Skip to main content
SpringerLink
Log in

On the importance of conjugation relations in fractional calculus

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

Abstract

It is useful to understand how the various operators of fractional calculus relate to each other, especially relations between newly defined operators and classical well-studied ones. This paper focuses on an important type of such relationship, namely conjugation relations, also called transmutation relations. We define a general abstract setting in which such relations are relevant, and indicate how they can be used to prove many results easily in general settings such as fractional calculus with respect to functions and weighted fractional calculus.

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

References

  • Agrawal OP (2012) Generalized multiparameters fractional variational calculus. Int J Differ Equ 2012:521750. https://doi.org/10.1155/2012/521750

    Article  MathSciNet  MATH  Google Scholar 

  • Agrawal OP (2012) Some generalized fractional calculus operators and their applications in integral equations. Fract Calc Appl Anal 15(4):700–711

    Article  MathSciNet  Google Scholar 

  • Almeida R (2017) A Caputo fractional derivative of a function with respect to another function. Commun Nonlinear Sci Numer Simul 44:460–481

    Article  MathSciNet  Google Scholar 

  • Almeida R, Malinowska AB, Monteiro MTT (2018) Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications. Math Methods Appl Sci 41(1):336–352

    Article  MathSciNet  Google Scholar 

  • Al-Refai M (2020) On weighted Atangana–Baleanu fractional operators. Adv Differ Equ 2020:3

    Article  MathSciNet  Google Scholar 

  • Al-Saqabi BN, Kiryakova VS (1998) Explicit solutions of fractional integral and differential equations involving Erdélyi-Kober operators. Appl Math Comput 95:1–13

    Article  MathSciNet  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  Google Scholar 

  • da Sousa CJV, de Oliveira EC (2018) On the \(\Psi \)-Hilfer fractional derivative. Commun Nonlinear Sci Numer Simul 60:72–91

    Article  MathSciNet  Google Scholar 

  • Diethelm K (2010) The analysis of fractional differential equations. Springer, Berlin

    Book  Google Scholar 

  • Erdelyi A (1964) An integral equation involving Legendre functions. J Soc Ind Appl Math 12(1):15–30

    Article  MathSciNet  Google Scholar 

  • Fahad HM, Fernandez A (2021) Operational calculus for Riemann–Liouville fractional calculus with respect to functions and the associated fractional differential equations. Fract Calc Appl Anal 24(2):518–540

    Article  MathSciNet  Google Scholar 

  • Fahad HM, Fernandez A (2021) Operational calculus for Caputo fractional calculus with respect to functions and the associated fractional differential equations. Appl Math Comput 409:126400

    Article  MathSciNet  Google Scholar 

  • Fahad HM, Ur Rehman M, Fernandez A (2021) On Laplace transforms with respect to functions and their applications to fractional differential equations. Math Methods Appl Sci. https://doi.org/10.1002/mma.7772

    Article  MATH  Google Scholar 

  • Fahad HM, Fernandez A, Ur Rehman M, Siddiqi M (2021) Tempered and Hadamard-type fractional calculus with respect to functions. Mediterr J Math 18:143

    Article  MathSciNet  Google Scholar 

  • Fernandez A, Fahad HM (2022) Weighted fractional calculus: a general class of operators. Fractal Fraction 6:208

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Hilfer R (ed) (2000) Applications of fractional calculus in physics. World Scientific, Singapore

    MATH  Google Scholar 

  • Hilfer R, Luchko Y (2019) Desiderata for fractional derivatives and integrals. Mathematics 7(2):149

    Article  Google Scholar 

  • Hilfer R, Luchko Y, Tomovski Ž (2009) Operational method for solution of fractional differential equations with generalized Riemann-Liouville fractional derivative. Fract Calc Appl Anal 12(3):299–318

    MathSciNet  MATH  Google Scholar 

  • Jarad F, Abdeljawad T, Shah K (2020) On the weighted fractional operators of a function with respect to another function. Fractals 28(08):2040011

    Article  Google Scholar 

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

    MATH  Google Scholar 

  • Kiryakova VS, Al-Saqabi BN (1997) Transmutation method for solving Erdélyi-Kober fractional differintegral equations. J Math Anal Appl 211:347–364

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Kolokoltsov VN (2019) The probabilistic point of view on the generalized fractional partial differential equations. Fract Calc Appl Anal 22(3):543–600

    Article  MathSciNet  Google Scholar 

  • Liu JG, Yang XJ, Feng YY, Geng LL (2021) Fundamental results to the weighted Caputo-type differential operator. Appl Math Lett 121:107421

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Luchko Y (2020) Fractional derivatives and the fundamental theorem of fractional calculus. Fract Calc Appl Anal 23(4):939–966

    Article  MathSciNet  Google Scholar 

  • Luchko Y (2021) Operational calculus for the general fractional derivative and its applications. Fract Calc Appl Anal 24(2):338–375

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Osler TJ (1970) Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM J Appl Math 18(3):658–674

    Article  MathSciNet  Google Scholar 

  • Oumarou CMS, Fahad HM, Djida J-D, Fernandez A (2021) On fractional calculus with analytic kernels with respect to functions. Comput Appl Math 40:244

    Article  MathSciNet  Google Scholar 

  • Restrepo JE, Ruzhansky M, Suragan D (2021) Explicit solutions for linear variable-coefficient fractional differential equations with respect to functions. Appl Math Comput 403:126177

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  • Shishkina E, Sitnik S (2020) Transmutations, singular and fractional differential equations with applications to mathematical physics. Academic Press, London

    MATH  Google Scholar 

  • Zaky MA, Hendy AS, Suragan D (2022) A note on a class of Caputo fractional differential equations with respect to another function. Math Comput Simul 196:289–295

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Arran Fernandez & Hafiz Muhammad Fahad

Authors
  1. Arran Fernandez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hafiz Muhammad Fahad
    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 conflicts of interest.

Additional information

Communicated by Vasily E. Tarasov.

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., Fahad, H.M. On the importance of conjugation relations in fractional calculus. Comp. Appl. Math. 41, 246 (2022). https://doi.org/10.1007/s40314-022-01925-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-01925-z

Keywords

Mathematics Subject Classification

Search

Navigation