A practical guide to Prabhakar fractional calculus

Abstract The Mittag–Leffler function is universally acclaimed as the Queen function of fractional calculus. The aim of this work is to survey the key results and applications emerging from the three-parameter generalization of this function, known as the Prabhakar function. Specifically, after reviewing key historical events that led to the discovery and modern development of this peculiar function, we discuss how the latter allows one to introduce an enhanced scheme for fractional calculus. Then, we summarize the progress in the application of this new general framework to physics and renewal processes. We also provide a collection of results on the numerical evaluation of the Prabhakar function.

[1]  Francesco Mainardi,et al.  On the propagation of transient waves in a viscoelastic Bessel medium , 2016 .

[2]  F. Polito.  Studies on generalized Yule models , 2018, Modern Stochastics: Theory and Applications.

[3]  Enrico Scalas,et al.  A renewal process of Mittag-Leffler type , 2004 .

[4]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[5]  F. Mainardi,et al.  Creep, relaxation and viscosity properties for basic fractional models in rheology , 2011, 1110.3400.

[6]  Aleksander Stanislavsky,et al.  The stochastic nature of complexity evolution in the fractional systems , 2007, 1111.3239.

[7]  M. M. Djrbashian,et al.  Harmonic analysis and boundary value problems in the complex domain , 1993 .

[8]  A. Chechkin,et al.  Generalized diffusion-wave equation with memory kernel , 2018, Journal of Physics A: Mathematical and Theoretical.

[9]  A. M. Mathai,et al.  On the H -Function With Applications , 2010 .

[10]  Francesco Mainardi,et al.  On infinite series concerning zeros of Bessel functions of the first kind , 2016, 1601.00563.

[11]  A. P. Riascos,et al.  Continuous time random walk and diffusion with generalized fractional Poisson process , 2019, Physica A: Statistical Mechanics and its Applications.

[12]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[13]  Francesco Mainardi,et al.  A class of linear viscoelastic models based on Bessel functions , 2016, 1602.04664.

[14]  G. H. Hardy,et al.  On the Theory of Linear Integral Equations , 1935, Mathematical Proceedings of the Cambridge Philosophical Society.

[15]  Photocatalytic degradation as Davidson–Cole relaxation in time domain , 2019, Journal of Advanced Dielectrics.

[16]  Francesco Mainardi,et al.  A dynamic viscoelastic analogy for fluid-filled elastic tubes , 2015, 1505.06694.

[17]  S. Havriliak,et al.  On the equivalence of dielectric and mechanical dispersions in some polymers; e.g. poly(n-octyl methacrylate)☆ , 1969 .

[18]  Francesco Mainardi,et al.  On Mittag-Leffler-type functions in fractional evolution processes , 2000 .

[19]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[20]  I. Podlubny.  Fractional differential equations , 1998 .

[21]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[22]  Federico Polito,et al.  Fractional Diffusion-Telegraph Equations and their Associated Stochastic Solutions , 2013, 1307.1696.

[23]  R. Cole,et al.  Dielectric Relaxation in Glycerol, Propylene Glycol, and n‐Propanol , 1951 .

[24]  Andrea Giusti.  General fractional calculus and Prabhakar's theory , 2020, Commun. Nonlinear Sci. Numer. Simul..

[25]  N. Balakrishnan,et al.  A class of weighted Poisson processes , 2008 .

[26]  R. Gorenflo,et al.  Time-fractional derivatives in relaxation processes: a tutorial survey , 2008, 0801.4914.

[27]  A. I. Saichev,et al.  Fractional Poisson Law , 2000 .

[28]  K. Cole.  ELECTRIC CONDUCTANCE OF BIOLOGICAL SYSTEMS , 1933 .

[29]  Zhili Lin.  On the FDTD Formulations for Biological Tissues With Cole–Cole Dispersion , 2010, IEEE Microwave and Wireless Components Letters.

