# Rudolf GorenfloFreie Universität Berlin | FUB · Department of Mathematics and Computer Science

Rudolf Gorenflo

Doctor rerum naturalium

## About

255

Publications

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Introduction

Additional affiliations

October 1973 - present

Education

October 1950 - February 1960

**Technische Hochschule Karlsruhe**

Field of study

- Mathematics

## Publications

Publications (255)

In this chapter we present the basic properties of the two-parametric Mittag-Leffler function Eα,β(z) (see (10.1007/978-3-662-61550-8_1)), which is the most straightforward generalization of the classical Mittag-Leffler function Eα(z) (see (10.1007/978-3-662-61550-8_3)). As in the previous chapter, the material can be formally divided into two part...

Gösta Magnus Mittag-LefflerMittag-Leffler, Gösta Magnus was born on March 16, 1846, in Stockholm, Sweden. His father, John Olof Leffler, was a school teacher, and was also elected as a member of the Swedish Parliament.

In this chapter we consider a number of integral equations and differential equations (mainly of fractional order). In representations of their solution, the Mittag-Leffler function, its generalizations and some closely related functions are used.

This chapter is devoted to the application of the Mittag-Leffler function and related special functions in the study of certain stochastic processes. As this topic is so wide, we restrict our attention to some basic ideas. For more complete presentations of the discussed phenomena we refer to some recent books and original papers which are mentione...

Consider the function defined for \(\alpha _1,\ \alpha _2\in {\mathbb R}\) \((\alpha _1^2+\alpha _2^2\ne 0)\) and \(\beta _1,\beta _2 \in {\mathbb C}\) by the series
$$\begin{aligned} E_{\alpha _1,\beta _1;\alpha _2,\beta _2}(z)\equiv \sum ^{\infty }_{k=0}\frac{z^k}{\varGamma (\alpha _1k+\beta _1) \varGamma (\alpha _2k+\beta _2)}\ \ (z\in {\mathbb...

In this chapter we present the basic properties of the classical Mittag-Leffler function Eα(z) (see (10.1007/978-3-662-61550-8_1)). The material can be formally divided into two parts. Starting from the basic definition of the Mittag-Leffler function in terms of a power series, we discover that for parameter α with positive real part the function E...

Here we present material illuminating the role of the Mittag-Leffler function and its generalizations in the study of deterministic models. It has already been mentioned that the Mittag-Leffler function is closely related to the Fractional Calculus (being called ‘The Queen Function of the Fractional Calculus’). This is why we focus our attention he...

This chapter deals with the classical Wright function. Like the functions of Mittag-Leffler type, the functions of Wright type are known to play fundamental roles in various applications of the fractional calculus. This is mainly due to the fact that they are interrelated with the Mittag-Leffler functions through Laplace and Fourier transformations...

The Prabhakar generalized Mittag-Leffler function [Pra71] is defined asPrabhakar function where (γ)n=γ(γ+1)…(γ+n-1) (see formula (A.17) in Appendix A).

The 2nd edition of this book is essentially an extended version of the 1st and provides a very sound overview of the most important special functions of Fractional Calculus. It has been updated with material from many recent papers and includes several surveys of important results known before the publication of the 1st edition, but not covered the...

In this chapter we consider the basic properties of some types of differential equations of fractional order that are related to relaxation and oscillation phenomena, based on functions of the Mittag-Leffler type. We finally generalize with fractional derivatives the Basset problem known in fluid-dynamics, so exploring other fractional relaxation p...

In this chapter, we consider simulation of spatially one-dimensional spacetime fractional diffusion. After a survey on the operators entering the basic fractional equation via Fourier-Laplace manipulations, we obtain the subordination integral formula that teaches us how a particle path can be constructed by first generating the operational time fr...

The chapter presents a survey of results on the properties and applications of the Mittag-Leffler function and its generalizations.

