Papers by Rudolf Gorenflo
The fractional oscillation equation is obtained from the classical equation for linear oscillatio... more The fractional oscillation equation is obtained from the classical equation for linear oscillations by replacing the second-order time derivative by a fractional derivative of order with 1 < < 2 : Using the method of the Laplace transform, it is shown that the fundamental solutions can be expressed in terms of Mittag-Le er functions, and exhibit a nite number of damped oscillations with an algebraic decay.
A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equ... more A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. By the spacetime fractional diffusion equation we mean an evolution equation obtained from the standard linear diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2] and skewness θ (|θ| ≤ min {α, 2 − α}), and the first-order time derivative with a Caputo derivative of order β ∈ (0, 1] . The fundamental solution (for the Cauchy problem) of the fractional diffusion equation can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process. We view it as a generalized diffusion process that we call fractional diffusion process, and present an integral representation of the fundamental solution. A more general approach to anomalous diffusion is however known to be provided by the master equation for a continuous time random walk (CTRW). We show how this equation reduces to our fractional diffusion equation by a properly scaled passage to the limit of compressed waiting times and jump widths. Finally, we describe a method of simulation and display (via graphics) results of a few numerical case studies.
Fractional Calculus and Applied Analysis, Jan 28, 2007
In this survey paper we consider some applications of the Wright function with special emphasis o... more In this survey paper we consider some applications of the Wright function with special emphasis of its key role in the partial differential equations of fractional order. It was found that the Green function of the time-fractional diffusion-wave equation can be represented in terms of the Wright function. Furthermore, extending the methods of Lie groups in partial differential equations to the partial differential equations of fractional order it was shown that some of the group-invariant solutions of these equations can be given in terms of the Wright and the generalized Wright functions.Finally, we discuss recent results about distribution of zeros of the Wright function, its order, type and indicator function.
Zeitschrift Fur Naturforschung a a Journal of Physical Sciences, 1987
Integral Transform Spec Funct, 1997
A modification of the mikusinski operational caculus is used to obtain explicit solution of some ... more A modification of the mikusinski operational caculus is used to obtain explicit solution of some classes of integral equations of the second kind of abel type .inparticular the solution of the generalized abel equation with the riemann-liouville fractional integral opereator is expressed in terms of the mittag-leffler function of several variables.
Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by r... more Fractional calculus allows one to generalize the linear, one-dimensional, diffusion equation by replacing either the first time derivative or the second space derivative by a derivative of fractional order. The fundamental solutions of these equations provide probability density functions, evolving on time or variable in space, which are related to the class of stable distributions. This property is a noteworthy generalization of what happens for the standard diffusion equation and can be relevant in treating financial and economical problems where the stable probability distributions play a key role.
... de This paper is a slightly revised version of the paper (with the same title) published on p... more ... de This paper is a slightly revised version of the paper (with the same title) published on pages 132-149 of the Conference Proceedings (Editors P. Rusev, I. Dimovski, V. Kiryakova) Transform ... 17. D. Stoyan and J. Mecke, Stochastische Geometric Akademie-Verlag, Berlin, 1983. ...
To offer a view into the rapidly developing theory of fractional diffusion processes we describe ... more To offer a view into the rapidly developing theory of fractional diffusion processes we describe in some detail three topics of present interest: (i) the well-scaled passage to the limit from continuous time random walk under power law assumptions to space-time fractional diffusion, (ii) the asymptotic universality of the Mittag-Leffler waiting time law in time-fractional processes, (iii) our method of parametric subordination for generating particle trajectories.
Zeitschrift Fur Angewandte Mathematik Und Mechanik, 1991
Eprint Arxiv 0705 0797, May 6, 2007
We show the asymptotic long-time equivalence of a generic power law waiting time distribution to ... more We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler waiting time distribution, characteristic for a time fractional CTRW. This asymptotic equivalence is effected by a combination of "rescaling" time and "respeeding" the relevant renewal process followed by a passage to a limit for which we need a suitable relation between the parameters of rescaling and respeeding. Turning our attention to spatially 1-D CTRWs with a generic power law jump distribution, "rescaling" space can be interpreted as a second kind of "respeeding" which then, again under a proper relation between the relevant parameters leads in the limit to the space-time fractional diffusion equation. Finally, we treat the `time fractional drift" process as a properly scaled limit of the counting number of a Mittag-Leffler renewal process.
Fractional Calculus and Applied Analysis, Feb 6, 2007
The Mellin transform is usually applied in probability theory to the product of independent rando... more The Mellin transform is usually applied in probability theory to the product of independent random variables. In recent times the machinery of the Mellin transform has been adopted to describe the L\'evy stable distributions, and more generally the probability distributions governed by generalized diffusion equations of fractional order in space and/or in time. In these cases the related stochastic processes are self-similar and are simply referred to as fractional diffusion processes. We provide some integral formulas involving the distributions of these processes that can be interpreted in terms of subordination laws.
Fractional Calculus and Applied Analysis, 2000
We deal with a partial differential equation of fractional order where the time derivative of ord... more We deal with a partial differential equation of fractional order where the time derivative of order β ∈ (0; 2] is defined in the Caputo sense and the space derivative of order α ∈ (0; 2] is given as a pseudo differential operator with the Fourier symbol −|κ| α , κ ∈ IR. This equation contains as particular cases the diffusion and the wave equations and it has already appeared both in mathematical papers and in applications. The main result of the paper consists in giving a mapping in the form of a linear integral operator between solutions of the equation with different parameters α, β and in presenting an explicit formula for the Green function of the Cauchy problem for the fractional diffusion-wave equation.
We show the asymptotic long-time equivalence of a generic power law waiting time distribution to ... more We show the asymptotic long-time equivalence of a generic power law waiting time distribution to the Mittag-Leffler distribution, the waiting time distribution characteristic for a time-fractional continuous time random walk. This asymptotic equivalence is effected by a combination of "rescaling" time and "respeeding" the relevant renewal process and subsequent passage to a limit for which we need a suitable relation
Archives of Mechanics, 1998
{ Fractional calculus allows one to generalize the linear (one dimensional) di usion equation by ... more { Fractional calculus allows one to generalize the linear (one dimensional) di usion equation by replacing either the rst time derivative or the second space derivative by a derivative of a fractional order. The fundamental solutions of these generalized di usion equations are shown to provide certain probability density functions, in space or time, which are related to the relevant class of stable distributions. For the space fractional di usion a random-walk model is also proposed.
A modi cation of the Mikusinski operational calculus is used to obtain explicit solutions of some... more A modi cation of the Mikusinski operational calculus is used to obtain explicit solutions of some classes of integral equations of the second kind of Abel type. In particular, the solution of the generalized Abel equation
We discuss some applications of the Mittag-Leffler function and related probability distributions... more We discuss some applications of the Mittag-Leffler function and related probability distributions in the theory of renewal processes and continuous time random walks. In particular we show the asymptotic (long time) equivalence of a generic power law waiting time to the Mittag-Leffler waiting time distribution via rescaling and respeeding the clock of time. By a second respeeding (by rescaling the spatial variable) we obtain the diffusion limit of the continuous time random walk under power law regimes in time and in space. Finally, we exhibit the time-fractional drift process as a diffusion limit of the fractional Poisson process and as a subordinator for space-time fractional diffusion.
We introduce the linear operators of fractional integration and fractional differentiation in the... more We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.
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Papers by Rudolf Gorenflo