TABLE OF CONTENTS In many transient calculations, one either undertakes to find the distribution of states as they approach steady state under conditions when the control points are fixed, or the dynamic calculations get to a point where this is the case. Once the transient in the control point dynamics has died out, the tedious domain-by-domain solution procedure can be abolished in favor of a more powerful method capable of providing a solution for arbitrary large times. The method can be used not just for situations in which the control points have become fixed but in any case where the control point dynamics has settled down to a simple repetitive behavior such as periodically changing control points.
A simple recursion formula between solutions at different times can often be obtained quite easily by perceptive guessing. The guessing is done after studying the solution for the first couple of domains after the control points have become fixed. For instance, for a chemostat with a binary fission organism that divides at the age a d, it is trivial to show that if the age distribution at time t 0 is W 0( a), then the age distribution at time t 0 + a d is 2 W 0( a) e -Dt. But the origin of the time axis is arbitrary, so one can certainly generalize this result to the following recursion formula:
or, written more compactly in a notation we will...
