Skip to main content
SpringerLink
Log in

A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

We define an analogue of the classical Mittag-Leffler function which is applied to two variables, and establish its basic properties. Using a corresponding single-variable function with fractional powers, we define an associated fractional integral operator which has many interesting properties. The motivation for these definitions is twofold: firstly, their link with some fundamental fractional differential equations involving two independent fractional orders, and secondly, the fact that they emerge naturally from certain applications in bioengineering.

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

Similar content being viewed by others

References

  • Andualem M, Debalkie B, Suthar DL (2019) A study on generalized multivariable Mittag–Leffler function via generalized fractional calculus operators. J Math 2019:1–7. https://doi.org/10.1155/2019/9864737

    Article  MathSciNet  MATH  Google Scholar 

  • Atangana A, Baleanu D (2016) New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Therm Sci 20(2):763–769

    Google Scholar 

  • Avci I, Mahmudov NI (2020) Numerical solutions for multi-term fractional order differential equations with fractional Taylor operational matrix of fractional integration. Mathematics 8(1):96

    Google Scholar 

  • Baleanu D, Fernandez A (2018) On some new properties of fractional derivatives with Mittag–Leffler kernel. Commun Nonlinear Sci Numer Simul 59:444–462

    MathSciNet  Google Scholar 

  • Baleanu D, Fernandez A (2019) On fractional operators and their classifications. Mathematics 7(9):830

    Google Scholar 

  • Baleanu D, Jajarmi A, Sajjadi SS, Mozyrska D (2019) A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator. Chaos 29:083127

    MathSciNet  MATH  Google Scholar 

  • Baleanu D, Fernandez A, Akgül A (2020) On a fractional operator combining proportional and classical differintegrals. Mathematics 8(3):360

    Google Scholar 

  • Bazhlekova E, Dimovski I (2013) Exact solution for the fractional cable equation with nonlocal boundary conditions. Cent Eur J Phys 11(10):1304–1313

    Google Scholar 

  • Bonfanti A, Fouchard J, Khalilgharibi N, Charras G, Kabla A (2019) A unified rheological model for cells and cellularised materials. R Soc Open Sci 7:190920. https://doi.org/10.1098/rsos.190920

    Article  Google Scholar 

  • Djida J-D, Fernandez A, Area I (2020) Well-posedness results for fractional semi-linear wave equations. Discret Contin Dyn Syst B 25(2):569–597

    MathSciNet  MATH  Google Scholar 

  • D’Ovidio M, Polito F (2018) Fractional diffusion-telegraph equations and their associated stochastic solutions. Theory Probab Appl 62(4):552–574

    MathSciNet  MATH  Google Scholar 

  • Fernandez A, Özarslan MA, Baleanu D (2019) On fractional calculus with general analytic kernels. Appl Math Comput 354:248–265

    MathSciNet  MATH  Google Scholar 

  • Fernandez A, Baleanu D, Srivastava HM (2019) Series representations for models of fractional calculus involving generalised Mittag–Leffler functions. Commun Nonlinear Sci Numer Simul 67:517–527

    MathSciNet  Google Scholar 

  • Ferreira M, Rodrigues MM, Vieira N (2018) First and second fundamental solutions of the time-fractional telegraph equation with Laplace or Dirac operators. Adv Appl Clifford Algebras 28:42

    MathSciNet  MATH  Google Scholar 

  • Garg M, Manohar P, Kalla SL (2013) A Mittag–Leffler-type function of two variables. Integral Transforms Spec Funct 24(11):934–944

    MathSciNet  MATH  Google Scholar 

  • Garra R, Garrappa R (2018) The Prabhakar or three parameter Mittag–Leffler function: theory and application. Commun Nonlinear Sci Numer Simul 56:314–329

    MathSciNet  Google Scholar 

  • Garra R, Gorenflo R, Polito F, Tomovski Ž (2014) Hilfer–Prabhakar derivatives and some applications. Appl Math Comput 242:576–589

