1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Heat in a Rod Fundamental Solution

  1. May 23, 2012 #1
    1. The problem statement, all variables and given/known data

    Screen_shot_2012_05_23_at_6_35_39_PM.png

    3. The attempt at a solution
    So I know that I must have boundary conditions u(0,t) = 0 and ux(L,t) = 0. My textbook recommends reducing the given boundary conditions to homogeneous ones by subtracting the steady state solution. But, I thought these were already homogenous boundary conditions (are they?). Is my steady state condition v''(x) = 0, but, then by the boundary conditions I know that this must be a trivial v. Am I thinking about this incorrectly?
     
  2. jcsd
  3. May 23, 2012 #2
    So I thought I might have to write something of the form:
    Assume the solution can be written [tex]u(x,t) = X(x)T(t)[/tex]. Thus, by the heat equation [tex]u_t = a^2 u_{xx}[/tex], we wind up with two linear differential equations. Namely, [tex]X'' + qX = 0[/tex] and [tex]T' + a^2 q T = 0[/tex]. Now I have to find which values of q make q an eigenvalue of the eigenfunction. We test three cases: q = 0; q > 0; q < 0.

    q = 0:
    We must have that [tex]X = C_1 x + C_2[/tex], but by the boundary conditions, this forces both of the arbitrary constants to be zero.
    q < 0:
    [tex]X = C_1 sinh(\sqrt{-q}x) + C_2 cosh(\sqrt{-q}x)[/tex], which implies that C_2 = 0, and, when one takes the derivative, we also have C_1 = 0 by the assumption that q was nonzero.

    q > 0:
    [tex]X = C_1 cos(\sqrt{q}x) + C_2 sin(\sqrt{q}x)[/tex], which implies that C_1 = 0, and, again, when one takes the derivative, we force the other constant to zero. This means that I am only getting trivial solutions here. What other approaches can I try? Or have I done something wrong?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Heat in a Rod Fundamental Solution
Loading...