The governing equation and boundary conditions in dimensionless form are:

, with and ,

(continuity of heat flux at the boundary),

(constant temperature at the boundary),

(same constant temperature at the boundary), and

(different constant temperature at the boundary),

with , , and .

The Chebyshev orthogonal collocation method implemented in Mathematica with collocation points gives the temperature distribution in the bar represented in the snapshots for user-set values of and .

The analytical solution of the differential equation obtained using the separation of variables technique [1] is given by:

,

where .

We have found excellent agreement between our numerical solution and the analytical solution.