The main purpose of this note is to point out the relevance of the Mittag-Leffler probability distribution in the so-called thinning theory for a renewal process with a queue of power law type. This theory, formerly considered by Gnedenko and Kovalenko in 1968 without the explicit reference to the Mittag-Leffler function, was used by the authors in...

Two diierential transforms involving the Gauss hypergeometric function in the kernels are considered. They generalize the classical Riemann–Liouville and Erdélyi–Kober fractional diierential operators. Formulas of compositions for such generalized fractional diierentials with the product of Bessel functions of the rst kind are proved. Special cases...

The Caputo time-derivative is usually defined pointwise for well-behaved functions, say, for the continuously differentiable functions. Accordingly, in the publications devoted to the theory of the partial fractional differential equations with the Caputo derivatives, the functional spaces where the solutions are looked for are often the spaces of...

We have provided a fractional generalization of the Poisson renewal processes by replacing the first time derivative in the relaxation equation of the survival probability by a fractional derivative of order α(0 < α ≤ 1). A generalized Laplacian model associated with the Mittag-Leffler distribution is examined. We also discuss some properties of th...

The Caputo time-derivative is usually defined pointwise for well-behaved
functions, say, for continuously differentiable functions. Accordingly, in the
theory of the partial fractional differential equations with the Caputo
derivatives, the functional spaces where the solutions are looked for are often
the spaces of the smooth functions that are to...

As a result of researchers and scientists increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions have recently caught the interest of the scientific community. Focusing on the theory of the Mittag-Leffler functions, the present volume offers a sel...

In this chapter we consider a number of integral equations and differential equations (mainly of fractional order). In representations of their solution, the Mittag-Leffler function, its generalizations and some closely related functions are used.

In this chapter we present the basic properties of the classical Mittag-Leffler function E
α
(z) (see (1.0.1)). The material can be formally divided into two parts.

Consider the function defined for \(\alpha _{1},\ \alpha _{2} \in \mathbb{R}\) (α
12 +α
22 ≠ 0) and \(\beta _{1},\beta _{2} \in \mathbb{C}\) by the series
$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \i...

In this chapter we present the basic properties of the two-parametric Mittag-Leffler function E
α, β
(z) (see (1. 0. 3)), which is the most straightforward generalization of the classical Mittag-Leffler function E
α
(z) (see (3. 1. 1)).

Gösta Magnus Mittag-Leffler
was born on March 16, 1846, in Stockholm, Sweden. His father, John Olof Leffler, was a school teacher, and was also elected as a member of the Swedish Parliament. His mother, Gustava Vilhelmina Mittag, was a daughter of a pastor, who was a person of great scientific abilities. At his birth Gösta was given the name Leffle...

Here we present material illuminating the role of the Mittag-Leffler function and its generalizations in the study of deterministic models. It has already been mentioned that the Mittag-Leffler function is closely related to the Fractional Calculus (being called ‘The Queen Function of the Fractional Calculus’). This is why we focus our attention he...

The Prabhakar generalized Mittag-Leffler function [Pra71] is defined as
$$\displaystyle{ E_{\alpha,\beta }^{\gamma }(z):=\sum _{ n=0}^{\infty } \frac{(\gamma )_{n}} {n!\varGamma (\alpha n+\beta )}\,z^{n}\,,\quad Re\,(\alpha ) > 0,\,Re\,(\beta ) > 0,\,\gamma > 0, }$$ (5.1.1) where (γ)n
= γ(γ + 1)…(γ + n − 1) (see formula (A.1.17)).

This chapter is devoted to the application of the Mittag-Leffler function and related special functions in the study of certain stochastic processes. As this topic is so wide, we restrict our attention to some basic ideas. For more complete presentations of the discussed phenomena we refer to some recent books and original papers which are mentione...

Fractal fault systems are analyzed mechanically by means of the fractional calculus. Small elastic deviations from equilibrium are captured by vectorial wave equations which imply elastic energy and conservation of momentum with spatio-temporal isofractality. Laplace and Fourier transformations lead to an eigenvalue problem which enables a diagonal...

