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Some Analytical and Numerical Properties of the Mittag-Leffler Functions

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Published/Copyright: February 10, 2015
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 18 Issue 1

Abstract

Some analytical properties of the Mittag-Leffler functions, eα(t)≢Eα(-tα), are established on some t-intervals. These are lower and upper bounds obtained in terms of simple rational functions (which are related to the Padé approximants of eα(t)) for t>0 and 0<α<1.. A new method, to compute such functions, solving numerically a Caputo-type fractional differential equation satisfied by them, is developed. This approach consists in an adaptive predictor-corrector method, based on the K. Diethelm's predictor-corrector algorithm, and is shown to outperform the current methods implemented by MATLAB® and by Mathematica when t is real and even possibly large

MSC 2010: Primary 33E12; Secondary 34A08; 65L99
Key Words and Phrases: Mittag-Leffler functions; fractional ordinary differential equations; predictor-corrector methods for fractional differential equations; adaptive methods for fractional differential equations

References

[1] M. Abramowitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, 55, U. S. Government Printing Office, Washington, D. C. (1964).10.1115/1.3625776Search in Google Scholar

[2] C. Alsina and M. S. Tomlaa, A geometrical proof of a new inequality for the gamma function. J. Ineq. Pure Appl. Math. 6, No 2 (2005), Article # 48.Search in Google Scholar

[3] K. Diethelm and A. D. Freed, The Frac PECE subroutine for the numerical solution of differential equations of fractional order. In: S. Heinzel, T. Plesser (Eds.), Forschung und Wissenschaftliches Rechnen 1998, Gessellschaft fur Wissenschaftliche Datenverarbeitung, Gottingen (1999), 57.71.Search in Google Scholar

[4] K. Diethelm, Efficient solution of multi-term fractional differential equations using P(EC)mE methods. Computing71 (2003), 305.319; at http://www.scielo.br/pdf/cam/v23n1/a02v23n1.pdf.10.1007/s00607-003-0033-3Search in Google Scholar

[5] K. Diethelm, N. J. Ford, and A. D. Freed, Detailed error analysis for a fractional Adams method. Numer. Algorithms36, No 1 (2004), 31.52.10.1023/B:NUMA.0000027736.85078.beSearch in Google Scholar

[6] A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions [Bateman Manuscript Project], Vols. 1, 2, and 3. McGraw-Hill, New York (1953).Search in Google Scholar

[7] R. Garrappa, On linear stability of predictor-corrector algorithms for fractional differential equations. Internat. J. Comput. Math. 87, No 10 (2010), 2281.2290.Search in Google Scholar

[8] R. Gorenflo, J. Loutchko, and Yu. Luchko, Computation of the Mittag- Leffler function E-z) and its derivative. Fract. Calc. Appl. Anal. 5, No 4 (2002), 491.518.Search in Google Scholar

[9] R. Gorenflo, J. Loutchko, and Yu. Luchko, Correction to Computation of the Mittag-Leffler function E-z) and its derivative [Fract. Calc. Appl. Anal. 5, No 4 (2002), 491.518]. Fract. Calc. Appl. Anal. 6, No 1 (2003), 111.112.Search in Google Scholar

[10] E. Hairer, C. Lubich, and M. Schlichte, Fast numerical solution of nonlinear Volterra convolution equations. SIAM J. Sci. Statist. Comput. 6, No 3 (1985), 532.541.Search in Google Scholar

[11] F. Mainardi, On some properties of the Mittag-Leffler function E completely monotone for t 0 with 0 Discrete Continuous Dynamical Systems - Series B (DCDS-B), To appear; see also FRACALMO Preprint at http://www.fracalmo.org; and http://arxiv.org/pdf/1305.0161.pdf.Search in Google Scholar

[12] http://reference.wolfram.com/mathematica/ref/MittagLefflerE.html.Search in Google Scholar

[13] http://www.mathworks.com/matlabcentral/fileexchange/8738-mittag-leffler-function.Search in Google Scholar

[14] F. W.J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark (Eds.), NIST Digital Library of Mathematical Functions. Cambridge University Press, Cambridge (2010).Search in Google Scholar

[15] T. Simon, Comparing Frechet and positive stable laws. Electron. J. Probab. 19, No 16 (2014), 1-25; see also at http://arxiv.org/pdf/1310.1888v1.pdf.10.1214/EJP.v19-3058Search in Google Scholar

