On asymptotics of discrete Mittag-Leffler function

Luděk Nechvátal

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 4, page 667-675
  • ISSN: 0862-7959

Abstract

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The (modified) two-parametric Mittag-Leffler function plays an essential role in solving the so-called fractional differential equations. Its asymptotics is known (at least for a subset of its domain and special choices of the parameters). The aim of the paper is to introduce a discrete analogue of this function as a solution of a certain two-term linear fractional difference equation (involving both the Riemann-Liouville as well as the Caputo fractional h -difference operators) and describe its asymptotics. Here, we shall employ our recent results on stability and asymptotics of solutions to the mentioned equation.

How to cite

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Nechvátal, Luděk. "On asymptotics of discrete Mittag-Leffler function." Mathematica Bohemica 139.4 (2014): 667-675. <http://eudml.org/doc/269833>.

@article{Nechvátal2014,
abstract = {The (modified) two-parametric Mittag-Leffler function plays an essential role in solving the so-called fractional differential equations. Its asymptotics is known (at least for a subset of its domain and special choices of the parameters). The aim of the paper is to introduce a discrete analogue of this function as a solution of a certain two-term linear fractional difference equation (involving both the Riemann-Liouville as well as the Caputo fractional $h$-difference operators) and describe its asymptotics. Here, we shall employ our recent results on stability and asymptotics of solutions to the mentioned equation.},
author = {Nechvátal, Luděk},
journal = {Mathematica Bohemica},
keywords = {discrete Mittag-Leffler function; fractional difference equation; asymptotics; backward $h$-Laplace transform; discrete Mittag-Leffler function; fractional difference equation; asymptotics; backward $h$-Laplace transform},
language = {eng},
number = {4},
pages = {667-675},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On asymptotics of discrete Mittag-Leffler function},
url = {http://eudml.org/doc/269833},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Nechvátal, Luděk
TI - On asymptotics of discrete Mittag-Leffler function
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 4
SP - 667
EP - 675
AB - The (modified) two-parametric Mittag-Leffler function plays an essential role in solving the so-called fractional differential equations. Its asymptotics is known (at least for a subset of its domain and special choices of the parameters). The aim of the paper is to introduce a discrete analogue of this function as a solution of a certain two-term linear fractional difference equation (involving both the Riemann-Liouville as well as the Caputo fractional $h$-difference operators) and describe its asymptotics. Here, we shall employ our recent results on stability and asymptotics of solutions to the mentioned equation.
LA - eng
KW - discrete Mittag-Leffler function; fractional difference equation; asymptotics; backward $h$-Laplace transform; discrete Mittag-Leffler function; fractional difference equation; asymptotics; backward $h$-Laplace transform
UR - http://eudml.org/doc/269833
ER -

References

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  1. Abdeljawad, T., On delta and nabla Caputo fractional differences and dual identities, Discrete Dyn. Nat. Soc. 2013 (2013), Article No. 406910, 12 pages. (2013) MR3081123
  2. Abdeljawad, T., 10.1016/j.camwa.2011.03.036, Comput. Math. Appl. 62 (2011), 1602-1611. (2011) Zbl1228.26008MR2824747DOI10.1016/j.camwa.2011.03.036
  3. Abdeljawad, T., Jarad, F., Baleanu, D., A semigroup-like property for discrete Mittag-Leffler functions, Adv. Difference Equ. (electronic only) 2012 (2012), Article No. 72, 7 pages. (2012) Zbl1292.39001MR2944466
  4. Atıcı, F. M., Eloe, P. W., 10.1216/RMJ-2011-41-2-353, Rocky Mt. J. Math. 41 (2011), 353-370. (2011) Zbl1218.39003MR2794443DOI10.1216/RMJ-2011-41-2-353
  5. Atıcı, F. M., Eloe, P. W., Discrete fractional calculus with the nabla operator, Electron. J. Qual. Theory Differ. Equ., Special Issue I (electronic only) 2009 (2009), Article No. 3, 12 pages. (2009) Zbl1189.39004MR2558828
  6. Atıcı, F. M., Eloe, P. W., 10.1090/S0002-9939-08-09626-3, Proc. Am. Math. Soc. 137 (2009), 981-989. (2009) Zbl1166.39005MR2457438DOI10.1090/S0002-9939-08-09626-3
  7. Bohner, M., Guseinov, G. S., 10.1016/j.jmaa.2009.09.061, J. Math. Anal. Appl. 365 (2010), 75-92. (2010) Zbl1188.44008MR2585078DOI10.1016/j.jmaa.2009.09.061
  8. Bohner, M., Peterson, A., Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser, Basel (2001). (2001) Zbl0978.39001MR1843232
  9. Čermák, J., Kisela, T., Nechvátal, L., 10.1016/j.amc.2012.12.019, Appl. Math. Comput. 219 (2013), 7012-7022. (2013) Zbl1288.34005MR3027865DOI10.1016/j.amc.2012.12.019
  10. Čermák, J., Kisela, T., Nechvátal, L., Discrete Mittag-Leffler functions in linear fractional difference equations, Abstr. Appl. Anal. 2011 (2011), Article No. 565067, 21 pages. (2011) Zbl1220.39010MR2817254
  11. Čermák, J., Nechvátal, L., 10.1142/S1402925110000593, J. Nonlinear Math. Phys. 17 (2010), 51-68. (2010) Zbl1189.26006MR2647460DOI10.1142/S1402925110000593
  12. Díaz, R., Pariguan, E., On hypergeometric functions and Pochhammer k -symbol, Divulg. Mat. 15 (2007), 179-192. (2007) Zbl1163.33300MR2422409
  13. Elaydi, S., An Introduction to Difference Equations (3rd edition), Undergraduate Texts in Mathematics Springer, New York (2005). (2005) MR2128146
  14. Gray, H. L., Zhang, N. F., 10.1090/S0025-5718-1988-0929549-2, Math. Comput. 50 (1988), 513-519. (1988) Zbl0648.39002MR0929549DOI10.1090/S0025-5718-1988-0929549-2
  15. Matignon, D., Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems and Application Multiconference 2 IMACS, IEEE-SMC, Lille, France, 1996 963-968. 
  16. Miller, K. S., Ross, B., Fractional difference calculus, Univalent Functions, Fractional Calculus, and Their Applications, Kōriyama, 1988 Ellis Horwood Ser. Math. Appl. Horwood, Chichester (1989), 139-152 H. M. Srivastava et al. (1989) Zbl0693.39002MR1199147
  17. Nagai, A., Discrete Mittag-Leffler function and its applications, Sūrikaisekikenkyūsho Kōkyūroku 1302 (2003), 1-20 New developments in the research of integrable systems that are continuous, discrete and ultradiscrete, Japanese, Kyoto, 2002. (2003) MR1986510
  18. Podlubny, I., Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering 198 Academic Press, San Diego (1999). (1999) Zbl0924.34008MR1658022
  19. Qian, D., Li, C., Agarwal, R. P., Wong, P. J. Y., 10.1016/j.mcm.2010.05.016, Math. Comput. Modelling 52 (2010), 862-874. (2010) Zbl1202.34020MR2661771DOI10.1016/j.mcm.2010.05.016

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