Please give a detailed explantion and answer in the form of an uploaded image as
ID: 2965586 • Letter: P
Question
Please give a detailed explantion and answer in the form of an uploaded image as I do not understand the computer notation. A satisfactory answer will be rewarded with points immediately. Please do not rush in to answer first, I would be rewardeing the best answer not the first.thanks.
The surface of a ball of radius A is kept at a temperature zero. If the initial temperature in the ball is f(r), write down the boundary conditions and show that the temperature in the ball at time t, u (r, t), is the solution to the equation:Explanation / Answer
Let du/dr = u'
We want to solve:
(1/r)d/dr(ru') = 0
=> d/dr(u') + (u'/r) = 0
Writing v for u', we get:
=> d/dr(v) + v/r = 0
=> dv/v = -(1/r)dr
=> ln |v| = (1/r^2) + C
=> v = exp(C)*exp(1/r^2)
=> v = A*exp(1/r^2)
where C and A are arbitrary constants.
Now v = du/dr.
So:
u' = A*exp(1/r^2)
=> u = -(2A/r^3)exp(1/r^2) + B
This can of course be written as:
u = (A/r^3) exp(1/r^2) + B