Category: Mathematics

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diffusion Fractional Calculus Porous Media Two-Phase Flow

Simulating the Saturation Front Using a Fractional Diffusion Model

E. Gerolymatou, I. Vardoulakis, R. Hilfer

in: Proceedings of the GRACM05 International Congress on Computational Mechanics, Limassol 2005
edited by: G. Georgiou, P. Papanastasiou, M. Papadrakakis
GRACM, Athens, 653 (2005)

submitted on
Thursday, June 30, 2005

In this paper the possibility of making use of fractional derivatives for the simulation of the flow of water through porous media and in particular through soils is considered. The Richards equation, which is a non-linear diffusion equation, will be taken as a basis and is used for the comparison of results. Fractional derivatives differ from derivatives of integer order in that they entail the whole history of the function in a weighted form and not only its local behavior, meaning that a different numerical approach is required. Previous work on the topic will be examined and a consistent approach based on fractional time evolutions will be presented.



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Mathematics Special Functions

Computation of the Generalized Mittag-Leffler Function and its Inverse in the Complex Plane

R. Hilfer, H.J. Seybold

Integral Transforms and Special Functions 17, 637 (2006)
https://doi.org/10.1080/10652460600725341

submitted on
Monday, March 21, 2005

The generalized Mittag-Leffler function Eα,β (z) has been studied for arbitrary complex argument and real parameters. This function plays a fundamental role in the theory of fractional differential equations and numerous applications in physics. The Mittag-Leffler function interpolates smoothly between exponential and algebraic functional behaviour. A numerical algorithm for its evaluation has been developed. The algorithm is based on integral representations and exponential asymptotics. Results of extensive numerical calculations in the complex z-plane are reported here. We find that all complex zeros emerge from the point z = 1 for small alpha. They diverge towards negative infinity for alpha approaching unity. All the complex zeros collapse pairwise onto the negative real axis for α approaching 2. We introduce and study also the inverse generalized Mittag-Leffler function. We determine its principal branch numerically.



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diffusion Fractional Calculus

On fractional diffusion and continuous time random walks

R. Hilfer

Physica A 329, 35 (2003)
https://doi.org/10.1016/S0378-4371(03)00583-1

submitted on
Thursday, May 22, 2003

A continuous time random walk model is presented with long-tailed waiting time density that approaches a Gaussian distribution in the continuum limit. This example shows that continuous time random walks with long time tails and di!usion equations with a fractional time derivative are in general not asymptotically equivalent.



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Mathematics Special Functions

Numerical Results for the Generalized Mittag-Leffler Function

H.J. Seybold, R. Hilfer

Fractional Calculus and Applied Analysis 8, 127 (2005)

submitted on
Wednesday, June 4, 2003

Results of extensive calculations for the generalized Mittag-Leffler function are presented in a region of the complex plane. This function is related to the eigenfunction of a fractional derivative.



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Fractional Calculus Fractional Time Theory of Time

Strange Kinetics

R. Hilfer, R. Metzler, A. Blumen, J. Klafter(eds)

Chemical Physics 284, 1 (2002)
https://doi.org/10.1016/S0301-0104(02)00801-7

submitted on
Monday, July 8, 2002

The term strange kinetics originally referred to the dynamics of Hamiltonian systems which, in the limit of weak chaos, display superdiffusion and Levy-walk characteristics. Here we employ the term strange kinetics in a generalized sense to denote all forms of slow kinetics or anomalous dynamics, such as sub-diffusion, superdiffusion, non-Debye relaxation, Levy walks or fractional time evolutions.



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Fractional Calculus Fractional Time Theory of Time

Remarks on Fractional Time

R. Hilfer

in: Time, Quantum and Information
edited by: L. Castell and O. Ischebeck
Springer, Berlin, 235 (2003)
10.1007/978-3-662-10557-3
ISBN: 978-3-540-44033-8

submitted on
Monday, July 1, 2002

It is not possible to repeat an experiment in the past. The underlying philosophical truth in this observation is the difference between certainty of the past and potentiality of the future. This difference is discussed, for example, in C.F.v. Weizsäcker’s papers and it was often pointed out by him in our discussions in the years 1983-1986 in the Starnberg institute. The perennial philosophical problem related to this difference between past and future is the question whether time is real or not.



