General solution of Bateman equations using Cauchy products and the Theory of Divided Differences

Abstract

The general solution of Bateman Equations is simplified and algorithmically improved in the present work. Such solution allows modeling successive transformations of nuclides due to radioactive decay and transmutation, considering the most general case where there are repeated nuclides or identical effective decay constants. It has been previously expressed in terms of nested sums and involving the Kronecker delta, which is a highly disadvantageous way in computational terms. An alternative procedure to overcome such disadvantages is developed in the present paper using Cauchy products and the theory of Divided Differences. Unlike the other approaches, this new procedure does not require specifying the position where repeated nuclides or repeated effective decay constants appear, simplifying considerably its computational implementation. Python codes and a set of numerical experiments are provided, where the new approach is compared with the original solution, showing that the proposed computational implementation is faster and simplified in a considerable way.