Numerical evaluation of two and three parameter Mittag-Leffler functions. The Mittag-Leffler function plays a fundamental role in fractional calculus. In the present paper, a method is introduced for the efficient computation of the Mittag-Leffler function based on the numerical inversion of its Laplace transform. The approach taken is to consider separate regions in which the Laplace transform is analytic and to look for the contour and discretization parameters allowing one to achieve a given accuracy. The optimal parabolic contour algorithm selects the region in which the numerical inversion of the Laplace transform is actually performed by choosing the one in which both the computational effort and the errors are minimized. Numerical experiments are presented to show accuracy and efficiency of the proposed approach. An application to the three parameter Mittag-Leffler function is also presented.

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  1. Baffet, Daniel: A Gauss-Jacobi kernel compression scheme for fractional differential equations (2019)
  2. Derakhshan, Mohammad Hossein; Ansari, Alireza: Numerical approximation to Prabhakar fractional Sturm-Liouville problem (2019)
  3. Górska, K.; Horzela, A.; Pogány, T. K.: A note on the article “Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel” [Z. angew. Math. Phys. (2019) 70: 42] (2019)
  4. Jia, Jia; Wang, Zhen; Huang, Xia; Wei, Yunliang: Some remarks on estimate of Mittag-Leffler function (2019)
  5. Moret, Igor; Novati, Paolo: Krylov subspace methods for functions of fractional differential operators (2019)
  6. Sene, Ndolane: Analytical solutions of Hristov diffusion equations with non-singular fractional derivatives (2019)
  7. Colombaro, Ivano; Garra, Roberto; Giusti, Andrea; Mainardi, Francesco: Scott-Blair models with time-varying viscosity (2018)
  8. Colombaro, Ivano; Giusti, Andrea; Vitali, Silvia: Storage and dissipation of energy in Prabhakar viscoelasticity (2018)
  9. Derakhshan, Mohammad Hossein; Ansari, Alireza: On Hyers-Ulam stability of fractional differential equations with Prabhakar derivatives (2018)
  10. D’Ovidio, Mirko; Loreti, Paola; Momenzadeh, Alireza; Ahrab, Sima Sarv: Determination of order in linear fractional differential equations (2018)
  11. Duo, Siwei; Ju, Lili; Zhang, Yanzhi: A fast algorithm for solving the space-time fractional diffusion equation (2018)
  12. Garrappa, Roberto: Numerical solution of fractional differential equations: a survey and a software tutorial (2018)
  13. Garrappa, Roberto; Popolizio, Marina: Computing the matrix Mittag-Leffler function with applications to fractional calculus (2018)
  14. Giusti, Andrea: A comment on some new definitions of fractional derivative (2018)
  15. Iyiola, O. S.; Asante-Asamani, E. O.; Wade, B. A.: A real distinct poles rational approximation of generalized Mittag-Leffler functions and their inverses: applications to fractional calculus (2018)
  16. Matychyn, Ivan; Onyshchenko, Viktoriia: Optimal control of linear systems with fractional derivatives (2018)
  17. Owolabi, Kolade M.: Numerical approach to fractional blow-up equations with Atangana-Baleanu derivative in Riemann-Liouville sense (2018)
  18. Polito, Federico: Studies on generalized Yule models (2018)
  19. Popolizio, Marina: Numerical solution of multiterm fractional differential equations using the matrix Mittag-Leffler functions (2018)
  20. Sadeghi, Amir; Cardoso, João R.: Some notes on properties of the matrix Mittag-Leffler function (2018)

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