Mainardi, Francesco; Pagnini, Gianni The role of the Fox-Wright functions in fractional sub-diffusion of distributed order. (English) Zbl 1120.35002 J. Comput. Appl. Math. 207, No. 2, 245-257 (2007). Summary: The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin-Barnes integral representation. Such solution is proved to be related via a Laplace-type integral to the Fox-Wright functions. A series expansion is also provided in order to point out the distribution of time-scales related to the distribution of the fractional orders. The results of the time fractional diffusion equation of a single order are also recalled and then re-obtained from the general theory. Cited in 1 ReviewCited in 52 Documents MSC: 35A08 Fundamental solutions to PDEs 35A22 Transform methods (e.g., integral transforms) applied to PDEs 26A33 Fractional derivatives and integrals 33E12 Mittag-Leffler functions and generalizations 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) 44A10 Laplace transform 45K05 Integro-partial differential equations Keywords:sub-diffusion; fractional derivatives; Mellin-Barnes integrals; Mittag-Leffler functions; Fox-Wright functions; integral transforms PDFBibTeX XMLCite \textit{F. Mainardi} and \textit{G. Pagnini}, J. Comput. Appl. Math. 207, No. 2, 245--257 (2007; Zbl 1120.35002) Full Text: DOI arXiv References: [1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1965), Dover: Dover New York · Zbl 0515.33001 [2] R.L. Bagley, P.J. Torvik, On the existence of the order domain and the solution of distributed order equations, Internat. J. Appl. Math. 2 (2000) 865-882, 965-987.; R.L. Bagley, P.J. Torvik, On the existence of the order domain and the solution of distributed order equations, Internat. J. Appl. Math. 2 (2000) 865-882, 965-987. · Zbl 1100.34006 [3] Braaksma, B. L.J., Asymptotic expansions and analytical continuations for a class of Barnes-integrals, Compos. Math., 15, 239-341 (1962-1963) · Zbl 0129.28604 [4] Butzer, P.; Westphal, U., Introduction to fractional calculus, (Hilfer, H., Fractional Calculus, Applications in Physics (2000), World Scientific: World Scientific Singapore), 1-85 · Zbl 0987.26005 [5] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent, Part II, Geophys. J. Roy. Astron. Soc., 13, 529-539 (1967) [6] Caputo, M., Elasticità e Dissipazione (1969), Zanichelli: Zanichelli Bologna, [in Italian] [7] Caputo, M., Mean fractional-order derivatives differential equations and filters, Ann. Univ. Ferrara, Sez VII, Sc. Mat., 41, 73-84 (1995) · Zbl 0882.34007 [8] Caputo, M., Distributed order differential equations modelling dielectric induction and diffusion, Fractional Calculus and Applied Analysis, 4, 421-442 (2001) · Zbl 1042.34028 [9] Caputo, M.; Mainardi, F., Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (Ser. II), 1, 161-198 (1971) [10] A.V. Chechkin, R. Gorenflo, I.M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E 66 (2002) 046129/1-6.; A.V. Chechkin, R. Gorenflo, I.M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E 66 (2002) 046129/1-6. [11] Chechkin, A. V.; Gorenflo, R.; Sokolov, I. M.; Gonchar, V. Yu., Distributed order time fractional diffusion equation, Fractional Calculus and Applied Analysis, 6, 259-279 (2003) · Zbl 1089.60046 [12] Chechkin, A. V.; Klafter, J.; Sokolov, I. M., Fractional Fokker-Planck equation for ultraslow kinetics, Europhysics Lett., 63, 326-332 (2003) [13] Craven, T.; Csordas, G., The Fox-Wright functions and Laguerre multiplier sequences, J. Mat. Anal. Appl., 314, 109-125 (2006) · Zbl 1081.30030 [14] Djrbashian, M. M., Integral Transforms and Representations of Functions in the Complex Plane (1966), Nauka: Nauka Moscow, [in Russian] · Zbl 0816.30023 [15] A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Bateman Project, vol. 