Skip to main content
SpringerLink
Log in

Accurate Padé Global Approximations for the Mittag-Leffler Function, Its Inverse, and Its Partial Derivatives to Efficiently Compute Convergent Power Series

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

The fractional derivative operator provides a mathematical means to concisely describe a heterogeneous and relatively complex system that exhibits non-local, power-law behavior. Discretization of a fractional partial differential equation is non-trivial and computationally intensive. Furthermore, the closed form solution, particularly in the case of the fractional time derivative, comes in the form of a convergent power series of the Mittag-Leffler type, which involves considerable computational burden for accurate and efficient calculation. Here, we extend the Padé global approximation technique in order to accurately compute the Mittag-Leffler function and its inverse operation. Furthermore, we introduce the use of the Padé technique for the partial derivatives of the Mittag-Leffler function, which holistically eliminates the need for the calculation of any convergent power series when fitting to data sets. The Padé approximations provide great utility, particularly in the field of diffusion-weighted magnetic resonance imaging, in which Mittag-Leffler function and its partial derivatives are computed on a voxel-wise basis, with each data set containing on the order of \(10^7\) voxels. Finally, the Padé approximation codes are provided freely for those interested in modeling and fitting data for processes encompassed by this convergent power series.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Metzler, R., Glackle, W.G., Nonnenmacher, T.F.: Fractional model equation for anomalous diffusion. Phys. A Stat. Mech. Appl. 211(1), 13–24 (1994)

    Article  Google Scholar 

  2. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Publishers, Danbury, CT (2006)

    Google Scholar 

  4. Zaslavsky, G.M.: Hamiltonian Chaos and Fractional Dynamics. Oxford University Press, New York (2005)

    MATH  Google Scholar 

  5. Ortigueira, M.D.: Fractional Calculus for Scientists and Engineers. Springer, New York (2011)

    Book  MATH  Google Scholar 

  6. Meerschaert, M.M., Sikorskii, A.: Stochastic Models for Fractional Calculus, vol. 43. De Gruyter, Berlin (2012)

    MATH  Google Scholar 

  7. Hilfer, R., Anton, L.: Fractional master equations and fractal time random walks. Phys. Rev. E 51, R848–R851 (1995)

    Article  Google Scholar 

  8. Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y., Vinagre Jara, B.M.: Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comput. Phys. 228, 3137–3153 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jafari, H., Tajadodi, H., Matikolai, S.A.: Homotopy perturbation pade technique for solving fractional Riccati differential equations. Int. J. Nonlinear Sci. Numer. Simul. 11, 271–276 (2010)

    MathSciNet  Google Scholar 

  10. Petras, I.: Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  11. Mainardi, F., Gorenflo, R.: On Mittag-Leffler-type functions in fractional evolution processes. J. Comput. Appl. Math. 118(1), 283–299 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gorenflo, R., Loutchko, J., Luchko, Y.: E\(\alpha,\,\beta \) (z) and its derivative. Fract. Calc. Appl. Anal. 5(4), 491–518 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Hilfer, R., Seybold, H.J.: Computation of the generalized Mittag-Leffler function and its inverse in the complex plane. Integral Transforms Spec. Funct. 17(9), 637–652 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang, S., Yu, Y., Wang, H.: Mittag-Leffler stability of fractional-order Hopfield neural networks. Nonlinear Anal. Hybrid Syst. 16, 104–121 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen, J., Zeng, Z., Jiang, P.: Global Mittag-Leffler stability and synchronization of memristor-based fractional-order neural networks. Neural Netw. 51, 1–8 (2014)

    Article  MATH  Google Scholar 

  16. Grebenkov, D.S., Vahabi, M., Bertseva, E., Forró, L., Jeney, S.: Hydrodynamic and subdiffusive motion of tracers in a viscoelastic medium. Phys. Rev. E 88(04), 071 (2013)

    Article  Google Scholar 

  17. Vandebroek, H., Vanderzande, C.: Transient behaviour of a polymer dragged through a viscoelastic medium. J. Chem. Phys. 141(11), 114910 (2014)

    Article  Google Scholar 

  18. Goychuk, I., Kharchenko, V.O., Metzler, R.: Molecular motors pulling cargos in the viscoelastic cytosol: how power strokes beat subdiffusion. Phys. Chem. Chem. Phys. 16(31), 16524–16535 (2014)

