General solutions to decay chain equations

Abstract

We present an explicit compact form of the general solution for linear decay chains with repeated eigenvalues. Comparing with previous solutions of the Bateman equations with identical decay constants, our expression brings new contributions in two ways. First, it is a closed analytical expression valid for arbitrary initial conditions, whereas available solutions typically concern the simplest case of null initial population for all but the first member of the chain. Second, our expression is written in terms of a minimal number of factors and sums, avoiding large nested sums, large products, complicated implicit conditions and high order derivatives. The efficiency of our expression is illustrated with computational implementations of both analytical and numerical applications. To complement our result for linear chains, we present also the solution, for arbitrary initial conditions, of the basic cyclic chain, restricted to the case of negative and distinct eigenvalues.