Extended higher order operators with the generalized cardi-nal sine kernel: analytical study and numerical simulations

Abstract

The extended higher-order fractional derivative operators with the generalized cardinal sine kernel are generalizations of classical higher-order operators, defined in broader function spaces and involving singular kernels. Integration by parts was employed to define these extended derivatives, and the corresponding integral operators were subsequently obtained by solving related fractional differential equations using the Laplace transform. Several properties of the new extended operators are discussed; in particular, it is shown that the extended integral operator serves as the inverse of the extended derivative. Multiple examples are provided to demonstrate the efficiency of the proposed operators. On one hand, it is shown that the extended Caputo-type derivative is well-defined for certain functions where the original derivative fails. On the other hand, solutions to fractional differential equations that are not solvable using traditional derivatives are successfully obtained. A fractional Duffing model involving the extended derivative is considered, and numerical simulations are presented. The proposed extended operators effectively address limitations related to the initialization of fractional differential equations, and the numerical results confirm the efficiency and applicability of the new approach.