  A simple and analytical formula is suggested to solve the problems of the
local burnup and the isotope distributions. The present method considers two
extreme conditions of neutrons penetrating the fuel rod. Based on these
considerations, the formula is obtained to calculate the reaction rates of
$^{235}$U, $^{238}$U, and $^{239}$Pu and straightforward the local burnup and
the isotope distributions. Starting from an initial burnup level, the
parameters of the formula are fitted to the reaction rates given by a Monte
Carlo (MC) calculation. Then the present formula independently gives very
similar results as the MC calculation from the starting to high burnup level,
but takes just a few minutes. The relative reaction rates are found to be
almost independent on the radius (except $(n,\gamma)$ of $^{238}$U) and the
burnup, providing a solid background for the present formula. A more realistic
examination is also performed when the fuel rods locate in an assembly. A
combination of the present formula and the MC calculation is expected to have a
nice balance on the accuracy and the cost on time.