We present a generalization of Hilfer derivatives in which Riemann--Liouville
integrals are replaced by more general Prabhakar integrals. We analyze and
discuss its properties. Further, we show some applications of these generalized
Hilfer-Prabhakar derivatives in classical equations of mathematical physics,
like the heat and the free electron lase...

We have provided a fractional generalization of the Poisson renewal processes
by replacing the first time derivative in the relaxation equation of the
survival probability by a fractional derivative of order $\alpha ~(0 < \alpha
\leq 1)$. A generalized Laplacian model associated with the Mittag-Leffler
distribution is examined. We also discuss some...

In this paper, the Cauchy problem for the spatially one-dimensional distributed order diffusion-wave equation
$\int_0^2 {p(\beta )D_t^\beta u(x,t)d\beta } = \frac{{\partial ^2 }}
{{\partial x^2 }}u(x,t)
$
is considered. Here, the time-fractional derivative D
t
β is understood in the Caputo sense and p(β) is a non-negative weight function with su...

We generate the fractional Poisson process by subordinating the standard
Poisson process to the inverse stable subordinator. Our analysis is based on
application of the Laplace transform with respect to both arguments of the
evolving probability densities.

The fractional Poisson process and the Wright process (as discretization of
the stable subordinator) along with their diffusion limits play eminent roles
in theory and simulation of fractional diffusion processes. Here we have
analyzed these two processes, concretely the corresponding counting number and
Erlang processes, the latter being the proce...

We consider simulation of spatially one-dimensional space-time fractional
diffusion. Whereas in an earlier paper of ours we have developed the basic
theory of what we call parametric subordination via three-fold splitting
applied to continuous time random walk with subsequent passage to the diffusion
limit, here we go the opposite way. Via Fourier-...

For the symmetric case of space-fractional diffusion processes (whose basic analytic theory has been developed in 1952 by Feller via inversion of Riesz potential operators) we present three random walk models discrete in space and time. We show that for properly scaled transition to vanishing space and time steps these models converge in distributi...

The uncoupled Continuous Time Random Walk (CTRW) in one space-dimension and under power law regime is splitted into three distinct random walks: (rw
1), a random walk along the line of natural time, happening in operational time; (w
2), a random walk along the line of space, happening in operational time; (rw
3), the inversion of (rw
1), namely a r...

We find the upper viscosity solutions to a nonlinear two-term time fractional diffusion-wave equation with time operator in the Caputo–Dzherbashyan sense and a nonlinear Lipschitz force term F∈Lloc∞([0,T)×R),T>0,x∈R,(1)b1D∗β1u(x,t)+b2D∗β2u(x,t)=∂2∂x2u(x,t)+F(t,u(x,t)),t≥0,b1+b2=1,β1β2∈(0,2), subject to the Cauchy conditions (2)u(x,0)=f(x),ut(x,0)=g...

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...

The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. This formula allows us to treat the CTRW as a discrete-space discrete-time random walk that in the c...

In this article, we discuss the solution of the space-fractional diffusion equation with and without central linear drift in the Fourier domain and show the strong connection between it and the alpha-stable Levy distribution, 0 < alpha < 2. We use some relevant transformations of the independent variables x and t, to find the solution of the space-...

The Silences of the Archives, the Reknown of the Story.
The Martin Guerre affair has been told many times since Jean de Coras and Guillaume Lesueur published their stories in 1561. It is in many ways a perfect intrigue with uncanny resemblance, persuasive deception and a surprizing end when the two Martin stood face to face, memory to memory, befor...

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 math...

The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the Caputo sense. After giving a necessary outline of the classical theory of linear viscoelasticity, we contrast th...

Starting from the model of continuous time random walk, we focus our interest on random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. By stating the relevant lemmata (of Tauber type) for the di...

To offer an insight into the rapidly developing theory of fractional diffusion processes, we describe in some detail three topics of current 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 waitin...

