# u(x,t) equation for a thin rod

• May 11th 2011, 06:25 AM
sublim25
u(x,t) equation for a thin rod
Consider a thin rod of length L with an initial temperature fix) throughout. Its ends are held at zero temperature for all time. The temperature u(x, t) in the rod is governed by the following equation, boundary and initial conditions

∂^2w = c2 ∂^2u (t ≥ 0, 0 ≤ x ≤ L)
∂x^2 ∂x^2

u(t,0) = 0, u(L,t) = 0, t ≥ 0
u(x,0) = f(x), 0 ≤ x ≤ L

(a) Solve for u(x,t)
(b) Find u(x, t) for the specific case of f(x) = 100, L = pi and c = 1.
• May 13th 2011, 03:39 PM
jamesdt
You must use a separation of variables, that is you assume the solution is on the form \$\displaystyle u(x,t)=X(x)T(t)\$. Then you can derive two ordinary differential equations which can easily be solved. This gives you a particular solution, then use the Fourier method. That is, the general solution is formed from linear combinations of the particular solutions. The initial condition will allow you to determine the coefficients of the general solution, which in this case will be the Fourier sine series of the initial function.

However, I think it's quite unfair of you to just post the problem in this way and expect a solution. You should show that you've at least made an attempt and explain what you have so far. That might make people more inclined to help.
• May 15th 2011, 04:45 AM
sublim25
Thank you for your help! I did not write anything because I did not know how to even begin to solve the problem or to find a helpful textbook to use as a reference. Thank you again!