Category: Special Functions

Fractional glassy relaxation and convolution modules of distributions

Post author By R Hilfer
Post date September 30, 2020

T. Kleiner, R. Hilfer

Analysis and Mathematical Physics 11, 130 (2021)
https://doi.org/10.1007/s13324-021-00504-5

submitted on
Wednesday, September 30, 2020

Solving fractional relaxation equations requires precisely characterized domains of definition for applications of fractional differential and integral operators. Determining these domains has been a longstanding problem. Applications in physics and engineering typically require extension from domains of functions to domains of distributions. In this work convolution modules are constructed for given sets of distributions that generate distributional convolution algebras. Convolutional inversion of fractional equations leads to a broad class of multinomial Mittag-Leffler type distributions. A comprehensive asymptotic analysis of these is carried out. Combined with the module construction the asymptotic analysis yields domains of distributions, that guarantee existence and uniqueness of solutions to fractional differential equations. The mathematical results are applied to anomalous dielectric relaxation in glasses. An analytic expression for the frequency dependent dielectric susceptibility is applied to broadband spectra of glycerol. This application reveals a temperature independent and universal dynamical scaling exponent.

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Fractional Calculus Mathematics Special Functions

Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions

Post author By R Hilfer
Post date November 9, 2009

Z. Tomovski, R. Hilfer, H.M. Srivastava

Integral Transforms and Special Functions 21, 797 (2010)
https://doi.org/10.1080/10652461003675737

submitted on
Monday, November 9, 2009

In this paper, we study a certain family of generalized Riemann–Liouville fractional derivative operators α,β Da± of order α and type β, which were introduced and investigated in several earlier works [R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000; R. Hilfer, Fractional time evolution, in Applications of Fractional Calculus in Physics, R. Hilfer, ed., World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 2000, pp. 87–130; R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, J. Chem. Phys. 284 (2002), pp. 399–408; R. Hilfer, Threefold introduction to fractional derivatives, in Anomalous Transport: Foundations and Applications, R. Klages, G. Radons, and I.M. Sokolov, eds., Wiley-VCH Verlag, Weinheim, 2008, pp. 17–73; R. Hilfer and L. Anton, Fractional master equations and fractal time random walks, Phys. Rev. E 51 (1995), pp. R848–R851; R. Hilfer,Y. Luchko, and Ž. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal. 12 (2009), pp. 299–318; F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey, Fract. Calc. Appl. Anal. 10 (2007), pp. 269–308; T. Sandev and Ž. Tomovski, General time fractional wave equation for a vibrating string, J. Phys. A Math. Theor. 43 (2010), 055204; H.M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (2009), pp. 198–210]. In particular, we derive various compositional properties, which are associated with Mittag–Leffler functions and Hardy-type inequalities for the generalized fractional α,β derivative operator Da± . Furthermore, by using the Laplace transformation methods, we provide solutions of many different classes of fractional differential equations with constant and variable coefficients and some general Volterra-type differintegral equations in the space of Lebesgue integrable functions. Particular cases of these general solutions and a brief discussion about some recently investigated fractional kinetic equations are also given.

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Mathematics Special Functions

Numerical Algorithm for Calculating the Generalized Mittag-Leffler Function

Post author By R Hilfer
Post date August 16, 2008

H.J. Seybold, R. Hilfer

SIAM Journal on Numerical Analysis 47, 69 (2008)
https://doi.org/10.1137/070700280

submitted on
Saturday, August 16, 2008

A numerical algorithm for calculating the generalized Mittag-Leffler function for arbitrary complex argument and real parameters is presented. The algorithm uses the Taylor series, the exponentially improved asymptotic series, and integral representations to obtain optimal stability and accuracy of the algorithm. Special care is applied to the limits of validity of the different schemes to avoid instabilities in the algorithm.

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Mathematics Special Functions

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

Post author By R Hilfer
Post date March 21, 2005

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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Mathematics Special Functions

Numerical Results for the Generalized Mittag-Leffler Function

Post author By R Hilfer
Post date May 4, 2003

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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dielectric relaxation Glasses Special Functions

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

Post author By R Hilfer
Post date June 28, 2001

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

Post author By R Hilfer
Post date April 13, 2001

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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