[30]  Karina Weron,et al.  HAVRILIAK-NEGAMI RESPONSE IN THE FRAMEWORK OF THE CONTINUOUS-TIME RANDOM WALK ∗ , 2005 .

[31]  Karina Weron,et al.  Numerical scheme for calculating of the fractional two-power relaxation laws in time-domain of measurements , 2012, Comput. Phys. Commun..

[32]  Enrico Scalas,et al.  Full characterization of the fractional Poisson process , 2011, 1104.4234.

[33]  A. P. Riascos,et al.  Generalized fractional Poisson process and related stochastic dynamics , 2019, Fractional Calculus and Applied Analysis.

[34]  Andrea Giusti,et al.  A comment on some new definitions of fractional derivative , 2017, Nonlinear Dynamics.

[35]  Virginia Kiryakova,et al.  Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus , 2000 .

[36]  E. Montroll,et al.  Random Walks on Lattices. II , 1965 .

[37]  Trifce Sandev,et al.  Fractional Equations and Models , 2019, Developments in Mathematics.

[38]  C. Lubich.  Convolution Quadrature Revisited , 2004 .

[39]  Virginia Kiryakova,et al.  MULTIINDEX MITTAG-LEFFLER FUNCTIONS, RELATED GELFOND-LEONTIEV OPERATORS AND LAPLACE TYPE INTEGRAL TRANSFORMS ⁄ , 1999 .

[40]  E. Hille,et al.  On the Theory of Linear Integral Equations. II , 1930 .

[41]  N. Laskin.  Fractional Poisson process , 2003 .

[42]  C. Lubich.  Convolution quadrature and discretized operational calculus. II , 1988 .

[43]  E. M. Wright,et al.  The asymptotic expansion of integral functions defined by Taylor series , 1940, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[44]  B. Braaksma,et al.  Asymptotic expansions and analytic continuations for a class of Barnes-integrals , 1964 .

[45]  H. Chamati,et al.  Generalized Mittag–Leffler functions in the theory of finite-size scaling for systems with strong anisotropy and/or long-range interaction , 2006 .

[46]  R. K. Saxena,et al.  Generalized mittag-leffler function and generalized fractional calculus operators , 2004 .

[47]  Leonard A. Dissado,et al.  Debye and non-Debye relaxation , 1985 .

[48]  Boris Gnedenko,et al.  Introduction to queueing theory , 1968 .

[49]  Rudolf Hilfer,et al.  Experimental evidence for fractional time evolution in glass forming materials , 2002 .

[50]  Trifce Sandev,et al.  Models for characterizing the transition among anomalous diffusions with different diffusion exponents , 2018, Journal of Physics A: Mathematical and Theoretical.

[51]  Gorjan Alagic,et al.  #p , 2019, Quantum information & computation.

[52]  J. Paneva-Konovska.  On the multi-index (3m-parametric) Mittag-Leffler functions, fractional calculus relations and series convergence , 2013 .

[53]  J. Trujillo,et al.  Differential equations of fractional order:methods results and problem —I , 2001 .

[54]  Alexander M. Krägeloh.  Two families of functions related to the fractional powers of generators of strongly continuous contraction semigroups , 2003 .

[55]  Trifce Sandev,et al.  Finite-velocity diffusion on a comb , 2018, EPL (Europhysics Letters).

[56]  A. M. Mathai,et al.  The H-Function: Theory and Applications , 2009 .

[57]  K. Diethelm.  Mittag-Leffler Functions , 2010 .

[58]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[60]  J. Klafter,et al.  First Steps in Random Walks: From Tools to Applications , 2011 .

[61]  C. T. White,et al.  On the origin of the universal dielectric response in condensed matter , 1979, Nature.

[62]  Piezo-catalytic degradation of Havriliak–Negami type , 2019, Journal of Advanced Dielectrics.

[63]  A. Wiman.  Über den Fundamentalsatz in der Teorie der FunktionenEa(x) , 1905 .

[64]  T. R. Prabhakar.  A SINGULAR INTEGRAL EQUATION WITH A GENERALIZED MITTAG LEFFLER FUNCTION IN THE KERNEL , 1971 .