    Article  Google Scholar 

  19. Qi, H., Guo, X.: Transient fractional heat conduction with generalized Cattaneo model. Int. J. Heat Mass Transf. 76, 535–539 (2014)

    Article  Google Scholar 

  20. Povstenko, Y.: Fractional radial heat conduction in an infinite medium with a cylindrical cavity and associated thermal stresses. Mech. Res. Commun. 37(4), 436–440 (2010)

    Article  MATH  Google Scholar 

  21. Povstenko, Y.Z.: Fractional Cattaneo-type equations and generalized thermoelasticity. J. Therm. Stress. 34(2), 97–114 (2011)

    Article  Google Scholar 

  22. Brockmann, D., Hufnagel, L., Geisel, T.: The scaling laws of human travel. Nature 439(7075), 462–465 (2006)

    Article  Google Scholar 

  23. Dokoumetzidis, A., MacHeras, P.: Fractional kinetics in drug absorption and disposition processes. J. Pharmacokinet. Pharmacodyn. 36, 165–178 (2009)

    Article  Google Scholar 

  24. Dokoumetzidis, A., Magin, R., MacHeras, P.: Fractional kinetics in multi-compartmental systems. J. Pharmacokinet. Pharmacodyn. 37, 507–524 (2010)

    Article  Google Scholar 

  25. Ionescu, C., Machado, J.T., De Keyser, R., Decruyenaere, J., Struys, M.M.R.F.: Nonlinear dynamics of the patients response to drug effect during general anesthesia. Commun. Nonlinear Sci. Numer. Simul. 20(3), 914–926 (2015)

    Article  Google Scholar 

  26. Ortigueira, M.D., Machado, J.A.T.: Fractional signal processing and applications. Signal Process. 83(11), 2285–2286 (2003)

    Article  Google Scholar 

  27. Metzler, R., Jeon, J.H., Cherstvy, A.G., Barkai, E.: Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16(44), 24128–24164 (2014)

    Article  Google Scholar 

  28. Atkinson, C., Osseiran, A.: Rational solutions for the time-fractional diffusion equation. SIAM J. Appl. Math. 71(1), 92–106 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Winitzki, S.: Uniform approximations for transcendental functions. Comput. Sci. Appl. ICCSA 2003. Springer 780–789 (2003)

  30. Webb, A.: Introduction to Biomedical Imaging. Wiley-IEEE Press, Hoboken (2003)

    Google Scholar 

  31. Stejskal, E.O., Tanner, J.E.: Spin diffusion measuremente spin echoes in the presence of a time-dependent field gradient. J. Chem. Phys. 42, 288–292 (1965)

    Article  Google Scholar 

  32. Tanner, J.E., Stejskal, E.O.: Restricted self-diffusion of protons in colloidal systems by the pulsed-gradient, spin-echo method. J. Chem. Phys. 49(4), 1768–1777 (1968)

    Article  Google Scholar 

  33. Le Bihan, D., Breton, E., Lallemand, D., Grenier, P., Cabanis, E., Laval-Jeantet, M.: MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders. Radiology 161(2), 401–407 (1986)

    Article  Google Scholar 

  34. Niendorf, T., Dijkhuizen, R.M., Norris, D.G., van Lookeren Campagne, M.: Biexponential diffusion attenuation in various states of brain tissue: implications for diffusion-weighted imaging. Magn. Reson. Med. 36(6), 847–857 (1996)

    Article  Google Scholar 

  35. Inglis, B.A., Bossart, E.L., Buckley, D.L., Wirth, E.D., Mareci, T.H.: Visualization of neural tissue water compartments using biexponential diffusion tensor MRI. Magn. Reson. Med. 45(4), 580–587 (2001)

    Article  Google Scholar 

  36. Bennett, K.M., Schmainda, K.M., Bennett, R.T., Rowe, D.B., Lu, H., Hyde, J.S.: Characterization of continuously distributed cortical water diffusion rates with a stretched-exponential model. Magn. Reson. Med. 50(4), 727–734 (2003)

    Article  Google Scholar 

  37. Hall, M.G., Barrick, T.R.: From diffusion-weighted MRI to anomalous diffusion imaging. Magn. Reson. Med. 59(3), 447–455 (2008)

    Article  Google Scholar 

  38. Magin, R.L., Abdullah, O., Baleanu, D., Zhou, X.J.: Anomalous diffusion expressed through fractional order differential operators in the Bloch–Torrey equation. J. Magn. Reson. 190(2), 255–270 (2008)