3, McGraw-Hill, New York, 1955, pp. 206-227 (Chapter 18: Miscellaneous Functions).; A. Erdélyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, Bateman Project, vol. 3, McGraw-Hill, New York, 1955, pp. 206-227 (Chapter 18: Miscellaneous Functions). [16] Fox, C., The \(G\) and \(H\) functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc., 98, 395-429 (1961) · Zbl 0096.30804 [17] Gorenflo, R.; Iskenderov, A.; Luchko, Yu., Mapping between solutions of fractional diffusion-wave equations, Fractional Calculus Appl. Anal., 3, 1, 75-86 (2000) · Zbl 1033.35161 [18] Gorenflo, R.; Louthcko, J.; Luchko, Yu., Computation of the Mittag-Leffler function \(E_{\alpha, \beta}(z)\) and its derivatives, Fractional Calculus Appl. Anal., 5, 4, 491-518 (2002) · Zbl 1027.33016 [19] Gorenflo, R.; Luchko, Yu.; Mainardi, F., Analytical properties and applications of the Wright function, Fractional Calculus Appl. Anal., 2, 383-414 (1999) · Zbl 1027.33006 [20] Gorenflo, R.; Luchko, Yu.; Mainardi, F., Wright functions as scale-invariant solutions of the diffusion-wave equation, J. Comput. Appl. Math., 118, 175-191 (2000) · Zbl 0973.35012 [21] Gorenflo, R.; Mainardi, F., Fractional calculus: integral and differential equations of fractional order, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer: Springer Wien), 223-276, (Reprinted in \(\langle\) http://www.fracalmo.org \(\rangle )\) · Zbl 1438.26010 [22] R. Gorenflo, F. Mainardi, H.M. Srivastava, Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena, in: D. Bainov (Eds.), Proceedings VIII International Colloquium on Differential Equations, Plovdiv 1997, VSP, Utrecht, 1998, pp. 195-202.; R. Gorenflo, F. Mainardi, H.M. Srivastava, Special functions in fractional relaxation-oscillation and fractional diffusion-wave phenomena, in: D. Bainov (Eds.), Proceedings VIII International Colloquium on Differential Equations, Plovdiv 1997, VSP, Utrecht, 1998, pp. 195-202. · Zbl 0921.33009 [23] Kilbas, A. A., Fractional calculus of the generalized Wright function, Fractional Calculus Appl. Anal., 8, 114-126 (2005) · Zbl 1144.26008 [24] Kilbas, A. A.; Saigo, M., On the \(H\) functions, J. Appl. Math. Stochastic Anal., 12, 191-204 (1999) · Zbl 0934.33018 [25] Kilbas, A. A.; Saigo, M., \(H\)-transforms, Theory and Applications (2004), CRC Press: CRC Press Boca Raton, FL · Zbl 1056.44001 [26] A Kilbas, A.; Saigo, M.; Trujillo, J. J., On the generalized Wright function, Fractional Calculus Appl. Anal., 5, 437-460 (2002) · Zbl 1027.33015 [27] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations (2006), Elsevier: Elsevier Amsterdam · Zbl 1092.45003 [28] Kiryakova, V., Generalized Fractional Calculus and Applications, vol. 301 (1994), Pitman Research Notes in Mathematics: Pitman Research Notes in Mathematics Longman, Harlow, UK · Zbl 0882.26003 [29] Klafter, J.; Sokolov, I. M., Anomalous diffusion spreads its wings, Phys. World, 18, 29-32 (2005) [30] Mainardi, F., On the initial value problem for the fractional diffusion-wave equation, (Rionero, S.; Ruggeri, T., Waves and Stability in Continuous Media (1994), World Scientific: World Scientific Singapore), 246-251 [31] Mainardi, F., The fundamental solutions for the fractional diffusion-wave equation, Appl. Math. Lett., 9, 6, 23-28 (1996) · Zbl 0879.35036 [32] Mainardi, F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons and Fractals, 7, 1461-1477 (1996) · Zbl 1080.26505 [33] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer: Springer Wien and New York), 291-348, (Reprinted in \(\langle\) http://www.fracalmo.org \(\rangle )\) · Zbl 0917.73004 [34] Mainardi, F.; Gorenflo, R., On Mittag-Leffler type functions in fractional evolution processes, J. Comput. Appl. Math., 118, 283-299 (2000) · Zbl 0970.45005 [35] Mainardi, F.; Luchko, Yu.