Category: Fractional Calculus

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Fractional Calculus Functional analysis

Fractional Calculus for Distributions

R. Hilfer, T. Kleiner

Fractional Calculus and Applied Analysis , (2024)
https://doi.org/10.1007/s13540-024-00306-z

submitted on
Friday, March 29, 2024

Fractional derivatives and integrals for measures and distributions are reviewed. The focus is on domains and co-domains for translation invariant fractional operators. Fractional derivatives and integrals interpreted as D’-convolution operators with power law kernels are found to have the largest domains of definition. As a result, extending domains from functions to distributions via convolution operators contributes to far reaching unifications of many previously existing definitions of fractional integrals and derivatives. Weyl fractional operators are thereby extended to distributions using the method of adjoints. In addition, discretized fractional calculus and fractional calculus of periodic distributions can both be formulated and understood in terms of D’-convolution.



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Fractional Calculus Functional analysis Mathematics

Convolution on Distribution Spaces Characterized by Regularization

T. Kleiner, R. Hilfer

Mathematische Nachrichten 296, 1938-1963 (2023)
https://doi.org/10.1002/mana.202100330

submitted on
Friday, October 15, 2021

Locally convex convolutor spaces are studied which consist of those distributions that define a continuous convolution operator mapping from the space of test functions into a given locally convex lattice of measures. The convolutor spaces are endowed with the topology of uniform convergence on bounded sets. Their locally convex structure is characterized via regularization and function-valued seminorms under mild structural assumptions on the space of measures. Many recent generalizations of classical distribution spaces turn out to be special cases of the general convolutor spaces introduced here. Recent topological characterizations of convolutor spaces via regularization are extended and improved. A valuable property of the convolutor spaces in applications is that convolution of distributions inherits continuity properties from those of bilinear convolution mappings between the locally convex latti\-ces of measures.



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Fractional Calculus Functional analysis Mathematics

Sequential generalized Riemann–Liouville derivatives based on distributional convolution

T. Kleiner, R. Hilfer

Fractional Calculus and Applied Analysis 25, 267-298 (2022)
https://doi.org/10.1007/s13540-021-00012-0

submitted on
Friday, October 15, 2021

Sequential generalized fractional Riemann-Liouville derivatives are introduced as composites of distributional derivatives on the right half axis and partially defined operators, called Dirac-function removers, that remove the component of singleton support at the origin of distributions that are of order zero on a neighborhood of the origin. The concept of Dirac-function removers allows to formulate generalized initial value problems with less restrictions on the orders and types than previous approaches to sequential fractional derivatives. The well-posedness of these initial value problems and the structure of their solutions are studied.



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Fractional Calculus Functional analysis Mathematics

On extremal domains and codomains for convolution of distributions and fractional calculus

T. Kleiner, R. Hilfer

Monatshefte für Mathematik 198, 122-152 (2022)
https://doi.org/10.1007/s00605-021-01646-1

submitted on
Wednesday, December 30, 2020

It is proved that the class of c-closed distribution spaces contains extremal domains and codomains to make convolution of distributions a well-defined bilinear mapping. The distribution spaces are systematically endowed with topologies and bornologies that make convolution hypocontinuous whenever defined. Largest modules and smallest algebras for convolution semigroups are constructed along the same lines. The fact that extremal domains and codomains for convolution exist within this class of spaces is fundamentally related to quantale theory. The quantale theoretic residual formed from two c-closed spaces is characterized as the largest c-closed subspace of the corresponding space of convolutors. The theory is applied to obtain maximal distributional domains for fractional integrals and derivatives, for fractional Laplacians, Riesz poten- tials and for the Hilbert transform. Further, maximal joint domains for families of these operators are obtained such that their composition laws are preserved.



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Fractional Calculus Functional analysis Glasses Mathematical Physics Mathematics Special Functions

Fractional glassy relaxation and convolution modules of distributions

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 Functional analysis Uncategorized

Maximal Domains for Fractional Derivatives and Integrals

R. Hilfer, T. Kleiner

Mathematics 8, 1107 (2020)
https://doi.org/10.3390/math8071107

submitted on
Wednesday, March 11, 2020

The purpose of this short communication is to announce the existence of fractional calculi on precisely specified domains of distributions. The calculi satisfy desiderata proposed above in Mathematics 7, 149 (2019). For the desiderata (a)–(c) the examples are optimal in the sense of having maximal domains with respect to convolvability of distributions. The examples suggest to modify desideratum (f) in the original list.



