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The numerical algorithms for discrete Mittag-Leffler functions approximation

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Veröffentlicht/Copyright: 19. März 2019
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Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 22 Heft 1

Abstract

Motivated essentially by the success of the applications of the discrete Mittag-Leffler functions (DMLF) in many areas of science and engineering, the authors present, in a unified manner, a detailed numerical implementation method of the Mittag-Leffler function. With the proposed method, the overflow problem can be well solved. To further improve the practicability, the state transition matrix described by discrete Mittag-Leffler functions are investigated. Some illustrative examples are provided to verify the effectiveness of the proposed theoretical results.

MSC 2010: Primary 65D20; Secondary 65D15; 33E12; 34A08; 33F05
Key Words and Phrases: discrete Mittag-Leffler function; numerical implementation; overflow phenomenon; state transition matrix

This paper won the “Grünwald-Letnikov Award: Best Student Paper” at the Conference ICFDA 2018.

Acknowledgements

The work described in this paper was fully supported by the National Natural Science Foundation of China (No. 61601431, No. 61573332), the Anhui Provincial Natural Science Foundation (No. 1708085QF141), the Fundamental Research Funds for the Central Universities (No. WK210010 0028) and the General Financial Grant from the China Postdoctoral Science Foundation (No. 2016M602032).

References

[1] T. Abdeljawad, F. Jarad and D. Baleanu, A semigroup-like property for discrete Mittag-Leffler functions. Advances in Difference Equations2012, No 1 (2012); 10.1186/1687-1847-2012-72.Suche in Google Scholar

[2] R.P. Agarwal, Certain fractional q-integrals and q-derivatives. Math. Proc. of the Cambridge Philos. Soc. 66, No 43 (1969), 365–370; 10.1017/S0305004100045060.Suche in Google Scholar

[3] I.D. Bassukas, Comparative Gompertzian analysis of alterations of tumor growth patterns. Cancer Research54, No 16 (1994), 4385–4392; 10.1016/0304-3835(94)90336-0.Suche in Google Scholar

[4] C. Goodrich and A.C. Peterson, Discrete Fractional Calculus. Springer (2015).10.1007/978-3-319-25562-0Suche in Google Scholar

[5] C.W. Granger and R. Joyeux, An introduction to long-memory time series models and fractional differencing. J. of Time Series Analysis1, No 1 (1980), 15–29; 10.1111/j.1467-9892.1980.tb00297.x.Suche in Google Scholar

[6] F. Jarad and D. Baleanu, Discrete variational principles for Lagrangians linear in velocities. Reports on Math. Physics59, No 1 (2007), 33–43; 10.1016/S0034-4877(07)80002-4.Suche in Google Scholar

[7] W.G. Kelley and A.C. Peterson, Difference Equations: An Introduction with Applications. Academic Press, New York (2001).Suche in Google Scholar

[8] Y. Li, Y.Q. Chen and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica45, No 8 (2009), 1965–1969; 10.1016/j.automatica.2009.04.003.Suche in Google Scholar

[9] J.A.T. Machado, A.M. Galhano, J.J. Trujillo, On development of fractional calculus during the last fifty years. Scientometrics98, No 1 (2014), 577–582; 10.1007/s11192-013-1032-6.Suche in Google Scholar

[10] G.M. Mittag-Leffler, Sur la nouvelle fonction Eα(x). Comptes Rendus de l’Acad. des Sci. 137, No 2 (1903), 554–558.Suche in Google Scholar

[11] R.N. Pillai and K. Jayakumar, Discrete Mittag-Leffler distributions. Statistics & Probability Letters23, No 3 (1995), 271–274; 10.1016/0167-7152(94)00124-Q.Suche in Google Scholar

[12] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1999).Suche in Google Scholar

[13] M. Saigo and A.A. Kilbas, On Mittag-Leffler type function and applications. Integr. Transf. Spec. Funct. 7, No 1–2 (1998), 97–112; 10.1080/10652469808819189.Suche in Google Scholar

