a spherical 3 dimensional egg of radius pi whose temperature satisfies
Ut - laplace(U) = 0
(t > 0), is boiled to a uniform temperature of 100 (Celsius)
everywhere inside, then placed into cold water of temperature 0.  Find
the temperature inside the egg at a point x and time t > 0.

Hi, awl-ga:

Note that if the solution, like the initial and boundary conditions,
is assumed to be radially symmetric, then taking radius x to agree
with your notation:

dU/dt = laplace(U) = (1/x^2) d/dx [ x^2 dU/dx ]

since derivatives with respect to the other polar coordinates vanish.

A "separation of variables" approach to solving this problem is
explained at:

[Spherical "roast"]

http://www.physics.ubc.ca/~birger/p312l23/index.html

Taking the radius of the spherical "egg" to be pi and the conduction
coefficient k = 1 makes the solution especially simple:

U(x,t) = (1/x) SUM a_n sin(nx) exp( - n^2 t) [FOR n = 1,2,3,...]

and it remains only to fit the initial condition at t=0 by choice of
a_n:

100 = (1/x) SUM a_n sin(nx) [FOR n = 1,2,3,...]

This bears an obvious relationship to the Fourier series expansion of
a sawtooth wave:

a_n = (2/pi) INTEGRAL 100x sin(nx) dx [OVER x = 0 to pi]

 = (200/pi) INTEGRAL x sin(nx) dx [OVER x = 0 to pi]

Using the fact that:

d(x cos(nx))/dx = -nx sin(nx) + cos(nx)

the indefinite integral may be found:

INTEGRAL x sin(nx) dx = -(x/n) cos(nx) + (1/n) sin(nx)

and therefore:

a_n = (200/pi) [ -(pi/n) cos(n*pi) ]

 = -(200/n) cos(n*pi)

 = (200/n) (-1)^(n+1)

for n = 1,2,3,... as required.

regards, mathtalk-ga

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Clarification of Answer by mathtalk-ga on 20 Nov 2002 06:02 PST

Hi, awl-ga:

Thanks for taking the time to rate my answer.  I think I speak for all
experts when I say that this means a lot to us, after we try our
ingenuities to come up with just the right response to a particular
question.

regards, mathtalk-ga

You might want to check out the URL
https://answers.google.com/answers/pricing.html and reconsider the
pricing.

how do i pay more to you without asking a new question, i want to play $10 dollars

do you think that you could take a look at the other math questions
that i have posted.  i don't believe that it is really hard of a
question but it's taking a long time for the other researcher to
answer. thanks

Hi, awl-ga:

I'll certainly take a look.  As I recall it was more a difficulty in
grasping your notation.  Perhaps I can post some requests for
clarification that will get the ball rolling on those.

regards, mathtalk-ga