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Fractional Calculus Mathematics

Desiderata for Fractional Derivatives and Integrals

R. Hilfer, Yu. Luchko

Mathematics 7, 149 (2019)
https://doi.org/10.3390/math7020149

submitted on
Friday, January 11, 2019

The purpose of this brief article is to initiate discussions in this special issue by proposing desiderata for calling an operator a fractional derivative or a fractional integral. Our desiderata are neither axioms nor do they define fractional derivatives or integrals uniquely. Instead they intend to stimulate the field by providing guidelines based on a small number of time honoured and well established criteria.



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Fractional Calculus Functional analysis Mathematical Physics Mathematics Stochastic Processes

Mathematical and physical interpretations of fractional derivatives and integrals

R. Hilfer

in: Handbook of Fractional Calculus with Applications: Basic Theory, Vol. 1
edited by: A. Kochubei and Y. Luchko
Walter de Gruyter GmbH, Berlin, 47-86 (2019)
https://doi.org/10.1515/9783110571622
ISBN: 9783110571622

submitted on
Saturday, June 2, 2018

Brief descriptions of various mathematical and physical interpretations of fractional derivatives and integrals have been collected into this chapter as points of reference and departure for deeper studies. “Mathematical interpretation” in the title means a brief description of the basic mathematical idea underlying a precise definition. “Physical interpretation” means a brief description of the physical theory underlying an identification of the fractional order with a known physical quantity. Numerous interpretations had to be left out due to page limitations. Only a crude, rough and ready description is given for each interpretation. For precise theorems and proofs an extensive list of references can serve as a starting point.



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Ergodic Theory Fractional Calculus Fractional Time Irreversibility Mathematics Theory of Time

Time Automorphisms on C*-Algebras

R. Hilfer

Mathematics 3, 623-643 (2015)
https://doi.org/10.3390/math3030626

submitted on
Tuesday, March 24, 2015

Applications of fractional time derivatives in physics and engineering require the existence of nontranslational time automorphisms on the appropriate algebra of observables. The existence of time automorphisms on commutative and noncommutative C∗-algebras for interacting many-body systems is investigated in this article. A mathematical framework is given to discuss local stationarity in time and the global existence of fractional and nonfractional time automorphisms. The results challenge the concept of time flow as a translation along the orbits and support a more general concept of time flow as a convolution along orbits. Implications for the distinction of reversible and irreversible dynamics are discussed. The generalized concept of time as a convolution reduces to the traditional concept of time translation in a special limit.



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diffusion Fractional Calculus

Experimental Implications of Bochner-Levy-Riesz Diffusion

R. Hilfer

Fractional Calculus and Applied Analysis 18, 333-341 (2015)
https://doi.org/10.1515/fca-2015-0022

submitted on
Monday, August 18, 2014

Fractional Bochner-Levy-Riesz diffusion arises from ordinary diffusion by replacing the Laplacean with a noninteger power of itself. Bochner- Levy-Riesz diffusion as a mathematical model leads to nonlocal boundary value problems. As a model for physical transport processes it seems to predict phenomena that have yet to be observed in experiment.



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dielectric relaxation Fractional Calculus Fractional Time Theory of Time

Applications and Implications of Fractional Dynamics for Dielectric Relaxation

R. Hilfer

in: Recent Advances in Broadband Dielectric Spectroscopy
edited by: Y. Kalmykov
Springer, Berlin, 123 (2012)
10.1007/978-94-007-5012-8
978-94-007-5011-1

