Determine the equilibrium temperature distribution for the r

Determine the equilibrium temperature distribution for the ring theta element of [0, 2pi) by both directly by setting u_t = 0, and finding the equilibrium solution, and finding the time-dependent solution and taking the limit t rightarrow infinity. Use the initial condition f(theta) = 1 + theta.

Solution

The equilibrium solution is then found by setting ut = 0,

from which: u_xx = 0 u(x) = C1_x + C2

The derivative is C1, so set it to zero, and we have: u(x) = C2.

Now, to find the value of C2, we note that ₀R^l u(x, t) dx is constant, and so it can be computed at any time- In particular, at time 0, then as t :

₀R^l u(x, t) dx =  ₀R^l u(x, 0) dx =  ₀R^l f(x) dx

And  ₀R^l u(x, t) dx =₀R^l u(x) dx = ₀R^l c2 dx = C2L

therefore, C2 = 1/ L ₀R^l f(x) dx