ML swMATH ID: 20907 Software Authors: Garrappa, R. Description: 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. Homepage: http://www.mathworks.com/matlabcentral/fileexchange/48154-the-mittag-leffler-function Dependencies: Matlab Related Software: Matlab; Mittag-Leffler; mlf; mftoolbox; DFOC; ma2dfc; GitHub; FracPECE; FOTF Toolbox; longmemo; Ninteger; CRONE; sysdfod; mlrnd; FDE12; FODE; funm; mctoolbox; Expokit; FSST Cited in: 114 Documents Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year Numerical evaluation of two and three parameter Mittag-Leffler functions. Zbl 1331.33043Garrappa, Roberto 2015 all top 5 Cited by 194 Authors 15 Garrappa, Roberto 7 Giusti, Andrea 6 Mainardi, Francesco 4 Băleanu, Dumitru I. 4 Fernandez, Arran 4 Garra, Roberto 4 Popolizio, Marina 3 Agahi, Hamzeh 3 Ansari, Alireza 3 Burrage, Kevin 3 Chen, Yangquan 3 Colombaro, Ivano 3 Sandev, Trifce 3 Tarasov, Vasily E. 2 Alipour, Mohsen 2 Baffet, Daniel H. 2 Cardone, Angelamaria 2 Chechkin, Aleksei V. 2 D’Ovidio, Mirko 2 Eshaghi, Shiva 2 Furati, Khaled M. 2 Górska, Katarzyna 2 Horzela, Andrzej 2 Khaliq, Abdul Q. M. 2 Li, Yu 2 Loreti, Paola 2 Matignon, Denis 2 McLean, William 2 Metzler, Ralf 2 Moret, Igor 2 Piot, Estelle 2 Polito, Federico 2 Rani, Noosheza 2 Sarumi, Ibrahim O. 2 Zeng, Fanhai 1 Abbaszadeh, D. 1 Abdeljawad, Thabet 1 Ahrab, Sima Sarv 1 Almeida, Ricardo 1 Aminikhah, Hossein 1 Antoniadis, Anestis 1 Araci, Serkan 1 Asante-Asamani, E. O. 1 Ashyralyev, Allaberen 1 Atanackovic, Teodor M. 1 Awad, Emad S. 1 Ayton, Lorna J. 1 Barabesi, Lucio 1 Baricz, Árpád 1 Beerenwinkel, Niko 1 Beghin, Luisa 1 Belgacem, Rachid 1 Bertaccini, Daniele 1 Biolek, Dalibor 1 Biolkova, Viera 1 Bokhari, Ahmed 1 Braga, Joao Paulo 1 Brossat, Xavier 1 Bueno-Orovio, Alfonso 1 Burrage, Pamela M. 1 Caccia, Gianluca Frasca 1 Cardoso, João R. 1 Carrer, José Antonio Marques 1 Chakraborty, Subrata 1 Colbrook, Matthew J. 1 D’Ambrosio, Raffaele 1 Dehghan, Sina 1 Deng, Weihua 1 Derakhshan, Mohammad Hossein 1 Dien, Nguyen Minh 1 Djilali, Salih 1 Dos Santos, José Paulo Carvalho 1 dos Santos, Maike A. F. 1 Duan, Beiping 1 Duc, Nguyen Van 1 Duo, Siwei 1 Durastante, Fabio 1 Esmaeili, Shahrokh 1 Fan, Yan 1 Frunzo, Luigi 1 Gabr, A. 1 Gajda, Janusz 1 Ghaziani, Reza Khoshsiar 1 Goude, Yannig 1 Guido, Maione 1 Guo, Ying 1 Hai, Dinh Nguyen Duy 1 Henry, Bruce I. 1 Hesthaven, Jan S. 1 Hicdurmaz, Betul 1 Higham, Nicholas J. 1 Hinze, Matthias 1 Huang, Xia 1 Iyiola, Olaniyi Samuel 1 Jia, Jia 1 Ju, Lili 1 Kapetina, Mirna N. 1 Kaslik, Eva 1 Kataria, Kuldeep Kumar 1 Khalili, Monavar ...and 94 more Authors all top 5 Cited in 50 Serials 13 Fractional Calculus & Applied Analysis 11 Communications in Nonlinear Science and Numerical Simulation 8 Journal of Scientific Computing 8 Computational and Applied Mathematics 7 Applied Numerical Mathematics 5 Computers & Mathematics with Applications 4 Journal of Computational Physics 4 Mathematics 3 Physica A 2 Mathematical Methods in the Applied Sciences 2 Chaos, Solitons and Fractals 2 Applied Mathematics and Computation 2 Journal of Computational and Applied Mathematics 2 Mathematics and Computers in Simulation 2 SIAM Journal on Numerical Analysis 2 Nonlinear Dynamics 2 Axioms 1 International Journal of Control 1 Journal of Statistical Physics 1 Mathematical Notes 1 ZAMP. Zeitschrift für angewandte Mathematik und Physik 1 Mathematics of Computation 1 Calcolo 1 Rendiconti del Circolo Matemàtico di Palermo. Serie II 1 Statistics & Probability Letters 1 Journal of Theoretical Probability 1 Applied Mathematics Letters 1 SIAM Journal on Matrix Analysis and Applications 1 Applied Mathematical Modelling 1 Communications in Statistics. Simulation and Computation 1 International Journal of Computer Mathematics 1 SIAM Journal on Applied Mathematics 1 Integral Transforms and Special Functions 1 Engineering Analysis with Boundary Elements 1 Mathematical Communications 1 Abstract and Applied Analysis 1 Chaos 1 Discrete Dynamics in Nature and Society 1 Analysis (München) 1 Journal of Applied Mathematics and Computing 1 Advances in Difference Equations 1 Journal of Statistical Mechanics: Theory and Experiment 1 Boundary Value Problems 1 European Series in Applied and Industrial Mathematics (ESAIM): Mathematical Modelling and Numerical Analysis 1 Mathematical Modelling of Natural Phenomena 1 Discrete and Continuous Dynamical Systems. Series S 1 Journal of Statistical Distributions and Applications 1 Journal of Function Spaces 1 Modern Stochastics. Theory and Applications 1 International Journal of Applied and Computational Mathematics all top 5 Cited in 31 Fields 49 Numerical analysis (65-XX) 43 Real functions (26-XX) 36 Special functions (33-XX) 35 Ordinary differential equations (34-XX) 32 Partial differential equations (35-XX) 11 Probability theory and stochastic processes (60-XX) 9 Integral transforms, operational calculus (44-XX) 7 Integral equations (45-XX) 5 Approximations and expansions (41-XX) 5 Mechanics of deformable solids (74-XX) 4 Statistics (62-XX) 4 Optics, electromagnetic theory (78-XX) 4 Statistical mechanics, structure of matter (82-XX) 4 Biology and other natural sciences (92-XX) 3 Linear and multilinear algebra; matrix theory (15-XX) 3 Operator theory (47-XX) 3 Calculus of variations and optimal control; optimization (49-XX) 3 Fluid mechanics (76-XX) 3 Systems theory; control (93-XX) 2 Functions of a complex variable (30-XX) 2 Classical thermodynamics, heat transfer (80-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 2 Information and communication theory, circuits (94-XX) 1 History and biography (01-XX) 1 Measure and integration (28-XX) 1 Potential theory (31-XX) 1 Several complex variables and analytic spaces (32-XX) 1 Difference and functional equations (39-XX) 1 Mechanics of particles and systems (70-XX) 1 Quantum theory (81-XX) 1 Geophysics (86-XX) Citations by Year