[65]  Arak M. Mathai,et al.  Mittag-Leffler Functions and Their Applications , 2009, J. Appl. Math..

[66]  Trifce Sandev,et al.  Generalized Langevin Equation and the Prabhakar Derivative , 2017 .

[67]  Harry Pollard,et al.  The completely monotonic character of the Mittag-Leffler function $E_a \left( { - x} \right)$ , 1948 .

[68]  C. Macci,et al.  Large deviations for fractional Poisson processes , 2012, 1204.1446.

[69]  B. Gross.  Ladder structures for representation of viscoelastic systems. II , 1956 .

[70]  M. Gurtin,et al.  On the linear theory of viscoelasticity , 1962 .

[71]  Dispersion and Absorption in Dielectrics 1 , 2022 .

[72]  Guido Maione,et al.  Fractional Prabhakar Derivative and Applications in Anomalous Dielectrics: A Numerical Approach , 2017 .

[73]  S. Havriliak,et al.  A complex plane analysis of α‐dispersions in some polymer systems , 2007 .

[74]  B. Gross,et al.  On Creep and Relaxation , 1947 .

[75]  S. Havriliak,et al.  Results from an unbiased analysis of nearly 1000 sets of relaxation data , 1994 .

[76]  A. Hanyga,et al.  On a Mathematical Framework for the Constitutive Equations of Anisotropic Dielectric Relaxation , 2008 .

[77]  Hilfer,et al.  Fractional master equations and fractal time random walks. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[78]  Federico Polito,et al.  Hilfer-Prabhakar derivatives and some applications , 2014, Appl. Math. Comput..

[79]  Francesco Mainardi,et al.  George William Scott Blair – the pioneer of fractional calculus in rheology , 2014, 1404.3295.

[80]  Hari M. Srivastava,et al.  Laplace type integral expressions for a certain three-parameter family of generalized Mittag-Leffler functions with applications involving complete monotonicity , 2014, J. Frankl. Inst..

[81]  A. Stanislavsky,et al.  Transient anomalous diffusion with Prabhakar-type memory. , 2018, The Journal of chemical physics.

[82]  Anatoly A. Kilbas,et al.  Solution of Volterra Integro-Differential Equations with Generalized Mittag-Leffler Function in the Kernels , 2002 .

[83]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[84]  Matthew F. Causley,et al.  Incorporating the Havriliak-Negami dielectric model in the FD-TD method , 2011, J. Comput. Phys..

[85]  Trifce Sandev,et al.  Asymptotic behavior of a harmonic oscillator driven by a generalized Mittag-Leffler noise , 2010 .

[86]  J. Paneva-Konovska.  Convergence of series in three parametric Mittag-Leffler functions , 2014 .

[87]  Francesco Mainardi,et al.  On transient waves in linear viscoelasticity , 2017 .

[88]  C. Lubich.  Convolution quadrature and discretized operational calculus. I , 1988 .

[89]  Andrea Giusti,et al.  Prabhakar-like fractional viscoelasticity , 2017, Commun. Nonlinear Sci. Numer. Simul..

[90]  E. Capelas de Oliveira,et al.  Solution of the fractional Langevin equation and the Mittag–Leffler functions , 2009 .

[91]  V. Kiryakova,et al.  The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus , 2010, Comput. Math. Appl..

[92]  P. Mieghem.  The Mittag-Leffler function , 2020, 2005.13330.

[93]  Federico Polito,et al.  Some properties of Prabhakar-type fractional calculus operators , 2015, 1508.03224.

[94]  Z. Tomovski,et al.  Probability distribution built by Prabhakar function. Related Turán and Laguerre inequalities , 2016, 1605.01981.

[95]  Trifce Sandev,et al.  Langevin equation for a free particle driven by power law type of noises , 2014 .

[96]  E. M. Wright.  The Asymptotic Expansion of the Generalised Hypergeometric Function , 1952 .

