1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

40 Chapter 0 Ordinary Differential Equations

Application of the special condition, thatu( 0 )andu′( 0 )be finite, immediately
tellsusthatc 1 =0; for both, ln(r)and its derivative 1/rbecome infinite asr
approaches 0.
The physical boundary condition, Eq. (2), says that

u(c)=−Hc

2

4 +c^2 =T.

Hence,c 2 =Hc^2 / 4 +T, and the complete solution is

u(r)=H

(c^2 −r^2 )
4 +T. (4)


From this example, it is clear that the “artificial” boundary condition,
boundedness ofu(r)at the singular pointr=0, works just the way an or-
dinary boundary condition works at an ordinary (not singular) point. It gives
one condition to be fulfilled by the unknown constantsc 1 andc 2 ,whichare
then completely determined by the second boundary condition.

Semi-Infinite and Infinite Intervals

Another type of singular boundary value problem is one for which the inter-
val of interest is infinite. (Of course, this is always a mathematical abstraction
that cannot be realized physically.) For instance, on the interval 0<x<∞,
sometimes called asemi-infinite interval, as it does have one finite endpoint, a
boundary condition would normally be imposed atx=0. At the other “end,”
no boundary condition is imposed, because no boundary exists. However, we
normally require that bothu(x)andu′(x)remain bounded asxincreases. In
precise terms, we require that there exist constantsMandM′for which
∣∣
u(x)

∣∣

≤M and

∣∣

u′(x)

∣∣

≤M′

are both satisfied for allx, no matter how large. We never identifyMorM′,
and the entire condition is usually written

u(x) and u′(x) bounded asx→∞.

Example: Cooling Fin.
A long cooling fin has one end held at a constant temperatureT 0 and ex-
changesheatwithamediumattemperatureTthrough convection. The tem-
peratureu(x)in the fin satisfies the requirements

d^2 u

dx^2

=hC

κA

(u−T), 0 <x, (5)

u( 0 )=T 0 (6)