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus Appl. Anal., 4, 153-192 (2001), (Reprinted in \(\langle\) http://www.fracalmo.org \(\rangle )\) · Zbl 1054.35156 [36] Mainardi, F.; Pagnini, G., The Wright functions as solutions of the time-fractional diffusion equations, Appl. Math. Comput., 141, 51-62 (2003) · Zbl 1053.35008 [37] Mainardi, F.; Pagnini, G., Salvatore Pincherle: the pioneer of the Mellin-Barnes integrals, J. Comput. Appl. Math., 153, 331-342 (2003) · Zbl 1050.33018 [38] Mainardi, F.; Pagnini, G.; Saxena, R. K., Fox \(H\) functions in fractional diffusion, J. Comput. Appl. Math., 178, 321-331 (2005) · Zbl 1061.33012 [39] Marichev, O. I., Handbook of Integral Transforms of Higher Transcendental Functions, Theory and Algorithmic Tables (1983), Chichester: Chichester Ellis Horwood · Zbl 0494.33001 [40] Mathai, A. M.; Saxena, R. K., Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Lecture Notes in Mathematics, vol. 348 (1973), Springer: Springer Berlin · Zbl 0272.33001 [41] Mathai, A. M.; Saxena, R. K., The \(H\)-function with Applications in Statistics and Other Disciplines (1978), Wiley Eastern Ltd.: Wiley Eastern Ltd. New Delhi · Zbl 0382.33001 [42] Naber, M., Distributed order fractional subdiffusion, Fractals, 12, 1, 23-32 (2004) · Zbl 1083.60066 [43] Paris, R. B.; Kaminski, D., Asymptotic and Mellin-Barnes Integrals (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0983.41019 [44] Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010 [45] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series, Vol 3: More Special Functions, Gordon and Breach, New York, 1990.; A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series, Vol 3: More Special Functions, Gordon and Breach, New York, 1990. [46] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach: Gordon and Breach New York · Zbl 0818.26003 [47] Schneider, W. R.; Wyss, W., Fractional diffusion and wave equations, J. Math. Phys., 30, 134-144 (1989) · Zbl 0692.45004 [48] Sokolov, I. M.; Chechkin, A. V.; Klafter, J., Distributed-order fractional kinetics, Acta Phys. Polon., 35, 1323-1341 (2004) [49] Sokolov, I. M.; Klafter, J.; Blumen, A., Fractional Kinetics, Phys. Today, 55, 48-54 (2002) [50] Srivastava, H. M.; Gupta, K. C.; Goyal, S. P., The \(H\)-Functions of One and Two Variables with Applications (1982), South Asian Publishers: South Asian Publishers New Delhi · Zbl 0506.33007 [51] Srivastava, H. M.; Saxena, R. K.; Ram, C., A unified presentation of the Gamma-type functions occurring in diffraction theory and associated probability distributions, Appl. Math. Comput., 162, 931-947 (2005) · Zbl 1063.33004 [52] M Temme, N., Special Functions: An Introduction to the Classical Functions of Mathematical Physics (1996), Wiley: Wiley New York · Zbl 0856.33001 [53] Wong, R.; Zaho, Y.-Q., Smoothing of Stokes’ discontinuity for the generalized Bessel function, Proc. Roy. Soc. London A, 455, 1381-1400 (1999) · Zbl 0953.65014 [54] Wong, R.; Zaho, Y.-Q., Smoothing of Stokes’ discontinuity for the generalized Bessel function, II, Proc. Roy. Soc. London A, 455, 3065-3084 (1999) · Zbl 0974.41021 [55] Wright, E. M., On the coefficients of power series having exponential singularities, J. London Math. Soc., 8, 71-79 (1933) · JFM 59.0383.01 [56] Wright, E. M., The asymptotic expansion of the generalized Bessel function, Proc. London Math. Soc. (Ser. II), 38, 257-270 (1935) · Zbl 0010.21103 [57] Wright, E. M., The asymptotic expansion of the generalized hypergeometric function, J. London Math. Soc., 10, 287-293 (1935) · Zbl 0013.02104 [58] Wright, E. M., The generalized Bessel function of order greater than one, Quart. J. Math., Oxford Ser., 11, 36-48 (1940) · Zbl 0023.14101 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.