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dielectric relaxation Fractional Calculus Fractional Time Glasses

Experimental Evidence for Fractional Time Evolution in Glass Forming Materials

R. Hilfer

Chem.Phys. 284, 399 (2002)
https://doi.org/10.1016/S0301-0104(02)00670-5

submitted on
Friday, December 7, 2001

The infinitesimal generator of time evolution in the standard equation for exponential (Debye) relaxation is replaced with the infinitesimal generator of composite fractional translations. Composite fractional translations are defined as a combination of translation and the fractional time evolution introduced in [Physica A, 221 (1995) 89]. The fractional differential equation for composite fractional relaxation is solved. The resulting dynamical susceptibility is used to fit broad band dielectric spectroscopy data of glycerol. The composite fractional susceptibility function can exhibit an asymmetric relaxation peak and an excess wing at high frequencies in the imaginary part. Nevertheless it contains only a single stretching exponent. Qualitative and quantitative agreement with dielectric data for glycerol is found that extends into the excess wing. The fits require fewer parameters than traditional fit functions and can extend over up to 13 decades in frequency.



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dielectric relaxation Fractional Calculus Fractional Time Glasses

Fitting the excess wing in the dielectric α-relaxation of propylene carbonate

R. Hilfer

Journal of Physics: Condensed Matter 14, 2297 (2002)
https://doi.org/10.1088/0953-8984/14/9/318

submitted on
Wednesday, November 28, 2001

A novel fitting function for the complex frequency-dependent dielectric susceptibility is introduced and compared against other fitting functions for experimental broadband dielectric loss spectra of propylene carbonate taken from Schneider et al (Schneider U, Lunkenheimer P, Brand R and Loidl A 1999 Phys. Rev. E 59 6924). The fitting function contains a single stretching exponent similar to the familiar Cole–Davidson or Kohlrausch stretched exponential fits. It is compared to these traditional fits as well as to the Havriliak–Negami susceptibility and a susceptibility for a two-step Debye relaxation. The results for the novel fit are found to give superior agreement.



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dielectric relaxation Glasses Special Functions

H-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems

R. Hilfer

Physical Review E 65, 061510 (2002)
https://doi.org/10.1103/PhysRevE.65.061510

submitted on
Thursday, June 28, 2001

Analytical expressions in the time and frequency domains are derived for non-Debye relaxation processes. The complex frequency-dependent susceptibility function for the stretched exponential relaxation function is given for general values of the stretching exponent in terms of H-functions. The relaxation functions corresponding to the complex frequency-dependent Cole-Cole, Cole-Davidson, and Havriliak-Negami susceptibilities are given in the time domain in terms of H-functions. It is found that a commonly used correspondence between the stretching exponent of Kohlrausch functions and the stretching parameters of Havriliak-Negami susceptibilities are not generally valid.



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dielectric relaxation Glasses Nonequilibrium Special Functions

Analytical representations for relaxation functions of glasses

R. Hilfer

Journal of Non-Crystalline Solids 305, 122 (2002)
https://doi.org/10.1016/S0022-3093(02)01088-8

submitted on
Friday, April 13, 2001

Analytical representations in the time and frequency domains are derived for the most frequently used phenomenological fit functions for non-Debye relaxation processes. In the time domain the relaxation functions corresponding to the complex frequency dependent Cole–Cole, Cole–Davidson and Havriliak–Negami susceptibilities are also rep- resented in terms of H-functions. In the frequency domain the complex frequency dependent susceptibility function corresponding to the time dependent stretched exponential relaxation function is given in terms of H-functions. The new representations are useful for fitting to experiment.



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Fractional Calculus Glasses

On Fractional Relaxation

R. Hilfer

Fractals 11, 251 (2003)
https://doi.org/10.1142/S0218348X03001914

submitted on
Monday, April 2, 2001

Generalized fractional relaxation equations based on generalized Riemann-Liouville derivatives are combined with a simple short time regularization and solved exactly. The solution involves generalized Mittag-Leffler functions. The associated frequency dependent susceptibilities are related to symmetrically broadened Cole-Cole susceptibilities occurring as Johari Goldstein β -relaxation in many glass formers. The generalized susceptibilities exhibit a high frequency wing and strong minimum enhancement.



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Fractional Calculus Glasses

On Fractional Relaxation

R. Hilfer

in: Scaling and Disordered Systems
edited by: F. Family and M. Daoud and H. Herrmann and H.E. Stanley
World Scientific, Singapore, 251 (2002)
https://doi.org/10.1142/9789812778109_0026
ISBN: 978-981-02-4838-3

submitted on
Monday, April 2, 2001

Generalized fractional relaxation equations based on generalized Riemann-Liouville derivatives are combined with a simple short time regularization and solved exactly. The solution involves generalized Mittag-Leffler functions. The associated frequency dependent susceptibilities are related to symmetrically broadened Cole-Cole susceptibilities occurring as Johari Goldstein β-relaxation in many glass formers. The generalized susceptibilities exhibit a high frequency wing and strong minimum enhancement.



