Dynamic-memory fractional calculus via generator-based memory construction: operational theory, semigroup structure, and applications

Abstract: Most generalized fractional operators rely on prescribed memory kernels, restricting hereditary behavior to predefined forms and limiting flexibility in modeling diverse memory effects. Motivated by these limitations, this paper develops a generator-based framework for fractional calculus in which memory laws are systematically generated through a dynamic memory generator in the Laplace domain. The resulting construction produces dynamic-memory kernels via inverse Laplace transforms, leading to generalized dynamic-memory fractional integrals together with Riemann--Liouville and Caputo dynamic-memory fractional derivatives. Fundamental analytical properties are established, including inverse relations, composition formulas, admissibility conditions, semigroup structures, and consistency principles. In addition, a unified convolution-symbol operational calculus and generalized dynamic-memory Mittag--Leffler functions are developed. Unlike fixed-kernel formulations, the proposed framework can generate singular, nonsingular, tempered, logarithmic, oscillatory, and multiscale memory behaviors within a single analytical setting. Numerous classical and modern fractional operators are recovered as special cases, demonstrating the unifying capability and flexibility of the developed theory.