On the Rigorous Correspondence Between Operator Fractional Powers and Fractional Derivatives via the Sonine Kernel

Traditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a universal framework for defining the fractional calculus operators using the generalized fractional calculus with the Sonine kernel. In this framework, we prove that the α-th power of a differential operator corresponds precisely to the α-th fractional derivative, ensuring both accuracy and consistency. The relationship between the fractional power operators and fractional calculus is not arbitrary, it must be determined by the specific operator form and the initial conditions. Furthermore, we provide operator representations of commonly used fractional derivatives and illustrate their applications with examples of fractional power operators in physical fractal spaces. A superposition principle is also introduced to simplify fractional differential equations with non-integer exponents by transforming them into zero-initial-condition problems. This framework offers new insights into the commutative properties of fractional calculus operators and their relevance in the study of fractal structures.