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diffusion Fractional Calculus

Fractional Diffusion based on Riemann-Liouville Fractional Derivatives

R. Hilfer

The Journal of Physical Chemistry B 104, 3914-3917 (2000)
DOI: 10.1021/jp9936289

submitted on
Tuesday, October 12, 1999

A fractional diffusion equation based on Riemann−Liouville fractional derivatives is solved exactly. The initial values are given as fractional integrals. The solution is obtained in terms of H-functions. It differs from the known solution of fractional diffusion equations based on fractional integrals. The solution of fractional diffusion based on a Riemann−Liouville fractional time derivative does not admit a probabilistic interpretation in contrast with fractional diffusion based on fractional integrals. While the fractional initial value problem is well defined and the solution finite at all times, its values for t → 0 are divergent.



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Fractional Calculus Fractional Time Irreversibility Theory of Time

Fractional Evolution Equations and Irreversibility

R. Hilfer

in: Traffic and Granular Flow’99
edited by: D. Helbing and H. Herrmann and M. Schreckenberg and D. Wolf
Springer, Berlin, 215 (2000)
10.1007/978-3-642-59751-0
ISBN: 978-3-642-64109-1

submitted on
Monday, September 27, 1999

The paper reviews a general theory predicting the general importance of fractional evolution equations. Fractional time evolutions are shown to arise from a microscopic time evolution in a certain long time scaling limit. Fractional time evolutions are generally irreversible. The infinitesimal generators of fractional time evolutions are fractional time derivatives. Evolution equations containing fractional time derivatives are proposed for physical, economical and traffic applications. Regular non-fractional time evolutions emerge as special cases from the results. Also for these regular time evolutions it is found that macroscopic irreversibility arises in the scaling limit.



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Equilibrium Fractional Calculus Statistical Physics

Fractional Calculus and Regular Variation in Thermodynamics

R. Hilfer

in: Applications of Fractional Calculus in Physics
edited by: R. Hilfer
World Scientific, Singapore, 429-463 (2000)
https://doi.org/10.1142/3779
ISBN: 978-981-02-3457-7

submitted on
Wednesday, May 5, 1999



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Fractional Calculus Fractional Time Theory of Time

Fractional Time Evolution

R. Hilfer

in: Applications of Fractional Calculus in Physics
edited by: R. Hilfer
World Scientific, Singapore, 87-130 (2000)
https://doi.org/10.1142/3779
ISBN: 978-981-02-3457-7

submitted on
Wednesday, May 5, 1999



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Fractional Calculus

Applications of Fractional Calculus in Physics

R. Hilfer

World Scientific Publ. Co., Singapore, 2000
https://doi.org/10.1142/3779
ISBN: 978-981-02-3457-7

submitted on
Wednesday, May 5, 1999

Although fractional calculus is a natural generalization of calculus, and although its mathematical history is equally long, it has, until recently, played a negligible role in physics. One reason could be that, until recently, the basic facts were not readily accessible even in the mathematical literature. This book intends to increase the accessibility of fractional calculus by combining an introduction to the mathematics with a review of selected recent applications in physics.



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diffusion Fractional Calculus Random Walks

On Fractional Diffusion and its Relation with Continuous Time Random Walks

R. Hilfer

in: Anomalous Diffusion: From Basis to Applications
edited by: R. Kutner, A. Pekalski and K. Sznajd-Weron
Lecture Notes in Physics, vol. 519,Springer, Berlin, 77 (1999)
10.1007/BFb0106828
978-3-662-14242-4

submitted on
Friday, May 22, 1998

Time evolutions whose infinitesimal generator is a fractional time derivative arise generally in the long time limit. Such fractional time evolutions are considered here for random walks. An exact relationship is established between the fractional master equation and a separable continuous time random walk of the Montroll-Weiss type. The waiting time density can be expressed using a generalized Mittag-Leffier function. The first moment of the waiting density does not exist.



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Fractional Calculus review article

Fractional Derivatives in Static and Dynamic Scaling

R. Hilfer

in: Scale Invariance and Beyond
edited by: B. Dubrulle and F. Graner and D. Sornette
Springer, Berlin, 53 (1997)
10.1007/978-3-662-09799-1
978-3-540-64000-4

submitted on
Tuesday, March 11, 1997

The paper is a brief review of recent applications of fractional calculus in physics with emphasis on static and dynamic scaling.



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Fractional Calculus Fractional Time Random Walks Stochastic Processes Theory of Time

Fractional Master Equations and Fractal Time Random Walks

R. Hilfer, L. Anton

Physical Review E, Rapid Communication 51, R848 (1995)
https://doi.org/10.1103/PhysRevE.51.R848

submitted on
Friday, October 28, 1994

Fractional master equations containing fractional time derivatives of order less than one are introduced on the basis of a recent classification of time generators in ergodic theory. It is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks. The corresponding waiting time density is obtained exactly in terms of the generalized Mittag-Leffler function. This waiting time distribution is singular both in the long time as well as in the short time limit.



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