Page 250 -

233

which is uniformly valid as from (5.92) we obtain

and so on.
The solution of (5.93a) is immediately seen to be

where is an arbitrary function; then (5.93b) becomes

This can be written as

and hence for to remain bounded as also ensures periodicity in
require

A solution of this equation can be expressed as

but very little headway can be made, at this stage, without some knowledge of v(x).
We do note, however, that for k < 0 the solutions for are essentially exponential
(growing and decaying), but for bounded and large enough, the solutions are
oscillatory.

Our final example is a problem of a gas flow but, because the density is so low, the
model is based on an approach that invokes the ideas of statistical mechanics.

E5.15 Low-pressure gas flow through a long tube
The flow of a gas through a long circular tube, i.e. radius/length is small, where
molecular collisions are assumed to occur only with the wall of the tube, can be
represented by the Clausing integral equation