Conduction in a Thin Rod (1-D)        (Continued)

 3 Strategy:   Since the heat conduction equation involves two independent variables, x and t, a method called “separation of variables” will be used to separate out the spatial variable, x, from the time variable, t.       Assume       u(x,t)  =  X(x) T(t)    (for separation of variables)    Put this expression into the heat conduction equation,                                                    ∂u/∂t   =   k ∂2u/∂x2         and taking the value for the thermal conductivity of the rod,  k  =  1   gives                     X dT/dt   =  d2X/dx2  T                    Let   d2X/dx2  =  X’’   and  dT/dt  =   T*   Then divide both sides by  X T  to separate the variables:                               T*/T  =  X’’/X  =  K  =  separation constant  =  - λ   Since T*/T  depends only on  t  and  X’’/X  depends only on x, the only way   T*/T  could equal  X’’/X  is for both to  be constant, the separation constant, K.   So to solve for heat conduction in a thin rod, one needs to solve an eigenvalue problem of the form:                    X’’  + λ X  =  0     and  T*  +  λ T  =  0   Click here for an example.

Copyright © 2011 Richard C. Coddington

All rights reserved