The Prabhakar or three parameter Mittag-Leffler function: theory and application. (English) Zbl 1524.33083
Summary: The Prabhakar function (namely, a three parameter Mittag-Leffler function) is investigated. This function plays a fundamental role in the description of the anomalous dielectric properties in disordered materials and heterogeneous systems manifesting simultaneous nonlocality and nonlinearity and, more generally, in models of Havriliak-Negami type. After reviewing some of the main properties of the function, the asymptotic expansion for large arguments is investigated in the whole complex plane and, with major emphasis, along the negative semi-axis. Fractional integral and derivative operators of Prabhakar type are hence considered and some nonlinear heat conduction equations with memory involving Prabhakar derivatives are studied.
MSC:
| 33E12 | Mittag-Leffler functions and generalizations |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
| 41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |
Keywords:
Prabhakar function; Mittag-Leffler function; asymptotic expansion; fractional calculus; Prabhakar derivative; Havriliak-Negami model; nonlinear heat equationSoftware:
MLReferences:
| [2] | Al-Bassam, M. A.; Luchko, Y. F., On generalized fractional calculus and its application to the solution of integro-differential equations, J Fract Calc, 7, 69-88 (1995) · Zbl 0840.44002 |
| [3] | An, J.; Van Hese, E.; Baes, M., Phase-space consistency of stellar dynamical models determined by separable augmented densities, Mon Not R Astron Soc, 422, 1, 652-664 (2012) |
| [4] | Askari, H.; Ansari, A., Fractional calculus of variations with a generalized fractional derivative, Fract Differ Calc, 6, 1, 57-72 (2016) · Zbl 1438.49035 |
| [5] | Bazhlekova, E.; Dimovski, I., Time-fractional Thornley’s problem, J Inequal Spec Funct, 4, 1, 21-35 (2013) · Zbl 1312.35174 |
| [6] | Bazhlekova, E.; Dimovski, I., Exact solution of two-term time-fractional Thornley’s problem by operational method, Integr Transforms Spec Funct, 25, 1, 61-74 (2014) · Zbl 1297.35265 |
| [7] | Braaksma, B. L.J., Asymptotic expansions and analytic continuations for a class of Barnes-integrals, Compositio Math, 15, 239-341 (1964) · Zbl 0129.28604 |
| [8] | Bulavatsky, V. M., Mathematical modeling of fractional differential filtration dynamics based on models of Hilfer-Prabhakar derivatives, Cybern Syst Anal, 53, 2, 204-216 (2017) · Zbl 1380.93038 |
| [9] | Capelas de Oliveira, E.; Mainardi, F.; Vaz, J. Jr., Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics, Eur Phys J Spec Top, 193, 161-171 (2011) |
| [10] | Chamati, H.; Tonchev, N. S., Generalized Mittag-Leffler functions in the theory of finite-size scaling for systems with strong anisotropy and/or long-range interaction, J Phys A Math Gen, 39, 3, 469 (2006) · Zbl 1088.82011 |
| [11] | Chaurasia, V. B.L.; Pandey, S. C., On the fractional calculus of generalized Mittag-Leffler function, Sci Ser A Math Sci (NS), 20, 113-122 (2010) · Zbl 1244.26010 |
| [12] | Derakhshan, M. H.; Ahmadi Darani, M.; Ansari, A.; Khoshsiar Ghaziani, R., On asymptotic stability of Prabhakar fractional differential systems, Comput Methods Differ Equ, 4, 4, 276-284 (2016) · Zbl 1438.34026 |
| [13] | Diethelm, K., The analysis of fractional differential equations; vol. 2004 of Lecture Notes in Mathematics (2010), Springer-Verlag: Springer-Verlag Berlin · Zbl 1215.34001 |