[97]  A. Pipkin,et al.  Lectures on Viscoelasticity Theory , 1972 .

[98]  R. Cole,et al.  Dielectric Relaxation in Glycerine , 1950 .

[99]  E. C. Oliveira,et al.  On some fractional Green’s functions , 2009 .

[100]  Roberto Garrappa,et al.  Numerical Evaluation of Two and Three Parameter Mittag-Leffler Functions , 2015, SIAM J. Numer. Anal..

[101]  Roberto Garrappa,et al.  Evaluation of generalized Mittag–Leffler functions on the real line , 2013, Adv. Comput. Math..

[102]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[103]  Holger Kantz,et al.  Distributed-order diffusion equations and multifractality: Models and solutions. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[104]  H. Konno,et al.  Stochastic modeling for neural spiking events based on fractional superstatistical Poisson process , 2018 .

[105]  E. Montroll.  Random walks on lattices , 1969 .

[106]  V. V. Novikov,et al.  Anomalous relaxation in dielectrics. Equations with fractional derivatives , 2005 .

[107]  G. Dattoli,et al.  The Havriliak–Negami relaxation and its relatives: the response, relaxation and probability density functions , 2016, 1611.06433.

[108]  R. Gorenflo,et al.  Mittag-Leffler Functions, Related Topics and Applications , 2014, Springer Monographs in Mathematics.

[109]  E. M. Wright,et al.  The Asymptotic Expansion of the Generalized Hypergeometric Function , 1935 .

[110]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[111]  B. Gross,et al.  ON CREEP AND RELAXATION. III , 1948 .

[112]  Roberto Garrappa,et al.  The Prabhakar or three parameter Mittag-Leffler function: Theory and application , 2017, Commun. Nonlinear Sci. Numer. Simul..

[113]  L. Beghin,et al.  Fractional Poisson processes and related planar random motions , 2009 .

[114]  Roberto Garrappa,et al.  Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models , 2016, Commun. Nonlinear Sci. Numer. Simul..

[115]  Roberto Garrappa,et al.  Computing the Matrix Mittag-Leffler Function with Applications to Fractional Calculus , 2018, Journal of Scientific Computing.

[116]  R. Hilfer,et al.  H-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[117]  A. Jonscher.  Dielectric relaxation in solids , 1983 .

[118]  J. Trujillo,et al.  Differential Equations of Fractional Order: Methods, Results and Problems. II , 2002 .

[119]  M. T. Cicero.  FRACTIONAL CALCULUS AND WAVES IN LINEAR VISCOELASTICITY , 2012 .

[120]  F. Mainardi,et al.  Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics , 2011, 1106.1761.

[121]  D. Fox.  Creep , 2019, Deformation and Evolution of Life in Crystalline Materials.

[122]  R. B. Paris,et al.  Exponentially small expansions in the asymptotics of the Wright function , 2010, J. Comput. Appl. Math..

[123]  S. Havriliak,et al.  A complex plane representation of dielectric and mechanical relaxation processes in some polymers , 1967 .

[124]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[125]  Dazhi Zhao,et al.  Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel , 2019, Zeitschrift für angewandte Mathematik und Physik.

[126]  F. Mainardi.  Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models , 2010 .

[127]  K. Cole,et al.  Dispersion and Absorption in Dielectrics I. Alternating Current Characteristics , 1941 .

[128]  Vasundara V. Varadan,et al.  Variation of Cole-Cole model parameters with the complex permittivity of biological tissues , 2009, 2009 IEEE MTT-S International Microwave Symposium Digest.

[129]  Bernhard Gross.  Electrical analogs for viscoelastic systems , 1956 .

[130]  A. Chechkin,et al.  Generalised Diffusion and Wave Equations: Recent Advances , 2019, 1903.01166.

[131]  M. Caputo,et al.  A new dissipation model based on memory mechanism , 1971 .

[132]  H. M. Srivastava,et al.  Fractional and operational calculus with generalized fractional derivative operators and Mittag–Leffler type functions , 2010 .

