Fractional differential equations with a \(\psi \)-Hilfer fractional derivative


In this paper, we consider a fractional differential equations involving a \(\psi \)-Hilfer fractional derivative. First, we give a correspondence between our problem and a Volterra-type integral equation. Next, sufficient conditions are given to ensure existence and uniqueness of solutions. Then, a numerical approximation method is used to approximate the solution of the problem. For an appropriate choice of the kernel \(\psi \), we recover most of all the previous results on fractional differential equations.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3


  1. Adjabi Y, Jarad F, Baleanu D, Abdeljawad T (2016) On Cauchy problems with Caputo Hadamard fractional derivatives. J Comput Anal Appl 21(4):661–681

    MathSciNet  MATH  Google Scholar 

  2. Almeida R (2020) Functional differential equations involving the \(\psi \)-Caputo fractional derivative. Fractal Fract 4:29.

  3. Almeida R, Malinowska A, Odzijewicz T (2016) Fractional differential equations with dependence on the CaputoKatugampola derivative. Comput Nonlinear Dyn 11(6):061017

    Article  Google Scholar 

  4. Almeida R, Malinowska A, Monteiro M (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

    MathSciNet  Article  Google Scholar 

  5. Burton T (2013) Fractional equations and a theorem of Brouwer–Schauder type. Fixed Point Theor 14(1):91–96

    MathSciNet  Article  Google Scholar 

  6. de Oliveir D, de Oliveira E (2018) Hilfer–Katugampola fractional derivatives. Comput Appl Math 37:3672–3690

    MathSciNet  Article  Google Scholar 

  7. Diethelm K (2010) The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type, lecture notes in mathematics. Springer, Berlin

    Google Scholar 

  8. Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations, North-Holland mathematics studies, vol 207. Elsevier, Amsterdam

    Google Scholar 

  9. Liu S, Li H, Dai Q, Liu J (2016) Existence and uniqueness results for nonlocal integral boundary value problems for fractional differential equations. Adv Differ Equ 2016:122.

    MathSciNet  Article  MATH  Google Scholar 

  10. Sousa J, de Oliveira E (2018) On the \(\psi \)-Hilfer fractional derivative. Commun Nonlinear Sci Numer Simul 60:72–91

    MathSciNet  Article  Google Scholar 

  11. Sousa J, de Oliveira E (2019a) Leibniz type rule: \(\psi \)-Hilfer fractional operator. Commun Nonlinear Sci Numer Simul 77:305–311

    MathSciNet  Article  Google Scholar 

  12. Sousa J, de Oliveira E (2019b) A Gronwall inequality and the Cauchy-type problem by means of \(\psi \)-Hilfer operator. Differ Equ Appl 11(1):87–106

    MathSciNet  MATH  Google Scholar 

  13. Sousa J, Kucche K, de Oliveira E (2019) Stability of \(\psi \)-Hilfer impulsive fractional differential equations. Appl Math Lett 88:73–80

    MathSciNet  Article  Google Scholar 

  14. Sousa J, Frederico G, de Oliveira E (2020a) \(\psi \)-Hilfer pseudo-fractional operator: new results about fractional calculus. Comput Appl Math 39:254.

    MathSciNet  Article  Google Scholar 

  15. Sousa J, Machado J, de Oliveira E (2020b) The \(\psi \)-Hilfer fractional calculus of variable order and its applications. Comput Appl Math 39:296.

    MathSciNet  Article  Google Scholar 

  16. Yang X (2019) General fractional derivatives: theory, methods and applications. CRC Press, New York

    Google Scholar 

  17. Yang X (2020) On traveling-wave solutions for the scaling-law telegraph equations. Therm Sci 24(6B):3861–3868

    Article  Google Scholar 

  18. Yang X, Machado J (2019) A new fractal nonlinear Burgers’ equation arising in the acoustic signals propagation. Math Methods Appl Sci 42(18):7539–7544

    MathSciNet  Article  Google Scholar 

  19. Yang X, Gao F, Srivastava H (2017a) Non-differentiable exact solutions for the nonlinear ODEs defined on fractal sets. Fractals 25(4):1740002

    MathSciNet  Article  Google Scholar 

  20. Yang X, Machado J, Nieto J (2017b) A new family of the local fractional PDEs. Fundamenta Informaticae 151(1–4):63–75

    MathSciNet  Article  Google Scholar 

  21. Yang X, Gao F, Ju Y (2020) General fractional derivatives with applications in viscoelasticity. Academic Press, Cambridge

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Wael Abdelhedi.

Additional information

Publisher's Note

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

Communicated by José Tenreiro Machado.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Abdelhedi, W. Fractional differential equations with a \(\psi \)-Hilfer fractional derivative. Comp. Appl. Math. 40, 53 (2021).

Download citation


  • Fractional differential equations
  • Fractional calculus

Mathematics Subject Classification

  • 26A33
  • 34A08