| [15] | Džrbašjan, M. M., Harmonic analysis and boundary value problems in the complex domain; vol. 65 of operator theory: advances and applications (1993), Birkhäuser Verlag, Basel · Zbl 0798.43001 |
| [16] | Eshaghi, S.; Ansari, A., Lyapunov inequality for fractional differential equations with Prabhakar derivative, Math Inequal Appl, 19, 1, 349-358 (2016) · Zbl 1337.34011 |
| [17] | Fabrizio, M.; Giorgi, C.; Morro, A., Modeling of heat conduction via fractional derivatives, Heat Mass Transfer, 1-13 (2017) |
| [18] | Figueiredo Camargo, R.; Capelas de Oliveira, E.; Vaz, J. Jr., On the generalized mittag-leffler function and its application in a fractional telegraph equation, Math Phys Anal Geom, 15, 1, 1-16 (2012) · Zbl 1245.33020 |
| [19] | Fox, C., The asymptotic expansion of integral functions defined by generalized hypergeometric functionss, Proc London Math Soc, s2-27, 1, 389-400 (1928) · JFM 54.0392.03 |
| [20] | Garra, R.; Gorenflo, R.; Polito, F.; Tomovski, Ž., Hilfer-Prabhakar derivatives and some applications, Appl Math Comput, 242, 576-589 (2014) · Zbl 1334.26008 |
| [21] | Garra, R.; Polito, F., On some operators involving Hadamard derivatives, Integr Transforms Spec Funct, 24, 10, 773-782 (2013) · Zbl 1282.33032 |
| [22] | Garrappa, R., Grünwald-Letnikov operators for fractional relaxation in Havriliak-Negami models, Commun Nonlinear Sci Numer Simul, 38, 178-191 (2016) · Zbl 1471.47032 |
| [23] | Garrappa, R., Numerical evaluation of two and three parameter Mittag-Leffler functions, SIAM J Numer Anal, 53, 3, 1350-1369 (2015) · Zbl 1331.33043 |
| [24] | Garrappa, R.; Mainardi, F.; Maione, G., Models of dielectric relaxation based on completely monotone functions, Fract Calc Appl Anal, 19, 5, 1105-1160 (2016) · Zbl 1499.78010 |
| [25] | Gerhold, S., Asymptotics for a variant of the Mittag-Leffler function, Integr Transforms Spec Funct, 23, 6, 397-403 (2012) · Zbl 1257.33050 |
| [26] | Giusti, A.; Colombaro, I., Prabhakar-like fractional viscoelasticity, Commun Nonlinear Sci Numer Simul, 56, 138-143 (2018) · Zbl 1510.74015 |
| [27] | Gorenflo, R.; Kilbas, A. A.; Mainardi, F.; Rogosin, S., Mittag-Leffler functions. theory and applications, Springer Monographs in Mathematics (2014), Springer, Berlin · Zbl 1309.33001 |
| [28] | Gorenflo, R.; Kilbas, A. A.; Rogosin, S. V., On the generalized Mittag-Leffler type functions, Integr Transform Spec Funct, 7, 3-4, 215-224 (1998) · Zbl 0935.33012 |
| [30] | Gupta, R. K.; Shaktawat, B. S.; Kumar, D., Certain relation of generalized fractional calculus associated with the generalized Mittag-Leffler function, J Rajasthan Acad Phys Sci, 15, 3, 117-126 (2016) · Zbl 1358.33016 |
| [31] | Gurtin, M. E.; Pipkin, A. C., A general theory of heat conduction with finite wave speeds, Arch Rational Mech Anal, 31, 2, 113-126 (1968) · Zbl 0164.12901 |
| [32] | Hanyga, A., Physically acceptable viscoelastic models, Trends in applications of mathematics to mechanics. Ber. Math., 125-136 (2005), Shaker Verlag, Aachen · Zbl 1074.74016 |
| [33] | Haubold, H. J.; Mathai, A. M.; Saxena, R. K., Mittag-Leffler functions and their applications, J Appl Math (2011) · Zbl 1218.33021 |
| [34] | Havriliak, S.; Negami, S., A complex plane representation of dielectric and mechanical relaxation processes in some polymers, Polymer, 8, 0, 161-210 (1967) |
| [35] | Henrici, P., Applied and computational complex analysis, 1 (1974), John Wiley & Sons: John Wiley & Sons New York · Zbl 0313.30001 |