[133]  Hari M. Srivastava,et al.  Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel , 2009, Appl. Math. Comput..

[134]  장윤희,et al.  Y. , 2003, Industrial and Labor Relations Terms.

[135]  S. Reich,et al.  Dielectric relaxation , 1977, Nature.

[136]  Richard B. Paris.  Asymptotics of the special functions of fractional calculus , 2019, Basic Theory.

[137]  Ram K. Saxena,et al.  Exact solutions of triple-order time-fractional differential equations for anomalous relaxation and diffusion I: The accelerating case , 2011 .

[138]  A. M. Mathai,et al.  Reaction-Diffusion Systems and Nonlinear Waves , 2006 .

[139]  K. Górska,et al.  Composition law for the Cole-Cole relaxation and ensuing evolution equations , 2018, Physics Letters A.

[140]  Luisa Beghin,et al.  Poisson-type processes governed by fractional and higher-order recursive differential equations , 2009, 0910.5855.

[141]  D. A. Dunnett.  Classical Electrodynamics , 2020, Nature.

[142]  Anatoly N. Kochubei,et al.  General Fractional Calculus, Evolution Equations, and Renewal Processes , 2011, 1105.1239.

[143]  Francesco Mainardi,et al.  Linear models of dissipation in anelastic solids , 1971 .

[144]  A. Horzela,et al.  A note on the article “Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel” [Z. Angew. Math. Phys. (2019) 70: 42] , 2019, Zeitschrift für angewandte Mathematik und Physik.

[145]  G. Fitzgerald,et al.  'I. , 2019, Australian journal of primary health.

[146]  Federico Polito,et al.  Renewal processes based on generalized Mittag-Leffler waiting times , 2013, Commun. Nonlinear Sci. Numer. Simul..

[147]  Overconvergence of series in generalized mittag-leffler functions , 2017 .

[148]  R. Nigmatullin,et al.  The justified data-curve fitting approach: recognition of the new type of kinetic equations in fractional derivatives from analysis of raw dielectric data , 2003 .

[149]  A. K. Jonscher,et al.  The ‘universal’ dielectric response , 1977, Nature.

[150]  A. Kochubei.  General fractional calculus , 2019, Basic Theory.

[151]  K. Cole,et al.  Dispersion and Absorption in Dielectrics II. Direct Current Characteristics , 1942 .

[152]  Francesco Mainardi,et al.  On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics , 2015, J. Comput. Phys..

[153]  M. Meerschaert,et al.  The Fractional Poisson Process and the Inverse Stable Subordinator , 2010, 1007.5051.

[154]  J. H. Barrett,et al.  Differential Equations of Non-Integer Order , 1954, Canadian Journal of Mathematics.

[155]  A. Chechkin,et al.  From continuous time random walks to the generalized diffusion equation , 2018 .

[156]  F. Mainardi,et al.  Models of dielectric relaxation based on completely monotone functions , 2016, 1611.04028.

[157]  Silvia Vitali,et al.  Storage and Dissipation of Energy in Prabhakar Viscoelasticity , 2017, 1712.09419.

[158]  Lloyd N. Trefethen,et al.  Parabolic and hyperbolic contours for computing the Bromwich integral , 2007, Math. Comput..

[159]  V. Uchaikin.  Relaxation Processes and Fractional Differential Equations , 2003 .

[160]  H. Kantz,et al.  Diffusion and Fokker-Planck-Smoluchowski Equations with Generalized Memory Kernel , 2015 .

[161]  F. Mainardi,et al.  for anomalous relaxation in dielectrics , 2011 .

[162]  R. Gorenflo,et al.  Multi-index Mittag-Leffler Functions , 2014 .

[163]  Guido Maione,et al.  A novel FDTD formulation based on fractional derivatives for dispersive Havriliak-Negami media , 2015, Signal Process..

[164]  Raoul R. Nigmatullin,et al.  Cole-Davidson dielectric relaxation as a self-similar relaxation process , 1997 .