Consider the partial differential equation for heat in a one-dimensional rod with temperature u(x,t):
du/dt = d^2u/dx^2
Assume initial condition:
u(x,0) = 6xe^-x/1,
and boundary conditions:
du/dx(0,t) = 1 du/dx(1,t) = β
For what values of β are there solutions to this heat equation?
β =
Determine the equilibrium temperature distribution.
u(x) =
In your homework assignment write a brief paragraph explaining what is occurring physically to allow the unique equilibrium solution.