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

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.

کلیدواژه ها:

Linear (\psi ، \phi)-fractional differential equations ، (\psi ، \phi)-fractional operators ، Laplace transform