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

  • Geoffrey T. Parks


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
$$ \frac{{\partial Fm}}{{\partial t}} = \frac{1}{l}\left[ { - x + go(x) + \sum\limits_{{i - 1}}^{I} {gi(x)yi + \sum\limits_{{j = 1}}^{J} {\varepsilon j(zj - x)} } } \right]\frac{{\partial Fm}}{{\partial x}} + \sum\limits_{{i - 1}}^{I} {\lambda i(x - yi)\frac{{\partial Fm}}{{\partial yi}} + S(f(x) - 1)F{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{m}}} $$
while the backward equation for the same system is
$$ - \frac{{\partial Fm}} {\partial s} = - \left[ {\frac{m}{l} + \sum\limits_{i = 1}^I {mi\lambda i + S} } \right]F\underline m + \frac{m} {l}\sum\limits_{v = o}^\infty {pvoF} \underline m + (v - 1)\underline \delta 0 + \frac{m} {l}\sum\limits_{i = 1}^I {\sum\limits_{v = 0}^\infty {pvi} F\underline m + (v - 1)\underline \delta 0 + \underline \delta i + \sum\limits_{i = 1}^I {mi\lambda iF\underline m } + \underline \delta 0 - \underline \delta i} + S\sum\limits_{v = 0}^\infty {qvF\underline m } + v\underline \delta 0 + \frac{m} {l}\sum\limits_{j = 1}^J {\varepsilon j} F\underline m - \underline \delta 0 + \underline \delta j $$


Convolution Ruby 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Faà de Bruno M., 1857, Note Sur Une Nouvelle Formule De Calcul Différentiel, Quart. J. Pure Appl. Math., 1:359.Google Scholar
  2. Hurwitz H., Jr., MacMillan D. B., Smith J. H., and Storm M. L., 1964, Numerical Solution Of The Forward Equation, in: “Naval Reactors Physics Handbook,” A. Radkowsky, ed., USAEC, Washington DC.Google Scholar
  3. Lewins J. D., 1988, Solving Low Power Stochastic Equations: Nonlinear Studies, in: “Proceedings of the NATO Advanced Workshop on Noise and Nonlinear Phenomena in Nuclear Systems,” J. L. Mufioz-Cobo, ed., Plenum, New York.Google Scholar
  4. Parks G. T., and Lewins J. D., 1985, An Exact Transient Stochastic Solution For Low-Power Neutron Multiplication, Ann. nucl. Energy, 12:65.CrossRefGoogle Scholar
  5. Ruby L., and McSwine T. L., 1986, Approximate Solution To The Kolmogorov Equation For A Fission Chain-Reacting System, Nucl. Sci. Eng., 94:271.Google Scholar

Copyright information

© Plenum Press, New York 1989

Authors and Affiliations

  • Geoffrey T. Parks
    • 1
  1. 1.Engineering DepartmentUniversity of CambridgeCambridgeEngland

Personalised recommendations