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. Bertaccini, D.; Durastante, F.: Computing functions of very large matrices with small TT/QTT ranks by quadrature formulas (2020)
  2. Kapetina, M. N.; Pisano, A.; Rapaić, M. R.; Usai, E.: Adaptive unit-vector law with time-varying gain for finite-time parameter estimation in LTI systems (2020)
  3. Macías-Díaz, J. E.: A fully explicit variational integrator for multidimensional systems of coupled nonlinear fractional hyperbolic equations (2020)
  4. Meoli, Alessandra; Beerenwinkel, Niko; Lebid, Mykola: The fractional birth process with power-law immigration (2020)
  5. Monteghetti, Florian; Matignon, Denis; Piot, Estelle: Time-local discretization of fractional and related diffusive operators using Gaussian quadrature with applications (2020)
  6. Sarumi, Ibrahim O.; Furati, Khaled M.; Khaliq, Abdul Q. M.: Highly accurate global Padé approximations of generalized Mittag-Leffler function and its inverse (2020)
  7. Baffet, Daniel: A Gauss-Jacobi kernel compression scheme for fractional differential equations (2019)
  8. Derakhshan, Mohammad Hossein; Ansari, Alireza: Numerical approximation to Prabhakar fractional Sturm-Liouville problem (2019)
  9. Górska, Katarzyna; Horzela, Andrzej; Garrappa, Roberto: Some results on the complete monotonicity of Mittag-Leffler functions of le Roy type (2019)
  10. 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)
  11. Hinze, Matthias; Schmidt, André; Leine, Remco I.: Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation (2019)
  12. Jia, Jia; Wang, Zhen; Huang, Xia; Wei, Yunliang: Some remarks on estimate of Mittag-Leffler function (2019)
  13. Moret, Igor; Novati, Paolo: Krylov subspace methods for functions of fractional differential operators (2019)
  14. Sene, Ndolane: Analytical solutions of Hristov diffusion equations with non-singular fractional derivatives (2019)
  15. Tarasov, Vasily E.; Tarasova, Valentina V.: Harrod-Domar growth model with memory and distributed lag (2019)
  16. Burrage, Kevin; Burrage, Pamela; Turner, Ian; Zeng, Fanhai: On the analysis of mixed-index time fractional differential equation systems (2018)
  17. Colombaro, Ivano; Garra, Roberto; Giusti, Andrea; Mainardi, Francesco: Scott-Blair models with time-varying viscosity (2018)
  18. Colombaro, Ivano; Giusti, Andrea; Vitali, Silvia: Storage and dissipation of energy in Prabhakar viscoelasticity (2018)
  19. Derakhshan, Mohammad Hossein; Ansari, Alireza: On Hyers-Ulam stability of fractional differential equations with Prabhakar derivatives (2018)
  20. D’Ovidio, Mirko; Loreti, Paola; Momenzadeh, Alireza; Ahrab, Sima Sarv: Determination of order in linear fractional differential equations (2018)

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