The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order β ∈ (0, 1). The fundamental solution for the Cauchy problem is interpreted as a probability density of a self-similar non-Markovian stochastic process related to a phenomenon of s...

We announce the forthcoming papers [4] and [5] in which we prove the existence and the uniqueness, find properties, asymptotic and regularity of the solution to diffusion-wave phenomena. We give explicit solutions for 1, 2, …, n -term time fractional diffusion-wave equations. For the distributed order equation and for the equation with n-term time...

A physical-mathematical approach to anomalous diffusion may be based on fractional diffusion equations and related random walk models. The fundamental solutions of these equations can be interpreted as probability densities evolving in time of peculiar self-similar stochastic processes: an integral representation of these solutions is here presente...

After sketching the basic principles of renewal theory and recalling the classical Poisson process, we discuss two renewal processes characterized by waiting time laws with the same power asymptotics defined by special functions of Mittag–Leffler and of Wright type. We compare these three processes with each other.

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 whic...

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 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 an...

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. The fundamental solution (for the {Cauchy} problem) of the fractional diffusion equations can be interpreted as a probability density evolving in ti...

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 g...

It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probabilit...

After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because special functions of Mittag-Leffler and of Wright type appear in the definition of the relevant waiting times....

The first-order differential equation of exponential relaxation can be generalized by using either the fractional derivative in the Riemann-Liouville (R-L) sense and in the Caputo (C) sense, both of a single order less than 1. The two forms turn out to be equivalent. When, however we use fractional derivatives of distributed order (between zero and...

The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general distribution of time orders we provide the fundamental solution, that is still a probability density, in terms of an integral...

The well-scaled transition to the diffusion limit in the framework of the theory of continuous-time random walk (CTRW) is presented starting from its representation as an infinite series that points out the subordinated character of the CTRW itself. We treat the CTRW as a combination of a random walk on the axis of physical time with a random walk...

The time-fractional diffusion equation is obtained by generalizing the standard diffusion equation by using a proper time-fractional
derivative of order 1 — β in the Riemann-Liouville (R-L) sense or of order β in the Caputo (C) sense, with β ∈ (0, 1). The two forms are equivalent and the fundamental solution of the associated Cauchy problem is inte...

After sketching the basic principles of renewal theory and recalling the classical Poisson process, we discuss two renewal processes characterized by waiting time laws with the same power asymptotics defined by special functions of Mittag-Leffler and of Wright type. We compare these three processes with each other.

A 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. A more general approach is however provided by the integral equation for the so-called continuous time random walk (CTRW), which can be understood as a rando...

The basic differential equations of exponential relaxation and Gaussian diffusion can be generalized by replacing the first-order time derivative with a fractional derivative in the Riemann-Liouville (R-L) sense and in the Caputo (C) sense, both of a single order less than 1. The two forms turn out to be equivalent. When, however we use fractional...

Starting from the continuous time random walk (CTRW) scheme with the space-dependent waiting-time probability density function (PDF) we obtain the time-fractional diffusion equation with varying in space fractional order of time derivative. As an example, we study the evolution of a composite system consisting of two separate regions with different...

In high-frequency financial data not only returns, but also waiting times between consecutive trades are random variables. Therefore, it is possible to apply continuous-time random walks (CTRWs) as phenomenological models of the high-frequency price dynamics. An empirical analysis performed on the 30 DJIA stocks shows that the waiting-time survival...

First a survey is presented on how space-time fractional diffusion processes can be obtained by well-scaled limiting from continuous time random walks under the sole assumption of asymptotic power laws (with appropriate exponents for the tail behaviour of waiting times and jumps). The spatial operator in the limiting pseudo-differential equation is...

Using bivariate generating functions, we prove convergence of the Grünwald-Letnikov difference scheme for the fractional diffusion equation (in one space dimension) with and without central linear drift in the Fourier-Laplace domain as the space and time steps tend to zero in a well-scaled way. This implies convergence in distribution (weak converg...

