Numerical algorithm for calculating the generalized Mittag-leffler function

SIAM Journal on Numerical Analysis

Volumn 47, Issue 1, 2008, Pages 69-88

  Seybold, Hansjörg ^a   Hilfer, Rudolf ^b,c  

^a ETH ZURICH   (Switzerland)

^b UNIVERSITY OF STUTTGART   (Germany)

^c JOHANNES GUTENBERG UNIVERSITY   (Germany)

Author keywords

Fractional calculus; Generalized Mittag Leffler functions; Numerical algorithms; Special functions of mathematical physics

Indexed keywords

DIFFERENTIATION (CALCULUS); PROBABILITY DENSITY FUNCTION;

ASYMPTOTIC SERIES; COMPLEX ARGUMENTS; FRACTIONAL CALCULUS; GENERALIZED MITTAG-LEFFLER FUNCTIONS; INTEGRAL REPRESENTATIONS; NUMERICAL ALGORITHMS; OPTIMAL STABILITIES; REAL PARAMETERS; SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS; TAYLOR-SERIES;

ALGORITHMS;

EID: 55349106156     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/070700280     Document Type: Article

References (24)

1. W. C. BOYD, Stieltjets transforms and the Stokes phenomenon, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 429 (1990), pp. 227-246.
2. A. ERDELYI, Higher Transcendental Functions, Vols. I-III, Krieger, Malabar, FL, 1981.
3. M. GALASSI, The GNU Scientific Library, Network Theory Ltd., Bristol, UK, 2006.
4. α,β(z) and its derivatives, Fract. Calc. Appl. Anal., 5 (2002), pp. 491-518. Erratum: Frac. Calc. Appl. Anal., 6 (2003).
5. R. HILFER AND L. ANTON, Fractional master equation and fractal time random walks, Phys. Rev. E, 51 (1995), pp. 848-851.
6. R. HILFER, R. METZLER, A. BLUMEN, AND J. KLAFTER, Strange kinetics, Chem. Phys., Special Issue, 284 (2002), pp. 1-2.
7. R. HILLER AND H. SEYBOLD, Computation of the generalized Mittag-Leffler function and its inverse in the complex plane, Integral Transforms Spec. Funct., 17 (2006), pp. 637-652.
8. R. HILFER, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
9. R. HILFER, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys., 284 (2002), pp. 399-408.
10. α(x), C. R. Math. Acad. Sci. Paris, 137 (1903), p. 554.
11. G. MITTAG-LEFFLER, Une généralisation de l'intégrale de Laplace-Abel, C. R. Math. Acad. Sci. Paris, 136 (1903), p. 537.
12. G. MITTAG-LEFFLER, Sur la représentation d'une branche uniforme d'une fonction monogène, Acta Math., 29 (1905), p. 101.
13. R. PARIS, Exponential asymptotics of the Mittag-Leffler function, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 458 (2002), pp. 3041-3052.
14. E. PIESSENS, E. DE DONCKER-KAPENGER, C. UEBERHUBER, AND D. KAHANER, Quadpack, a subroutine package for automatic integration, Springer, New York, 1983.
15. I. POLDUBNY, Fractional Differential Equations, 1st ed., Academic Press, New York, 1999.
16. H. POLLARD, The completely monotonic character of the Mittag-Leffler function, Bull. Amer. Math. Soc., 54 (1948), pp. 1115-1116.
17. S. SAMKO, K. KILBAS, AND O. MARICHEV, Fractional Integrals and Derivatives, Gordon and Breach, Berlin, 1993.
18. W. SCHNEIDER, Completely monotone generalized Mittag-Leffler functions, Expo. Math., 14 (1996), p. 3.
19. H. SEYBOLD AND R. HILFER, Numerical results for the generalized Mittag-Leffler function, Fract. Calc. Appl. Anal., 8 (2005), pp. 127-139.
20. H. SEYBOLD AND R. HILFER, http://www.icp.uni-stuttgart. de/~hilfer/forschung.
21. α(x), Acta Math., 29 (1905), pp. 191-201.
22. A. WIMAN, Über die Nullstellen der Funktionen Eα(x), Acta Math., 29 (1905), pp. 217-234.
23. R. WONG AND Y. ZHAO, Exponential asymptotics of the Mittag-Leffler function, Constr. Approx., 18 (2002), pp. 355-385.
24. W. WYSS, The fractional diffusion equation, J. Math. Phys., 27 (1986), pp. 2782-2785.