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. Carrer, J. A. M.; Solheid, B. S.; Trevelyan, J.; Seaid, M.: A boundary element method formulation based on the Caputo derivative for the solution of the anomalous diffusion problem (2021)
  2. Derakhshan, MohammadHossein: New numerical algorithm to solve variable-order fractional integrodifferential equations in the sense of Hilfer-Prabhakar derivative (2021)
  3. Baricz, Árpád; Prajapati, Anuja: Radii of starlikeness and convexity of generalized Mittag-Leffler functions (2020)
  4. Bertaccini, D.; Durastante, F.: Computing functions of very large matrices with small TT/QTT ranks by quadrature formulas (2020)
  5. Dien, Nguyen Minh; Hai, Dinh Nguyen Duy; Viet, Tran Quoc; Trong, Dang Duc: On Tikhonov’s method and optimal error bound for inverse source problem for a time-fractional diffusion equation (2020)
  6. Eshaghi, Shiva; Ghaziani, Reza Khoshsiar; Ansari, Alireza: Stability and dynamics of neutral and integro-differential regularized Prabhakar fractional differential systems (2020)
  7. Fernandez, Arran; Kürt, Cemaliye; Özarslan, Mehmet Ali: A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators (2020)
  8. Garrappa, Roberto; Kaslik, Eva: On initial conditions for fractional delay differential equations (2020)
  9. Giusti, Andrea: General fractional calculus and Prabhakar’s theory (2020)
  10. Giusti, Andrea; Colombaro, Ivano; Garra, Roberto; Garrappa, Roberto; Polito, Federico; Popolizio, Marina; Mainardi, Francesco: A practical guide to Prabhakar fractional calculus (2020)
  11. 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)
  12. Lam, P. H.; So, H. C.; Chan, C. F.: Exponential sum approximation for Mittag-Leffler function and its application to fractional Zener wave equation (2020)
  13. Li, Yu; Cao, Yang; Fan, Yan: Generalized Mittag-Leffler quadrature methods for fractional differential equations (2020)
  14. Macías-Díaz, J. E.: A fully explicit variational integrator for multidimensional systems of coupled nonlinear fractional hyperbolic equations (2020)
  15. Meoli, Alessandra; Beerenwinkel, Niko; Lebid, Mykola: The fractional birth process with power-law immigration (2020)
  16. Michelitsch, Thomas M.; Riascos, Alejandro P.: Generalized fractional Poisson process and related stochastic dynamics (2020)
  17. Monteghetti, Florian; Matignon, Denis; Piot, Estelle: Time-local discretization of fractional and related diffusive operators using Gaussian quadrature with applications (2020)
  18. Sarumi, Ibrahim O.; Furati, Khaled M.; Khaliq, Abdul Q. M.: Highly accurate global Padé approximations of generalized Mittag-Leffler function and its inverse (2020)
  19. Tarasov, Vasily E.: Cagan model of inflation with power-law memory effects (2020)
  20. Agahi, Hamzeh; Alipour, Mohsen: Mittag-Leffler-Gaussian distribution: theory and application to real data (2019)

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