    MathSciNet  MATH  Google Scholar 

  • Garrappa R (2016) Grünwald–Letnikov operators for fractional relaxation in Havriliak–Negami models. Commun Nonlinear Sci Numer Simul 38:178–191

    MathSciNet  Google Scholar 

  • Garrappa R, Mainardi F, Maione G (2016) Models of dielectric relaxation based on completely monotone functions. Fraction Calc Appl Anal 19(5):1105–1160

    MathSciNet  MATH  Google Scholar 

  • Giusti A, Colombaro I, Garra R, Garrappa R, Polito F, Popolizio M, Mainardi F (2020) A practical guide to Prabhakar fractional calculus. Fraction Calc Appl Anal 23(1):9–54

    MathSciNet  MATH  Google Scholar 

  • Gorenflo R, Kilbas AA, Mainardi F, Rogosin SV (2016) Mittag–Leffler functions, related topics and applications. Springer, Berlin

    MATH  Google Scholar 

  • Hilfer R, Luchko Y (2019) Desiderata for fractional derivatives and integrals. Mathematics 7:149

    Google Scholar 

  • Huseynov IT, Ahmadova A, Fernandez A, Mahmudov NI Explicit analytic solutions of incommensurate fractional differential equation systems (under review)

  • Kaczorek T (2010) Positive linear systems with different fractional orders. Bull Pol Acad Sci Tech Sci 58(3):453–458

    MathSciNet  MATH  Google Scholar 

  • Kilbas AA, Saigo M, Saxena RK (2004) Generalized Mittag–Leffler function and generalized fractional calculus operators. Integral Transf Spec Funct 15(1):31–49

    MathSciNet  MATH  Google Scholar 

  • Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Kumar D, Singh J, Tanwar K, Baleanu D (2019) A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag–Leffler laws. Int J Heat Mass Transf 138:1222–1227

    Google Scholar 

  • Kürt C, Özarslan MA, Fernandez A (2020) On a certain bivariate Mittag–Leffler function analysed from a fractional-calculus point of view. Math Methods Appl Sci 20:1–21. https://doi.org/10.1002/mma.6324

    Article  Google Scholar 

  • Lavault C (2018) Integral representations and asymptotic behaviour of a Mittag- -Leffler type function of two variables. Adv Oper Theory 3(2):40–48

    MathSciNet  MATH  Google Scholar 

  • Luchko Y (1999) Operational method in fractional calculus. Fraction Calc Appl Anal 2(4):463–488

    MathSciNet  MATH  Google Scholar 

  • Luchko Y, Gorenflo R (1999) An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math Vietnam 24(2):207–233

    MathSciNet  MATH  Google Scholar 

  • Mainardi F (2014) On some properties of the Mittag–Leffler function \(E_{\alpha }(-t^{\alpha })\), completely monotone for \(t>0\) with \(0<\alpha <1\). Discret Contin Dyn Sys B 19(7):2267–2278

    MathSciNet  MATH  Google Scholar 

  • Mathai AM, Haubold HJ (2008) Mittag–Leffler functions and fractional calculus. Spec Funct Appl Sci 20:79–134

    Google Scholar 

  • Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York

    MATH  Google Scholar 

  • Mittag-Leffler MG (1903) Sur la nouvelle fonction \(E(x)\). Comptes Rendus Acad Sci 137:554–558

    MATH  Google Scholar 

  • Momani S, Odibat Z (2006) Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method. Appl Math Comput 177(2):488–494

    MathSciNet  MATH  Google Scholar 

  • Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, San Diego

    MATH  Google Scholar 

  • Özarslan MA (2014) On a singular integral equation including a set of multivariate polynomials suggested by Laguerre polynomials. Appl Math Comput 229:350–358

