Research article

On $ \psi $-Hilfer generalized proportional fractional operators

  • Received: 13 July 2021 Accepted: 22 September 2021 Published: 30 September 2021
  • MSC : 26A33, 34A12, 34A43, 34D20

  • In this paper, we introduce a generalized fractional operator in the setting of Hilfer fractional derivatives, the $ \psi $-Hilfer generalized proportional fractional derivative of a function with respect to another function. The proposed operator can be viewed as an interpolator between the Riemann-Liouville and Caputo generalized proportional fractional operators. The properties of the proposed operator are established under some classical and standard assumptions. As an application, we formulate a nonlinear fractional differential equation with a nonlocal initial condition and investigate its equivalence with Volterra integral equations, existence, and uniqueness of solutions. Finally, illustrative examples are given to demonstrate the theoretical results.

    Citation: Ishfaq Mallah, Idris Ahmed, Ali Akgul, Fahd Jarad, Subhash Alha. On $ \psi $-Hilfer generalized proportional fractional operators[J]. AIMS Mathematics, 2022, 7(1): 82-103. doi: 10.3934/math.2022005

    Related Papers:

  • In this paper, we introduce a generalized fractional operator in the setting of Hilfer fractional derivatives, the $ \psi $-Hilfer generalized proportional fractional derivative of a function with respect to another function. The proposed operator can be viewed as an interpolator between the Riemann-Liouville and Caputo generalized proportional fractional operators. The properties of the proposed operator are established under some classical and standard assumptions. As an application, we formulate a nonlinear fractional differential equation with a nonlocal initial condition and investigate its equivalence with Volterra integral equations, existence, and uniqueness of solutions. Finally, illustrative examples are given to demonstrate the theoretical results.



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