# Exact and Inexact, Explicit and Implicit Solution Techniques for the Forward and Backward Kolmogorov Equations

• Geoffrey T. Parks

## Abstract

This paper considers possible solution methods for both the forward and backward Kolmogorov equations of a fission chain-reacting system. The notation, based on that used by Ruby and McSwine (1986), is identical to that defined in the preceding paper (Lewins, 1988). Thus the forward equation for a system with one group of neutrons, I groups of precursors and J groups of ‘detectrons’ is
$$\frac{{\partial Fm}}{{\partial t}} = \frac{1}{l}\left[ { - x + go(x) + \sum\limits_{{i - 1}}^{I} {gi(x)yi + \sum\limits_{{j = 1}}^{J} {\varepsilon j(zj - x)} } } \right]\frac{{\partial Fm}}{{\partial x}} + \sum\limits_{{i - 1}}^{I} {\lambda i(x - yi)\frac{{\partial Fm}}{{\partial yi}} + S(f(x) - 1)F{\underset{\raise0.3em\hbox{\smash{\scriptscriptstyle-}}}{m}}}$$
(1)
while the backward equation for the same system is
$$- \frac{{\partial Fm}} {\partial s} = - \left[ {\frac{m}{l} + \sum\limits_{i = 1}^I {mi\lambda i + S} } \right]F\underline m + \frac{m} {l}\sum\limits_{v = o}^\infty {pvoF} \underline m + (v - 1)\underline \delta 0 + \frac{m} {l}\sum\limits_{i = 1}^I {\sum\limits_{v = 0}^\infty {pvi} F\underline m + (v - 1)\underline \delta 0 + \underline \delta i + \sum\limits_{i = 1}^I {mi\lambda iF\underline m } + \underline \delta 0 - \underline \delta i} + S\sum\limits_{v = 0}^\infty {qvF\underline m } + v\underline \delta 0 + \frac{m} {l}\sum\limits_{j = 1}^J {\varepsilon j} F\underline m - \underline \delta 0 + \underline \delta j$$
(2)
.

### Keywords

Convolution Ruby

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### References

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