Simplified recursive relations for the derivatives of Bateman linear chain solution and their application to sensitivity and multi-point analysis

Bateman linear chain solution is widely used for fusion activation calculations. Its first order derivatives with respect to the decay and cross section constants are implemented for the sensitivity analysis. The general formula for these derivatives is a recursive one which is lengthy and cumbersome to use. The large numbers of arithmetic operations consume considerable computational effort and grow as k2, where 'k' is the size of a linear chain. However, they provide a complete analytical treatment of the sensitivity analysis. In the present paper, the general formula for the derivatives has been reworked and further simplified without any approximations. The resulting derivatives are a set of simple yet exact recursive relations that require almost no computational labour and grow almost linearly with size 'k' of the chain. The derived relations can be easily used for a quicker sensitivity analysis and results are presented within the paper. Higher order derivatives can be derived using simple algebra from the first order derivatives. Further, it is shown in the paper that the first and second order derivatives of the Bateman solution can be used within a Taylor series expansion to perform multi-point analysis on fine meshes over a large geometry. Most of the current multi-point algorithms focus on the coupling or algorithmic aspects to accelerate the calculations. In the present scheme, we use taylor expansion of the Bateman solution itself to directly affect the increase in computational speed. The solution of the Bateman equations at a given reference point within the geometry can be used to estimate the results at neighbouring points through a Taylor expansion. While the method in itself is straightforward, it is made feasible with the simplified derivative relations. This methodology has been successfully implemented within the linear chain solution algorithm and a few results and inferences are provided along with the computational accuracy and efficiency.