[14] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publ., London (1993).Suche in Google Scholar

[15] Y.H. Wei, Y. Chen, S. Cheng and Y. Wang, Completeness on the stability criterion of fractional order LTI systems. Fract. Calc. Appl. Anal. 20, No 1 (2017), 159–172; 10.1515/fca-2017-0008; https://www.degruyter.com/view/j/fca.2017.20.issue-1/issue-files/fca.2017.20.issue-1.xml.Suche in Google Scholar

[16] Y.H. Wei, Y.Q. Chen, S. Cheng and Y. Wang, A note on short memory principle of fractional calculus. Fract. Calc. Appl. Anal. 20, No 6 (2017), 1382–1404; 10.1515/fca-2017-0073; https://www.degruyter.com/view/j/fca.2017.20.issue-6/issue-files/fca.2017.20.issue-6.xml.Suche in Google Scholar

[17] Y.H. Wei, Y.Q. Chen, J.C. Wang and Y. Wang, Analysis and description of the infinite-dimensional nature for nabla discrete fractional order systems. Commun. in Nonlin. Sci. and Numer. Simul. 72 (2019), 472–492; 10.1016/j.cnsns.2018.12.023.Suche in Google Scholar

[18] Y.H. Wei, Q. Gao, D.Y. Liu and Y. Wang, On the series representation of nabla discrete fractional calculus. Commun. in Nonlin. Sci. and Numer. Simul. 69, (2019), 198–218; 10.1016/j.cnsns.2018.09.024.Suche in Google Scholar

[19] C.B. Zeng and Y.Q. Chen, Global Padé approximations of the generalized Mittag-Leffler function and its inverse. Fract. Calc. Appl. Anal. 18, No 6 (2016), 1492–1506; 10.1515/fca-2015-0086; https://www.degruyter.com/view/j/fca.2015.18.issue-6/issue-files/fca.2015.18.issue-6.xml.Suche in Google Scholar

Received: 2018-10-25
Published Online: 2019-03-19
Published in Print: 2019-02-25

© 2019 Diogenes Co., Sofia

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 22–1–2019)
  4. Survey Paper
  5. Ranking the scientific output of researchers in fractional calculus
  6. A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications
  7. Research Paper
  8. From power laws to fractional diffusion processes with and without external forces, the non direct way
  9. Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem
  10. The numerical algorithms for discrete Mittag-Leffler functions approximation
  11. New interpretation of fractional potential fields for robust path planning
  12. Stable distributions and green’s functions for fractional diffusions
  13. Fractional calculus in economic growth modelling of the group of seven
  14. Relationship between controllability and observability of standard and fractional different orders discrete-time linear system
  15. Optimal control of linear systems of arbitrary fractional order
  16. Fractional impulsive differential equations: Exact solutions, integral equations and short memory case
  17. Fractional-order modelling and parameter identification of electrical coils
  18. Fractional-order value identification of the discrete integrator from a noised signal. part I
https://doi.org/10.1515/fca-2019-0006
Schlagwörter für diesen Artikel
discrete Mittag-Leffler function; numerical implementation; overflow phenomenon; state transition matrix

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 22–1–2019)
  4. Survey Paper
  5. Ranking the scientific output of researchers in fractional calculus
  6. A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications
  7. Research Paper
  8. From power laws to fractional diffusion processes with and without external forces, the non direct way
  9. Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem
  10. The numerical algorithms for discrete Mittag-Leffler functions approximation
  11. New interpretation of fractional potential fields for robust path planning
  12. Stable distributions and green’s functions for fractional diffusions
  13. Fractional calculus in economic growth modelling of the group of seven
  14. Relationship between controllability and observability of standard and fractional different orders discrete-time linear system
  15. Optimal control of linear systems of arbitrary fractional order
  16. Fractional impulsive differential equations: Exact solutions, integral equations and short memory case
  17. Fractional-order modelling and parameter identification of electrical coils
  18. Fractional-order value identification of the discrete integrator from a noised signal. part I
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