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

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.