Novel $(\psi,\phi)$-fractional operators with exponential kernels: properties and applications to linear differential equations

Sadek, Lakhlifa

doi:10.22034/cmde.2025.62416.2748

Novel $(\psi,\phi)$-fractional operators with exponential kernels: properties and applications to linear differential equations

Document Type : Research Paper

Author

Lakhlifa Sadek

1. Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai 602105, Tamil Nadu, India. 2. Department of Mathematics, Faculty of Sciences and Technology, BP 34. Ajdir 32003 Al-Hoceima, Abdelmalek Essaadi University, Tetouan, Morocco.

10.22034/cmde.2025.62416.2748

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 integrals, 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.

Keywords