| [36] | James, L. F., Lamperti-type laws, Ann Appl Probab, 20, 4, 1303-1340 (2010) · Zbl 1204.60024 |
| [37] | Kilbas, A. A.; Koroleva, A. A.; Rogosin, S. V., Multi-parametric Mittag-Leffler functions and their extension, Fract Calc Appl Anal, 16, 2, 378-404 (2013) · Zbl 1312.33058 |
| [38] | Kilbas, A. A.; Saigo, M., On solution of integral equation of Abel-Volterra type, Differ Integr Equ, 8, 5, 993-1011 (1995) · Zbl 0823.45002 |
| [39] | Kilbas, A. A.; Saigo, M.; Saxena, R. K., Solution of Volterra integrodifferential equations with generalized Mittag-Leffler function in the kernels, J Integr Equ Appl, 14, 4, 377-396 (2002) · Zbl 1041.45011 |
| [40] | Kilbas, A. A.; Saigo, M.; Saxena, R. K., Generalized Mittag-Leffler function and generalized fractional calculus operators, Integr Transforms Spec Funct, 15, 1, 31-49 (2004) · Zbl 1047.33011 |
| [41] | Kiryakova, V., The multi-index Mittag-Leffler functions as an important class of special functions of fractional calculus, Comput Math Appl, 59, 5, 1885-1895 (2010) · Zbl 1189.33034 |
| [43] | Kurulay, M.; Bayram, M., Some properties of the Mittag-Leffler functions and their relation with the Wright functions, Adv Differ Equ, 2012, 1, 181 (2012) · Zbl 1377.33012 |
| [45] | Liemert, A.; Sandev, T.; Kantz, H., Generalized Langevin equation with tempered memory kernel, Phys A, 466, 356-369 (2017) · Zbl 1400.82195 |
| [46] | Luchko, Y. F.; Srivastava, H. M., The exact solution of certain differential equations of fractional order by using operational calculus, Comput Math Appl, 29, 8, 73-85 (1995) · Zbl 0824.44011 |
| [47] | Mainardi, F.; Garrappa, R., On complete monotonicity of the Prabhakar function and non-Debye relaxation in dielectrics, J Comput Phys, 293, 70-80 (2015) · Zbl 1349.65085 |
| [48] | Mathai, A. M.; Saxena, R. K.; Haubold, H. J., The H-function: theory and applications (2010), Springer · Zbl 1181.33001 |
| [49] | Miskinis, P., The Havriliak-Negami susceptibility as a nonlinear and nonlocal process, Phys Scripta, 2009, T136, 014019 (2009) |
| [50] | Mittag-Leffler, M. G., Sur l’intégrale de laplace-abel, C R Acad Sci Paris (Ser II), 136, 937-939 (1902) · JFM 33.0408.01 |
| [51] | Mittag-Leffler, M. G., Sopra la funzione \(E_α}(x)\), Rend Acad Lincei, 13, 5, 3-5 (1904) · JFM 35.0448.02 |
| [52] | Nemes, G., On the coefficients of the asymptotic expansion of \(n\)!, J Integer Seq, 13, 6 (2010) · Zbl 1216.11023 |
| [53] | Nigmatullin, R. R.; Khamzin, A. A.; Baleanu, D., On the Laplace integral representation of multivariate Mittag-Leffler functions in anomalous relaxation, Math Methods Appl Sci, 39, 11, 2983-2992 (2016) · Zbl 1383.78012 |
| [54] | Nigmatullin, R.; Ryabov, Y., Cole-Davidson dielectric relaxation as a self-similar relaxation process, Phys Solid State, 39, 1, 87-90 (1997) |
| [55] | Novikov, V.; Wojciechowski, K.; Komkova, O.; Thiel, T., Anomalous relaxation in dielectrics. Equations with fractional derivatives, Mater Sci Poland, 23, 4, 977-984 (2005) |
| [56] | Pandey, S. C., The Lorenzo-Hartley’s function for fractional calculus and its applications pertaining to fractional order modelling of anomalous relaxation in dielectrics, Comput Appl Math (2017) |
| [57] | Paneva-Konovska, J., Comparison between the convergence of power and generalized Mittag-Leffler series, Complex analysis and applications ’13, 166-173 (2013), Bulgarian Acad. Sci., Sofia · Zbl 1310.40002 |
| [58] | Paneva-Konovska, J., Convergence of series in three parametric Mittag-Leffler functions, Math Slovaca, 64, 1, 73-84 (2014) · Zbl 1324.33015 |