    Article  Google Scholar 

  39. Palombo, M., Gabrielli, A., De Santis, S., Cametti, C., Ruocco, G., Capuani, S.: Spatio-temporal anomalous diffusion in heterogeneous media by nuclear magnetic resonance. J. Chem. Phys. 135(3), 34504 (2011)

    Article  Google Scholar 

  40. Ingo, C., Magin, R.L., Colon-Perez, L., Triplett, W., Mareci, T.H.: On random walks and entropy in diffusion-weighted magnetic resonance imaging studies of neural tissue. Magn. Reson. Med. 71, 617–627 (2014)

    Article  Google Scholar 

  41. Gorenflo, R., Vivoli, A., Mainardi, F.: Discrete and continuous random walk models for space-time fractional diffusion. Nonlinear Dyn. 38(1), 101–116 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  42. Mittag-Leffler, G.M.: Sur la nouvelle fonction E\(\alpha \) (x). C. R. Acad. Sci. Paris 137, 554–558 (1903)

    MATH  Google Scholar 

  43. Mittag-Leffler, G.: Sur la representation analytique d’une branche uniforme d’une fonction monogene. Acta Math. 29(1), 101–181 (1905)

    Article  MathSciNet  MATH  Google Scholar 

  44. Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag-Leffler functions and their applications. J. Appl. Math. 2011, 298628 (2011). doi:10.1155/2011/298628

    Article  MathSciNet  MATH  Google Scholar 

  45. Ingo, C., Magin, R.L., Parrish, T.B.: New insights into the fractional order diffusion equation using entropy and kurtosis. Entropy 16(11), 5838–5852 (2014)

    Article  Google Scholar 

  46. Ingo, C., Sui, Y., Chen, Y., Parrish, T., Webb, A., Ronen, I.: Parsimonious continuous time random walk models and kurtosis for diffusion in magnetic resonance of biological tissue. Front. Phys. (2015). doi:10.3389/fphy.2015.00011

    Google Scholar 

  47. Goychuk, I., Heinsalu, E., Patriarca, M., Schmid, G., Hänggi, P.: Current and universal scaling in anomalous transport. Phys. Rev. E 73(2), 020101 (2006)

    Article  Google Scholar 

  48. He, Y., Burov, S., Metzler, R., Barkai, E.: Random time-scale invariant diffusion and transport coefficients. Phys. Rev. Lett. 101(058), 101 (2008)

    Google Scholar 

  49. Pollard, H.: The completely monotonic character of the Mittag-Leffler function. Bull. Am. Math. Soc. 54(12), 1115–1116 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  50. Feller, W.: Fluctuation theory of recurrent events. Trans. Am. Math. Soc. 67(1), 98–119 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  51. Van Essen, D.C., Ugurbil, K., Auerbach, E., Barch, D., Behrens, T.E.J., Bucholz, R., Chang, A., Chen, L., Corbetta, M., Curtiss, S.W., et al.: The Human Connectome Project: a data acquisition perspective. NeuroImage 62(4), 2222–2231 (2012)

    Article  Google Scholar 

  52. Zhu, X., Zhang, D.: Efficient parallel Levenberg–Marquardt model fitting towards real-time automated parametric imaging microscopy. PloS One 8(10), e76665 (2013)

    Article  Google Scholar 

  53. Podlubny, I.: The Mittag-Leffler function. (2009). www.mathworks.com/matlabcentral/fileexchange/8738

Download references

Author information

Authors and Affiliations

  1. C.J. Gorter Center for High Field MRI, Department of Radiology, Leiden University Medical Center, Leiden, The Netherlands

    Carson Ingo, Andrew G. Webb & Itamar Ronen

  2. Neuroscience Research Centre, Cardiovascular and Cell Sciences Research Institute, St George’s, University of London, London, UK

    Thomas R. Barrick

Authors
  1. Carson Ingo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas R. Barrick
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrew G. Webb
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Itamar Ronen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carson Ingo.

Additional information

This work has been funded by a grant from the Whitaker International Program of the Institute of International Education.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ingo, C., Barrick, T.R., Webb, A.G. et al. Accurate Padé Global Approximations for the Mittag-Leffler Function, Its Inverse, and Its Partial Derivatives to Efficiently Compute Convergent Power Series. Int. J. Appl. Comput. Math 3, 347–362 (2017). https://doi.org/10.1007/s40819-016-0158-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40819-016-0158-7

Keywords

Search

Navigation