Introducing novel $(\psi,\phi)$-fractional operators advances in fractional calculus

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Sciences and Technology, BP 34. Ajdir 32003 Al-Hoceima, Abdelmalek Essaadi University, Tetouan, Morocco.

Abstract

This study introduces novel generalized fractional derivatives known as $(\psi,\phi)$-fractional derivatives of the Riemann-Liouville and Caputo types, each incorporating exponential function kernels. These new operators offer distinct advantages, including a semi-group property and a seamless extension of the Riemann-Liouville (RL-FD) and Caputo fractional derivatives (C-FD), as well as integrals (RL-FI). We explore the Laplace transform of these $(\psi,\phi)$-fractional derivatives and $(\psi,\phi)$-integral, leveraging them to address linear $(\psi,\phi)$-fractional differential equations. Moreover, these fractional operators are general to classical fractional operators, cotangent fractional operators, and generalized proportional operators.

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Articles in Press, Accepted Manuscript
Available Online from 19 May 2025
  • Receive Date: 09 July 2024
  • Revise Date: 29 April 2025
  • Accept Date: 11 May 2025