We treat the question of existence, uniqueness and construction of a solution to the Cauchy and multi-point problems for a general linear evolution equa-tion with (in general) temporal fractional derivatives with distributed orders. Such equations have met great interest in recent years among researchers in viscoelasticity and in anomalous diffusio...

Continuous-time random walks can be used as phenomenological models of high-frequency time dynamics in financial markets. Empirical analyses show that the intertrade durations (or waiting-times) are non-exponentially distributed. This fact imposes constraints on agent-based models of financial markets based on continuous-double auctions.

A 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. A more general approach is however provided by the integral equation for the so-called continuous time random walk (CTRW), which can be understood as a rando...

Contents. 1.1 Banach spaces 1.2 Hilbert spaces 1.3 Some useful function spaces 1.3.1 Spaces of continuous functions 1.3.2 Spaces of integrable functions 1.3.3 Sobolev spaces 1.4 Analytic functions and harmonic functions 1.5 Fourier transform and Laplace transform

Contents. 4.1 Finite moment approximation of (4.1) 4.1.1 Proof of Theorem 4.1. 4.1.2 Proof of Theorem 4.2. 4.2 A moment problem from Laplace transform 4.3 Notes and remarks

6.1 Analyticity of harmonic functions
6.2 Cauchy’s problem for the Laplace equation
6.3 Surface temperature determination from borehole measurements (steady case)

5.1 Reconstruction of functions in H
2(U): approximation by polynomials
5.2 Reconstruction of an analytic function: a problem of optimal recovery
5.3 Cardinal series representation and approximation: reformulation of moment problems
5.3.1 Two-dimensional Sinc theory
5.3.2 Approximation theorems

Contents. 2.1 Method of truncated expansion 2.1.1 A construction of regularized solutions 2.1.2 Convergence of regularized solutions and error estimates 2.1.3 Error estimates using eigenvalues of the Laplacian 2.2 Method of Tikhonov 2.2.1 Case 1: exact solutions in L2(W)L^2(\Omega ) 2.2.2 Case 2: exact solutions in
La*(W), 1 \leqq a* \leqq ¥L^{\a...

7.1 The backward heat equation
7.2 Surface temperature determination from borehole measurements: a two-dimensional problem
7.3 An inverse two-dimensional Stefan problem: identification of boundary values
7.4 Notes and remarks

The purpose of this paper is to provide a generalization of the Poisson renewal process and the related Erlang distribution via fractional calculus. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the counting function and its average, the survival probability. If the waiting time is exp...

A detailed study is presented for a large class of uncoupled continuous-time random walks. The master equation is solved for the Mittag-Leffler survival probability. The properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed. Finally, some common objections found in the l...

In financial markets not only returns, but also waiting times between consecutive trades are random variables and it is possible to apply continuous- time random walks (CTRWs) as phenomenological models of high-frequency prices. Based on these considerations, in this extended abstract, some results are outlined which can be useful for speculative o...

For space–time fractional diffusion equations a theory of discrete-space discrete-time random walks, analogous to the theory of continuous-time random walks, is presented. The essential assumption is that the probabilities for waiting times and jump-widths behave asymptotically like powers with negative exponents related to the orders of the fracti...

In high-frequency financial data not only returns, but also waiting times between consecutive trades are random variables. Therefore, it is possible to apply continuous-time random walks (CTRWs) as phenomenological models of the high-frequency price dynamics. An empirical analysis performed on the 30 DJIA stocks shows that the waiting-time survival...

The fractional diffusion equation is derived from the master equation of continuous-time random walks (CTRWs) via a straightforward application of the Gnedenko-Kolmogorov limit theorem. The Cauchy problem for the fractional diffusion equation is solved in various important and general cases. The meaning of the proper diffusion limit for CTRWs is di...

We propose diffusionlike equations with time and space fractional derivatives of the distributed order for the kinetic description of anomalous diffusion and relaxation phenomena, whose diffusion exponent varies with time and which, correspondingly, cannot be viewed as self-affine random processes possessing a unique Hurst exponent. We prove the po...