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. Almeida, Ricardo; Morgado, M. Luísa: Optimality conditions involving the Mittag-Leffler tempered fractional derivative (2022)
  2. Bokhari, Ahmed; Belgacem, Rachid; Kumar, Sunil; Baleanu, Dumitru; Djilali, Salih: Projectile motion using three parameter Mittag-Leffler function calculus (2022)
  3. Colbrook, Matthew J.; Ayton, Lorna J.: A contour method for time-fractional PDEs and an application to fractional viscoelastic beam equations (2022)
  4. Marasi, H. R.; Derakhshan, M. H.: Haar wavelet collocation method for variable order fractional integro-differential equations with stability analysis (2022)
  5. Moufid, Ilyes; Matignon, Denis; Roncen, Rémi; Piot, Estelle: Energy analysis and discretization of the time-domain equivalent fluid model for wave propagation in rigid porous media (2022)
  6. Oliveira, D. S.: Properties of (\psi)-Mittag-Leffler fractional integrals (2022)
  7. Rani, Noosheza; Fernandez, Arran: Solving Prabhakar differential equations using Mikusiński’s operational calculus (2022)
  8. Abbaszadeh, D.; Tavassoli Kajani, M.; Momeni, M.; Zahraei, M.; Maleki, M.: Solving fractional Fredholm integro-differential equations using Legendre wavelets (2021)
  9. 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)
  10. Derakhshan, MohammadHossein: New numerical algorithm to solve variable-order fractional integrodifferential equations in the sense of Hilfer-Prabhakar derivative (2021)
  11. Duan, Beiping; Zhang, Zhimin: A rational approximation scheme for computing Mittag-Leffler function with discrete elliptic operator as input (2021)
  12. Gabr, A.; Abdel Kader, A. H.; Abdel Latif, M. S.: The effect of the parameters of the generalized fractional derivatives on the behavior of linear electrical circuits (2021)
  13. Gajda, Janusz; Beghin, Luisa: Prabhakar Lévy processes (2021)
  14. Garrappa, Roberto; Giusti, Andrea; Mainardi, Francesco: Variable-order fractional calculus: a change of perspective (2021)
  15. Higham, Nicholas J.; Liu, Xiaobo: A multiprecision derivative-free Schur-Parlett algorithm for computing matrix functions (2021)
  16. Kataria, K. K.; Khandakar, M.: On the long-range dependence of mixed fractional Poisson process (2021)
  17. McLean, William: Numerical evaluation of Mittag-Leffler functions (2021)
  18. Miyajima, Shinya: Computing enclosures for the matrix Mittag-Leffler function (2021)
  19. Sarumi, Ibrahim O.; Furati, Khaled M.; Khaliq, Abdul Q. M.; Mustapha, Kassem: Generalized exponential time differencing schemes for stiff fractional systems with nonsmooth source term (2021)
  20. Van Thang, Nguyen; Van Duc, Nguyen; Minh, Luong Duy Nhat; Thành, Nguyen Trung: Identifying an unknown source term in a time-space fractional parabolic equation (2021)

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