On the composition of fractional derivatives with singular kernels

Fractional derivatives are widely used to model memory and hereditary effects, but their composition is difficult due to the nonlocal nature of these operators. This paper studies the composition of fractional derivatives in different settings, including formulations with fixed and variable terminal conditions. It is shown that the classical composition rule D γ 1 D γ 2 Φ = D γ 1 + γ 2 Φ does not hold in general, and the conditions under which it is valid are identified. In particular, the validity of this rule depends on the structure of the operator, the choice of terminal, and the regularity of the function. Several counterexamples illustrate the limitations of existing results, and precise conditions are provided to ensure correct composition. The variable order case is also discussed, with an emphasis on the main difficulties that arise. These results clarify common misconceptions in the literature and provide a reliable framework for the use of fractional derivatives in both theoretical and applied contexts. This study focuses on singular kernel fractional derivatives, with particular emphasis on the Riemann–Liouville and Caputo types. Additionally, it proposes novel concepts for alternative fractional operators, including non-singular kernels and nonlinear scaling operators, to facilitate their advancement and future applications.