poisson processes is presented.
Abstract: We present three different fractional versions of the Poisson process and some related results concerning the distribution of order statistics and the compound Poisson process. The main version is constructed by considering the difference-differential equation governing the distribution of the standard Poisson process, $ N(t),t>0$, and by replacing the time-derivative with the fractional Dzerbayshan-Caputo derivative of order $
u\in(0,1]$. For this process, denoted by $\mathcal{N}_
u(t),t>0,$ we obtain an interesting probabilistic representation in terms of a composition of the standard Poisson process with a random time, of the form $\mathcal{N}_
u(t)= N(\mathcal{T}_{2
u}(t)),$ $t>0$. The time argument $\mathcal{T}_{2
u }(t),t>0$, is itself a random process whose distribution is related to the fractional diffusion equation. We also construct a planar random motion described by a particle moving at finite velocity and changing direction at times spaced by the fractional Poisson process $\mathcal{N}_
u.$ For this model we obtain the distributions of the random vector representing the position at time $t$, under the condition of a fixed number of events and in the unconditional case. For some specific values of $
u\in(0,1]$ we show that the random position has a Brownian behavior (for $
u =1/2$) or a cylindrical-wave structure (for $
u =1$).
233 citations
Additional excerpts
...…(2.18) and (2.19), we obtain the inverse Laplace transform as Gν(u, t) = 1 Γ(1− ν) ∫ +∞ 0 d y ∫ t 0 (t −w)−ν fν(w; y)eλy(u−1)dw. (2.20) It has been proved in Orsingher and Beghin (2004) that the solution v2ν to (2.12) can be expressed as v2ν(y, t) = 1 2Γ(1− ν) ∫ t 0 (t −w)−ν fν(w; |y |)dw, y…...
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Book•
08 Dec 1993
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.
Abstract: Fractional integrals and derivatives on an interval fractional integrals and derivatives on the real axis and half-axis further properties of fractional integrals and derivatives other forms of fractional integrals and derivatives fractional integrodifferentiation of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations fo the first kind with special function kernels applications to differential equations.
7,096 citations
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.
Abstract: Diffusion and wave equations together with appropriate initial condition(s) are rewritten as integrodifferential equations with time derivatives replaced by convolution with tα−1/Γ(α), α=1,2, respectively. Fractional diffusion and wave equations are obtained by letting α vary in (0,1) and (1,2), respectively. The corresponding Green’s functions are obtained in closed form for arbitrary space dimensions in terms of Fox functions and their properties are exhibited. In particular, it is shown that the Green’s function of fractional diffusion is a probability density.
1,046 citations