    MathSciNet  MATH  Google Scholar 

  • Özarslan MA, Kürt C (2017) On a double integral equation including a set of two variables polynomials suggested by Laguerre polynomials. J Comput Anal Appl 22(7):1198–1207

    MathSciNet  Google Scholar 

  • Özarslan MA, Kürt C (2019) Bivariate Mittag–Leffler functions arising in the solutions of convolution integral equation with 2D-Laguerre–Konhauser polynomials in the kernel. Appl Math Comput 347:631–644

    MathSciNet  MATH  Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, San Diego

    MATH  Google Scholar 

  • Pollard H (1948) The completely monotonic character of the Mittag–Leffler function \(E_{\alpha }(-x)\). Bull Am Math Soc 54(12):1115–1116

    MATH  Google Scholar 

  • Prabhakar TR (1970) On a set of polynomials suggested by Laguerre polynomials. Pac J Math 35(1):213–219

    MathSciNet  MATH  Google Scholar 

  • Prabhakar TR (1971) A singular integral equation with a generalized Mittag Leffler function in the kernel. Yokohama Math J 19:7–15

    MathSciNet  MATH  Google Scholar 

  • Sakamoto K, Yamamoto M (2011) Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J Math Anal Appl 382:426–447

    MathSciNet  MATH  Google Scholar 

  • Samko SG, Kilbas AA, Marichev OI (1987) Fractional integrals and derivatives: theory and applications. Taylor & Francis, London [orig. ed. in Russian; Nauka i Tekhnika, Minsk, 1987]

    MATH  Google Scholar 

  • Sandev T (2017) Generalized Langevin equation and the Prabhakar derivative. Mathematics 5(4):66

    MATH  Google Scholar 

  • Saxena RK, Kalla SL, Saxena R (2011) Multivariate analogue of generalised Mittag–Leffler function. Integral Transforms Spec Funct 22(7):533–548

    MathSciNet  MATH  Google Scholar 

  • Srivastava HM, Daoust MC (1972) A note on the convergence of Kampé de Fériet’s Double Hypergeometric series. Math Nach 53:151–159

    MATH  Google Scholar 

  • Suthar DL, Amsalu H, Godifey K (2019) Certain integrals involving multivariate Mittag–Leffler function. J Inequal Appl 2019:208

    MathSciNet  Google Scholar 

  • Teodoro GS, Tenreiro Machado JA, de Oliveira EC (2019) A review of definitions of fractional derivatives and other operators. J Comput Phys 388:195–208

    MathSciNet  Google Scholar 

  • Wang JR, Zhou Y, O’Regan D (2018) A note on asymptotic behaviour of Mittag–Leffler functions. Integral Transforms Spec Funct 29(2):81–94

    MathSciNet  MATH  Google Scholar 

  • Whittaker ET, Watson GN (1927) A course of modern analysis, 4th edn. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Yusuf A, Inc M, Aliyu AI, Baleanu D (2018) Efficiency of the new fractional derivative with nonsingular Mittag–Leffler kernel to some nonlinear partial differential equations. Chaos Solitons Fractals 116:220–226

    MathSciNet  Google Scholar 

  • Zhao D, Sun H (2019) Anomalous relaxation model based on the fractional derivative with a Prabhakar-like kernel. Z Angew Math Phys 70:42

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Alessandra Bonfanti and Louis Kaplan for stimulating discussions about interdisciplinary applications.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, via Mersin 10, Famagusta, Northern Cyprus, Turkey

    Arran Fernandez, Cemaliye Kürt & Mehmet Ali Özarslan

Authors
  1. Arran Fernandez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cemaliye Kürt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mehmet Ali Özarslan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arran Fernandez.

Additional information

Communicated by Roberto Garrappa.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fernandez, A., Kürt, C. & Özarslan, M.A. A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators. Comp. Appl. Math. 39, 200 (2020). https://doi.org/10.1007/s40314-020-01224-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-020-01224-5

Keywords

Search

Navigation