Skip to main content
SpringerLink
Log in

Do the Mittag–Leffler Functions Preserve the Properties of Their Matrix Arguments?

  • Conference paper
  • First Online:
Fractional Differential Equations (INDAM 2021)

Part of the book series: Springer INdAM Series ((SINDAMS,volume 50))

  • 293 Accesses

Abstract

The matrix Mittag–Leffler (ML) functions are receiving great attention at the moment. As a matter of fact, in many applications, the matrix argument has special features and/or structure and is important to know if the application of ML functions preserves them. We collect results here that can help researchers working with this topic and who usually struggle to find them because they are scattered around or not explicitly derived for the special case of ML functions. Furthermore, the treatment is also suitable for non-experts in linear algebra, giving adequate references to the appropriate literature. In particular, nonnegativity and circulant structure are addressed, which may be of great interest in the analysis of systems of fractional differential equations or in the context of graph theory.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    www.mathworks.com/matlabcentral/fileexchange/66272-mittag-leffler-function-with-matrix-arguments.

References

  1. Arrigo, F., Durastante, F.: Mittag–Leffler functions and their applications in network science. SIAM J. Matrix Anal. Appl. 42(4), 1581–1601 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  2. Benzi, M., Bertaccini, D., Durastante, F., Simunec, I.: Non-local network dynamics via fractional graph Laplacians. J. Complex Netw. 8(3), cnaa017 (2020)

    Google Scholar 

  3. Bharali, G., Holtz, O.: Functions preserving nonnegativity of matrices. SIAM J. Matrix Anal. Appl. 30(1), 84–101 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bianchi, D., Donatelli, M., Durastante, F., Mazza, M.: Compatibility, embedding and regularization of non-local random walks on graphs. J. Math. Anal. Appl. 511(1), 126020 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bini, D.A., Massei, S., Meini, B.: On functions of quasi-Toeplitz matrices. Sb. Math. 208(11), 1628 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Davis, P.J.: Circulant Matrices. John Wiley&Sons, New York (1979)

    MATH  Google Scholar 

  7. Diaz-Diaz, F., Estrada, E.: Time and space generalized diffusion equation on graph/networks. Chaos, Solitons Fractals 156, 111791 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  8. Diethelm, K.: The analysis of fractional differential equations. Lecture Notes in Mathematics, vol. 2004. Springer, Berlin (2010)

    Google Scholar 

  9. Dubourdieu, M.J.: Sur un théorème de M. S. Bernstein relatif à la transformation de Laplace-Stieltjes. Comp. Math. 7, 96–111 (1940)

    MathSciNet  MATH  Google Scholar 

  10. Elagan, S.: On the invalidity of semigroup property for the Mittag–Leffler function with two parameters. J. Egypt. Math. Soc. 24(2), 200–203 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  11. Esmaeili, S., Garrappa, R.: A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation. Int. J. Comput. Math. 92(5), 980–994 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Estrada, E.: Generalized walks-based centrality measures for complex biological networks. J. Theor. Biol. 263(4), 556–565 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Estrada, E.: Networks virology. Focus on Covid-19. Modeling and Simulation in Science, Engineering and Technology (2021)

    Google Scholar 

  14. Estrada, E., Knight, P.: A First Course in Network Theory. OUP, Oxford (2015)

    MATH  Google Scholar 

  15. Garrappa, R.: Exponential integrators for time–fractional partial differential equations. Eur. Phys. J. Special Topics 222(8), 1915–1927 (2013)

    Article  Google Scholar 

  16. Garrappa, R.: A family of Adams exponential integrators for fractional linear systems. Comput. Math. Appl. 66(5), 717–727 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Garrappa, R., Politi, T., Popolizio, M.: Numerical approximation of the Mittag–Leffler function for large sparse low rank matrices. In preparation (2022)

    Google Scholar 

  18. Garrappa, R., Popolizio, M.: Computing the matrix Mittag–Leffler function with applications to fractional calculus. J. Sci. Comput. 77(1), 129–153 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Golub, G., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  20. Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.: Mittag–Leffler Functions. Theory and Applications. Springer Monographs in Mathematics. Springer, Berlin (2014)

    Google Scholar 

  21. Grindrod, P., Higham, D.J.: A dynamical systems view of network centrality. Proc. R. Soc. A: Math. Phys. Eng. Sci. 470(2165), 20130835 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Higham, N.J.: Functions of matrices. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)

    Google Scholar 

  23. Higham, N.J., Mackey, D.S., Mackey, N., Tisseur, F.: Functions preserving matrix groups and iterations for the matrix square root. SIAM J. Matrix Anal. Appl. 26(3), 849–877 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Holme, P., Saramoki, J.: Temporal networks. Phys. Rep. 519(3), 97–125 (2012)

    Article  Google Scholar 

  25. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  26. Lopez, L., Pellegrino, S.F.: A spectral method with volume penalization for a nonlinear peridynamic model. Int. J. Numer. Methods Eng. 122(3), 707–725 (2021)

    Article  MathSciNet  Google Scholar 

  27. Luenberger, D.G.: Introduction to Dynamic Systems: Theory, Models, and Applications. Wiley, London (1979)

    MATH  Google Scholar 

  28. Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)

    Book  MATH  Google Scholar 

  29. Micchelli, C.A., Willoughby, R.A.: On functions which preserve the class of Stieltjes matrices. Linear Algebra Appl. 23, 141–156 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  30. Peng, J., Li, K.: A note on property of the Mittag–Leffler function. J. Math. Anal. Appl. 370(2), 635–638 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Popolizio, M.: Numerical solution of multiterm fractional differential equations using the matrix Mittag–Leffler functions. Mathematics 6(1), 7 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Popolizio, M.: On the matrix Mittag–Leffler function: theoretical properties and numerical computation. Mathematics 7(12), 1140 (2019)

    Article  Google Scholar 

  33. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon (1993)

    MATH  Google Scholar 

  34. Varga, R.S.: Nonnegatively posed problems and completely monotonic functions. Linear Algebra Appl. 1(3), 329–347 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  35. Weaver, J.R.: Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors. Am. Math. Monthly 92(10), 711–717 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author wishes to thank the anonymous referees for the accurate review of the manuscript and for the suggestions which helped to improve the presentation.

The work of M. Popolizio is partially supported by PRIN2017-MIUR project.

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Elettrica e dell’Informazione, Politecnico di Bari, Bari, Italy

    Marina Popolizio

  2. Member of the INdAM Research Group GNCS, Roma, Italy

    Marina Popolizio

Authors
  1. Marina Popolizio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marina Popolizio .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Salerno, Fisciano, Salerno, Italy

    Angelamaria Cardone

  2. Department of Science and High Technology, University of Insubria, Como, Italy

    Marco Donatelli

  3. Department of Mathematics, University of Pisa, Pisa, Italy

    Fabio Durastante

  4. Department of Mathematics, University of Bari Aldo Moro, Bari, Italy

    Roberto Garrappa

  5. Department of Science and High Technology, University of Insubria, Como, Italy

    Mariarosa Mazza

  6. Department of Electrical and Information Engineering, Polytechnic University of Bari, Bari, Italy

    Marina Popolizio

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Popolizio, M. (2023). Do the Mittag–Leffler Functions Preserve the Properties of Their Matrix Arguments?. In: Cardone, A., Donatelli, M., Durastante, F., Garrappa, R., Mazza, M., Popolizio, M. (eds) Fractional Differential Equations. INDAM 2021. Springer INdAM Series, vol 50. Springer, Singapore. https://doi.org/10.1007/978-981-19-7716-9_5

Download citation

Publish with us

Policies and ethics

Search

Navigation