[16] D. Valerio and J. T. Machado, On the numerical computation of the Mittag-Leffler function. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 3419-3424; doi:10.1016/j.cnsns.2014.03.014.10.1016/j.cnsns.2014.03.014Search in Google Scholar

[17] Caibin Zeng and YangQuan Chen, Global Pade approximations of the generalized Mittag-Leffler function and its inverse; at http://arxiv.org/abs/1310.5592.Search in Google Scholar

Received: 2014-3-21
Published Online: 2015-2-10

© 2015 Diogenes Co., Sofia

Articles in the same Issue

  1. FCAA Related News, Events and Books (FCAA-Volume 18-1-2015)
  2. Electrolytic Fractional Derivative and Integral of Two-Terminal Element
  3. Initial Value Problem of Fractional Integro-Differential Equations in Banach Space
  4. Examples of Analytical Solutions by Means of Mittag-Leffler Function of Fractional Black-Scholes Option Pricing Equation
  5. Multiple Solutions for a Class of Fractional Hamiltonian Systems
  6. Some Analytical and Numerical Properties of the Mittag-Leffler Functions
  7. Sobolev Type Fractional Dynamic Equations and Optimal Multi-Integral Controls with Fractional Nonlocal Conditions
  8. Systems of Nonlinear Fractional Differential Equations
  9. Existence and Multiplicity of Solutions for Nonlinear Elliptic Problems with the Fractional Laplacian
  10. Invariant Subspace Method and Exact Solutions of Certain Nonlinear Time Fractional Partial Differential Equations
  11. Filippov Lemma for a Class of Hadamard-Type Fractional Differential Inclusions
  12. New Stability Results for Partial Fractional Differential Inclusions with Not Instantaneous Impulses
  13. Several Results of Fractional Derivatives in Dʹ(R+)
  14. Modeling Extreme-Event Precursors with the Fractional Diffusion Equation
  15. Multiplicity Results for Integral Boundary Value Problems of Fractional Order with Parametric Dependence
  16. Improvements on Flat Output Characterization for Fractional Systems
  17. Nonlocal Fractional Boundary Value Problems with Slit-Strips Boundary Conditions
  18. Remarks to the Paper “On the Existence of Blow UP Solutions for a Class of Fractional Differential Equations” by Z. Bai et al., In “FCAA”, VOL. 17, NO 4 (2014), 1175-1187
https://doi.org/10.1515/fca-2015-0006
Keywords for this article
Mittag-Leffler functions; fractional ordinary differential equations; predictor-corrector methods for fractional differential equations; adaptive methods for fractional differential equations

Articles in the same Issue

  1. FCAA Related News, Events and Books (FCAA-Volume 18-1-2015)
  2. Electrolytic Fractional Derivative and Integral of Two-Terminal Element
  3. Initial Value Problem of Fractional Integro-Differential Equations in Banach Space
  4. Examples of Analytical Solutions by Means of Mittag-Leffler Function of Fractional Black-Scholes Option Pricing Equation
  5. Multiple Solutions for a Class of Fractional Hamiltonian Systems
  6. Some Analytical and Numerical Properties of the Mittag-Leffler Functions
  7. Sobolev Type Fractional Dynamic Equations and Optimal Multi-Integral Controls with Fractional Nonlocal Conditions
  8. Systems of Nonlinear Fractional Differential Equations
  9. Existence and Multiplicity of Solutions for Nonlinear Elliptic Problems with the Fractional Laplacian
  10. Invariant Subspace Method and Exact Solutions of Certain Nonlinear Time Fractional Partial Differential Equations
  11. Filippov Lemma for a Class of Hadamard-Type Fractional Differential Inclusions
  12. New Stability Results for Partial Fractional Differential Inclusions with Not Instantaneous Impulses
  13. Several Results of Fractional Derivatives in Dʹ(R+)
  14. Modeling Extreme-Event Precursors with the Fractional Diffusion Equation
  15. Multiplicity Results for Integral Boundary Value Problems of Fractional Order with Parametric Dependence
  16. Improvements on Flat Output Characterization for Fractional Systems
  17. Nonlocal Fractional Boundary Value Problems with Slit-Strips Boundary Conditions
  18. Remarks to the Paper “On the Existence of Blow UP Solutions for a Class of Fractional Differential Equations” by Z. Bai et al., In “FCAA”, VOL. 17, NO 4 (2014), 1175-1187
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