On a Generalized Class of Nonsingular Kernel Operators and Their Singular Kernel Extensions: Useful Modeling Insights

Abstract Some possible definitions of fractional derivative operators with nonsingular analytic kernels have been introduced. In this paper, we propose a new generalized class of fractional derivative operators of Caputo-type with nonsingular analytic kernels which includes some known operators as special cases. We demonstrate a relationship between the fractional derivative operators of the proposed generalized class and the Riemann–Liouville (RL) fractional integral operator. We also, using this relationship, introduce the corresponding fractional integral operators. Then, mainly, we provide extensions to the fractional derivative operators of the proposed generalized class that display integrable singular kernels. The extended fractional derivative operators provide useful insights regarding the modeling issue so that the initialization problem can be overcome. Finally, we discuss some basic properties of the proposed operators that are expected to be widely used in fractional calculus.