A novel representation of the mth level fractional derivative and its Laplace transforms

This paper presents a new formulation of the second level fractional derivative that aligns more closely with the well-established Hilfer fractional derivative framework. Building on this formulation, we extend the concept to define a broader class of mth level fractional derivatives, designed to incorporate various levels of fractional differentiation within a unified representation. Appropriate functional spaces are identified to ensure the well-posedness of these operators. A detailed comparison between the proposed and existing definitions of the second level fractional derivative is conducted, highlighting differences in parameter constraints and flexibility. The new formulation proves greater flexibility than the existing approach, as it imposes fewer restrictions on the parameters. Furthermore, we explore essential properties of the new operators and employ the Laplace transform to solve selected Cauchy problems involving the proposed second level fractional derivative. These solutions naturally reduce to well-known first-level fractional models under specific parameter choices. Although our primary focus is on the second-level derivative to facilitate interpretation and foundational development, the generalization to the mth level fractional derivative is straightforward and systematically addressed. Overall, the results provide a refined and adaptable framework for modeling and solving fractional differential equations.