1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

0.5 Green’s Functions 43

6. Inside a nuclear fuel rod, heat is constantly produced by nuclear reaction.
   Atypicalrodisabout3mlongandabout1cmindiameter,sotemperature
   variationalongthelengthismuchlessthanalongaradius.Thus,wetreat
   the temperature in such a rod as a function of the radial variable alone. Find
   this temperatureu(r), which is the solution of the boundary value problem
   1
   r

d

dr

(

rdu

dr

)

=−g

κ

, 0 <r<a,

u(a)=T 0.

7. For the problem of Exercise 6, find the temperature at the center of the
   rod,u( 0 ), using these values for the parameters:a= 0 .5cm,thepower
   densityg=418 W/cm^3 =100 cal/s cm^3 ,conductivityκ= 0 .01 cal/s cm◦C,
   and the surface temperatureT 0 = 325 ◦C.


8. A model for microwave heating of food uses this equation for the tempera-
   tureu(x)in a large solid object:
   d^2 u
   dx^2

=−Ae−x/L, 0 <x.

Here,Ais a constant representing the strength of the radiation and prop-

erties of the object, andLis a characteristic length, known as penetration

depth, that depends on frequency of the radiation and properties of the ob-

ject. (Typically,Lis about 12 cm in frozen raw beef or 2 cm thawed.) Show

that the boundary conditionu′( 0 )=0 is incompatible with the condition

thatu(x)be bounded asxgoes to infinity. [See C.J. Coleman, The mi-

crowave heating of frozen substances,Applied Math. Modeling, 14 (1990):

439–443.]

9. Solve the differential equation in Exercise 8 subject to the conditions

u( 0 )=T 0 , u(x) bounded.

0.5 Green’s Functions

The most important features of the solution of the boundary value problem,^1

d^2 u

dx^2 +k(x)

du

dx+p(x)u=f(x), l<x<r, (1)

αu(l)−α′u′(l)= 0 , (2)

βu(r)+β′u′(r)= 0 , (3)

(^1) The primes on the constantsα′,β′are not to indicate differentiation, of course, but to
show that they are coefficients of derivatives.