submitted on
Friday, September 23, 2011

This article summarizes briefly the presentation given by the author at the NATO Advanced Research Workshop on “Broadband Dielectric Spectroscopy and its Advanced Technological Applications”, held in Perpignan, France, in September 2011. The purpose of the invited presentation at the workshop was to review and summarize the basic theory of fractional dynamics (Hilfer, Phys Rev E 48:2466, 1993; Hilfer and Anton, Phys Rev E Rapid Commun 51:R848, 1995; Hilfer, Fractals 3(1):211, 1995; Hilfer, Chaos Solitons Fractals 5:1475, 1995; Hilfer, Fractals 3:549, 1995; Hilfer, Physica A 221:89, 1995; Hilfer, On fractional diffusion and its relation with continuous time random walks. In: Pekalski et al. (eds) Anomalous diffusion: from basis to applications. Springer, Berlin, p 77, 1999; Hilfer, Fractional evolution equations and irreversibility. In: Helbing et al. (eds) Traffic and granular flow’99. Springer, Berlin, p 215, 2000; Hilfer, Fractional time evolution. In: Hilfer (ed) Applications of fractional calculus in physics. World Scientific, Singapore, p 87, 2000; Hilfer, Remarks on fractional time. In: Castell and Ischebeck (eds) Time, quantum and information. Springer, Berlin, p 235, 2003; Hilfer, Physica A 329:35, 2003; Hilfer, Threefold introduction to fractional derivatives. In: Klages et al. (eds) Anomalous transport: foundations and applications. Wiley-VCH, Weinheim, pp 17– 74, 2008; Hilfer, Foundations of fractional dynamics: a short account. In: Klafter et al. (eds) Fractional dynamics: recent advances. World Scientific, Singapore,207, 2011) and demonstrate its relevance and application to broadband dielectric spectroscopy (Hilfer, J Phys Condens Matter 14:2297, 2002; Hilfer, Chem Phys 284:399, 2002; Hilfer, Fractals 11:251, 2003; Hilfer et al., Fractional Calc Appl Anal 12:299, 2009). It was argued, that broadband dielectric spectroscopy might be useful to test effective field theories based on fractional dynamics.



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Fractional Calculus Fractional Time

Foundations of Fractional Dynamics: A Short Account

R. Hilfer

in: Fractional Dynamics: Recent Advances
edited by: J. Klafter and S. Lim and R. Metzler
World Scientific, Singapore, 207 (2011)
https://doi.org/10.1142/8087
ISBN: 978-981-4340-58-8

submitted on
Tuesday, March 22, 2011

Applications of fractional dynamics have received a steadily increasing amount of attention during the past decade. Its foundations have found less interest. This chapter briefly reviews the physical foundations of fractional dynamics.



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Fractional Calculus Mathematics Special Functions

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

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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Fractional Calculus Mathematics

Operational Method for the Solution of Fractional Differential Equations with Generalized Riemann-Liouville Fractional Derivatives

R. Hilfer, Y. Luchko, Z. Tomovski

Fractional Calculus and Applied Analysis 12, 299 (2009)

submitted on
Wednesday, December 17, 2008

The operational calculus is an algorithmic approach for the solution of initial-value problems for differential, integral, and integro-differential equations. In this paper, an operational calculus of the Mikusiński type for a generalized Riemann-Liouville fractional differential operator with types introduced by one of the authors is developed. The traditional Riemann-Liouville and Liouville-Caputo fractional derivatives correspond to particu lar types of the general one-parameter family of fractional derivatives with the same order. The operational calculus constructed in this paper is used to solve the corresponding initial value problem for the general n-term linear equation with these generalized fractional derivatives of arbitrary orders and types with constant coefficients. Special cases of the obtained solutions are presented.



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Fractional Calculus Mathematics review article

Threefold Introduction to Fractional Derivatives

R. Hilfer

in: Anomalous Transport: Foundations and Applications
edited by: R. Klages and G. Radons and I. Sokolov
Wiley-VCH, Weinheim, 17-74 (2008)
ISBN: 978-3-527-40722-4

submitted on
Wednesday, January 2, 2008

Historical, mathematical and physical introduction to fractrional derivatives.



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diffusion Fractional Calculus Porous Media Two-Phase Flow

Modeling Infiltration by Means of a Nonlinear Fractional Diffusion Model

E. Gerolymatou, I. Vardoulakis, R. Hilfer

Journal of Physics D: Applied Physics 39, 4104 (2006)

submitted on
Thursday, May 18, 2006

The classical Richards equation describes infiltration into porous soil as a nonlinear diffusion process. Recent experiments have suggested that this process exhibits anomalous scaling behaviour. These observations suggest generalizing the classical Richards equation by introducing fractional time derivatives. The resulting fractional Richards equation with appropriate initial and boundary values is solved numerically in this paper. The numerical code is tested against analytical solutions in the linear case. Saturation profiles are calculated for the fully nonlinear fractional Richards equation. Isochrones and isosaturation curves are given. The cumulative moisture intake is found as a function of the order of the fractional derivative. These results are compared against experiment.



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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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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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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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