# 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

Implicit Method Explicit Function Kolmogorov Equation Forward Equation General Solution Technique
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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