[go: up one dir, main page]

Skip to main content
Log in

Generalized Mellin transform and its applications in fractional calculus

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

Abstract

In this paper, we introduce a generalized Mellin transform in the framework of fractional operators with respect to functions. The generalized Mellin transform is derived form the generalized Fourier transform with respect to functions. We prove several fundamental properties of the generalized Mellin transform. Relation of the generalized Mellin transform with various fractional calculus operators is investigated. Generalized Mellin convolutions are defined and applied for solution of a certain differential equation.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others

References

  • Almeida R (2017) A Caputo fractional derivative of a function with respect to another function. Commun Nonlinear Sci Numer Simul 44:460–81

    Article  MathSciNet  Google Scholar 

  • Arran F, Joel R, Jean-Daniel D (2021) On the fractional Laplacian of a function with respect to another function. hal-03318401

  • Atangana A, Bildik N (2013) The use of fractional order derivative to predict the groundwater flow. Math Probl Eng 2013

  • Butzer PL, Kilbas AA, Trujillo JJ (2002) Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J Math Anal Appl 269(1):1–27

    Article  MathSciNet  Google Scholar 

  • Butzer PL, Kilbas AA, Trujillo JJ (2002) Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J Math Anal Appl 270(1):1–5

    Article  MathSciNet  Google Scholar 

  • Debnath L, Bhatta D (2016) Integral transforms and their applications. Chapman and Hall/CRC, New York

    Book  Google Scholar 

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

    Book  Google Scholar 

  • Duffy DG (2004) Transform methods for solving partial differential equations. Chapman and Hall/CRC, New York

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Fahad HM, Fernandez A (2019) On Laplace transforms with respect to functions and their applications to fractional differential equations. arXiv preprint arXiv:1907.04541

  • Fahad HM, Fernandez A, Rehman M, Siddiqi M (2021) Tempered and Hadamard-type fractional calculus with respect to functions. Mediterr J Math 18(4):1–28

    Article  MathSciNet  Google Scholar 

  • Hardy GH, Littlewood JE (1916) Contributions to the theory of the Riemann zeta-function and the theory of the distribution of primes. Acta Math 41(1):119–96

    Article  MathSciNet  Google Scholar 

  • Hilfer R (2000) editor. Applications of fractional calculus in physics. World scientific

  • Jarad F, Abdeljawad T (2020) Generalized fractional derivatives and Laplace transform. Discrete Contin Dyn Syst 13(3):709

    MathSciNet  MATH  Google Scholar 

  • Katugampola UN (2011) New approach to a generalized fractional integral. Appl Math Comput 218(3):860–5

    MathSciNet  MATH  Google Scholar 

  • Katugampola UN (2015) Mellin transforms of generalized fractional integrals and derivatives. Appl Math Comput 257:566–80

    MathSciNet  MATH  Google Scholar 

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

  • Kilbas AA (2001) Hadamard-type fractional calculus. J Korean Math Soc 38

  • Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. North-Holland mathematics studies, 204. Amsterdam: Elsevier Science B.V

  • Kılıçman A, Omran M (2016) Note on fractional Mellin transform and applications. Springerplus 5(1):1–8

    Article  Google Scholar 

  • Osler TJ (1970) The fractional derivative of a composite function. SIAM J Math Anal 1(2):288–93

    Article  MathSciNet  Google Scholar 

  • Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, Amsterdam

    MATH  Google Scholar 

  • Prudnikov AP (2006) Operational calculus and related topics. Chapman and Hall/CRC, New York

    Book  Google Scholar 

  • Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives, translated from the 1987 Russian original. Gordon and Breach, Yverdon

    MATH  Google Scholar 

  • Sousa JV, da C., Capelas De Oliveira E (2019) Leibniz type rule: \(\psi \)-Hilfer fractional operator. Commun Nonlinear Sci Numer Simul 77:305–311

  • Sousa J, da Vanterler C, Capelas De Oliveira E (2018) On the \(\psi \)-Hilfer fractional derivative. Commun Nonlinear Sci Numer Simul 60:72–91

    Article  MathSciNet  Google Scholar 

  • Tarasov VE (2011) Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media. Springer, New York

    Google Scholar 

  • Wheatcraft SW, Meerschaert MM (2008) Fractional conservation of mass. Adv Water Resour 31(10):1377–81

    Article  Google Scholar 

Download references

Acknowledgements

We are very grateful to the anonymous reviewers for their useful comments that led to improvement of the manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mujeeb ur Rehman.

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

Aziz, T., Rehman, M.u. Generalized Mellin transform and its applications in fractional calculus. Comp. Appl. Math. 41, 88 (2022). https://doi.org/10.1007/s40314-022-01802-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-022-01802-9

Keywords

Mathematics Subject Classification