Skip to main content
SpringerLink
Log in

On univariate fractional calculus with general bivariate analytic kernels

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

Abstract

Several fractional integral and derivative operators have been defined recently with a bivariate structure, acting on functions of a single variable but with kernels defined using double power series. We propose a general structure to contain all such operators, and establish some important mathematical facts, such as a series formula, a Leibniz rule, a fundamental theorem of calculus, and Laplace and Fourier transform relations, which are applicable to all operators within our general structure.

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

Similar content being viewed by others

Data availability

There is no data associated with this research.

References

  • Abilassan A, Restrepo JE, Suragan D (2023) On a variant of multivariate Mittag–Leffler’s function arising in the Laplace transform method. Integr Transform Spec Funct 34(3):244–260

    Article  MathSciNet  MATH  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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Bazhlekova E (2021) Completely monotone multinomial Mittag–Leffler type functions and diffusion equations with multiple time-derivatives. Fract Calc Appl Anal 24:88–111

    Article  MathSciNet  MATH  Google Scholar 

  • Bazhlekova E, Bazhlekov I (2021) Identification of a space-dependent source term in a nonlocal problem for the general time-fractional diffusion equation. J Comput Appl Math 386:113213

    Article  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, Kürt C, Özarslan MA (2020) A naturally emerging bivariate Mittag–Leffler function and associated fractional-calculus operators. Comput Appl Math 39:200

    Article  MathSciNet  MATH  Google Scholar 

  • Fernandez A, Özarslan MA, Kürt C (2022) A catalogue of semigroup properties for integral operators with Fox–Wright kernel functions. Stud Appl Math 148:1477–1518

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Gorenflo R, Luchko Y (1997) Operational method for solving generalized Abel integral equation of second kind. Integr. Transform Spec Funct 5:47–58

    Article  MathSciNet  MATH  Google Scholar 

  • Gorenflo R, Kilbas AA, Mainardi F, Rogosin SV (2020) Mittag–Leffler functions, related topics and applications, 2nd edn. Springer, Berlin

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Isah SS, Fernandez A, Özarslan MA (2023) On bivariate fractional calculus with general univariate analytic kernels. Chaos Solitons Fract 171:113495

    Article  MathSciNet  Google Scholar 

  • Jleli M, Kirane M, Samet B (2019) A derivative concept with respect to an arbitrary kernel and applications to fractional calculus. Math Methods Appl Sci 42:137–160

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Kiryakova V (2010) The special functions of fractional calculus as generalized fractional calculus operators of some basic functions. Comput Math Appl 59(3):1128–1141

    MathSciNet  MATH  Google Scholar 

  • Kiryakova V (2019) Fractional calculus of some “new’’ but not new special functions: \(K\)-, multi-index-, and \(S\)-analogues. AIP Conf Proc 2172:050008

    Article  Google Scholar 

  • Kochubei AN (2011) General fractional calculus, evolution equations, and renewal processes. Integr Eqn Oper Theory 71:583–600

    Article  MathSciNet  MATH  Google Scholar 

  • Kürt C, Fernandez A, Özarslan MA (2023) Two unified families of bivariate Mittag–Leffler functions. Appl Math Comput 443:127785

    MathSciNet  MATH  Google Scholar 

  • Ledesma CT, da Sousa JVC, Cruz AM (2021) Hardy–Littlewood type inequality for \(\psi \)-Riemann–Liouville fractional integrals. Hal Open Sci hal-03189239

  • Luchko Y (2021) General fractional integrals and derivatives with the Sonine kernels. Mathematics 9:594

    Article  Google Scholar 

  • Luchko Y (2021) General fractional integrals and derivatives of arbitrary order. Symmetry 13(5):755

    Article  Google Scholar 

  • Martínez-Fuentes O, Meléndez-Vázquez F, Fernández-Anaya G, Gómez-Aguilar JF (2021) Analysis of fractional-order nonlinear dynamic systems with general analytic kernels: Lyapunov stability and inequalities. Mathematics 9(17):2084

    Article  Google Scholar 

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

    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, Fernandez A (2022) On the fractional calculus of multivariate Mittag–Leffler functions. Int J Comput Math 99(2):247–273

    Article  MathSciNet  MATH  Google Scholar 

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

    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 

  • Raina RK (2005) On generalized Wright’s hypergeometric functions and fractional calculus operators. East Asian Math J 21(2):191–203

    MATH  Google Scholar 

  • Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives: theory and applications. Gordon & Breach, Yverdon

    MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  • Srivastava HM, Harjule P, Jain R (2015) A general fractional differential equation associated with an integral operator with the \(H\)-function in the kernel. Russ J Math Phys 22(1):112–126

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Tuan VK (2020) Fractional integro-differential equations in Wiener spaces. Fract Calc Appl Anal 23(5):1300–1328

    Article  MathSciNet  MATH  Google Scholar 

  • Zhao D, Luo M (2019) Representations of acting processes and memory effects: general fractional derivative and its application to theory of heat conduction with finite wave speeds. Appl Math Comput 346:531–544

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Sunday Simon Isah, Arran Fernandez & Mehmet Ali Özarslan

Authors
  1. Sunday Simon Isah
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arran Fernandez
    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.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Isah, S.S., Fernandez, A. & Özarslan, M.A. On univariate fractional calculus with general bivariate analytic kernels. Comp. Appl. Math. 42, 228 (2023). https://doi.org/10.1007/s40314-023-02363-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-023-02363-1

Keywords

Mathematics Subject Classification

Search

Navigation