Fractional Calculus in Nuclear Multistep Decay: Analytical Solutions, Existence and Uniqueness Analysis of the Actinium Series

This paper provides a thorough examination of the Actinium radioactive decay series, which converts Uranium-235 into the stable Lead-207 isotope via a succession of alpha, beta, and gamma decays. For the first time, the series is modeled using fractional calculus, employing two innovative analytical methods: the Sumudu Residual Power Series Method (SRPSM) and the Temimi Ansari Method (TAM). The study discusses the well-posedness of the fractional-order model in the Caputo sense within a Banach space setting. These fractional models capture complex, non-ideal decay behaviors more accurately than traditional exponential models. Mathematica is used to do numerical computations for four different Actinium series scenarios. The results are tabulated and visually depicted to show how radionuclide concentrations change over time. The findings demonstrate that SRPSM and TAM effectively simplify the complex differential equations governing nuclear decay, offering enhanced precision and flexibility. This work provides a robust framework for modeling the Actinium series, with potential applications in nuclear physics, radiometric dating, and radiation safety studies.