| [59] | Paris, R. B., Exponentially small expansions in the asymptotics of the Wright function, J Comput Appl Math, 234, 2, 488-504 (2010) · Zbl 1215.33003 |
| [60] | Paris, R. B.; Kaminski, D., Asymptotics and mellin-barnes integrals; vol. 85 of encyclopedia of mathematics and its applications (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0983.41019 |
| [61] | Pogány, T. K.; Tomovski, Ž., Probability distribution built by Prabhakar function. Related Turán and Laguerre inequalities, Integr Transforms Spec Funct, 27, 10, 783-793 (2016) · Zbl 1357.26044 |
| [62] | Polito, F.; Scalas, E., A generalization of the space-fractional Poisson process and its connection to some Lévy processes, Electron Commun Probab, 21 (2016) · Zbl 1338.60129 |
| [63] | Polito, F.; Tomovski, Ž., Some properties of Prabhakar-type fractional calculus operators, Fract Differ Calc, 6, 1, 73-94 (2016) · Zbl 1424.26017 |
| [64] | Polyanin, A. D.; Zaitsev, V. F., Handbook of nonlinear partial differential equations (2004), Chapman & Hall/CRC, Boca Raton, FL · Zbl 1053.35001 |
| [65] | Povstenko, Y., Fractional thermoelasticity; vol. 219 of solid mechanics and its applications (2015), Springer, Cham · Zbl 1316.74001 |
| [66] | Prabhakar, T. R., A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math J, 19, 1, 7-15 (1971) · Zbl 0221.45003 |
| [67] | Sahadevan, R.; Prakash, P., Exact solution of certain time fractional nonlinear partial differential equations, Nonlinear Dynam, 85, 1, 659-673 (2016) · Zbl 1349.35406 |
| [68] | Saxena, R. K.; Saigo, M., Certain properties of fractional calculus operators associated with generalized Mittag-Leffler function, Fract Calc Appl Anal, 8, 2, 141-154 (2005) · Zbl 1144.26010 |
| [69] | Stanislavsky, A.; Weron, K., Atypical case of the dielectric relaxation responses and its fractional kinetic equation, Fract Calc Appl Anal, 19, 1, 212-228 (2016) · Zbl 1381.82026 |
| [70] | Stanislavsky, A.; Weron, K., Numerical scheme for calculating of the fractional two-power relaxation laws in time-domain of measurements, Comput Phys Commun, 183, 2, 320-323 (2012) · Zbl 1268.65031 |
| [71] | Straughan, B., Heat waves; vol. 177 of applied mathematical sciences (2011), Springer-Verlag: Springer-Verlag New York · Zbl 1232.80001 |
| [72] | Tomovski, Ž.; Pogány, T.; Srivastava, H. M., Laplace type integral expressions for a certain three-parameter family of generalized Mittag-Leffler functions with applications involving complete monotonicity, J Franklin Inst, 351, 12, 5437-5454 (2014) · Zbl 1393.93060 |
| [73] | Tonchev, N. S., Finite-size scaling in anisotropic systems, Phys Rev E, 75, 031110 (2007) |
| [74] | Tricomi, F. G.; Erdélyi, A., The asymptotic expansion of a ratio of gamma functions, Pacific J Math, 1, 133-142 (1951) · Zbl 0043.29103 |
| [75] | Wright, E. M., The asymptotic expansion of the generalised hypergeometric function, J London Math Soc, s1-10, 4, 286-293 (1935) · Zbl 0013.02104 |
| [76] | Wright, E. M., The asymptotic expansion of integral functions defined by Taylor series, Philos Trans Roy Soc London, Ser-A, 238, 423-451 (1940) · Zbl 0023.14002 |
| [77] | Wright, E. M., The asymptotic expansion of the generalized hypergeometric function, Proc London Math Soc (Ser 2), 46, 389-408 (1940) · JFM 66.1243.01 |
| [78] | Xu, J., Time-fractional particle deposition in porous media, J Phys A, 50, 19 (2017) · Zbl 1369.82034 |
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