Consider a box of length L = 1 that is initially in contact with a heat reservoir on the left at T = 1 and a heat reservoir on the right with T = 0. When this system reaches equilibrium, it will have a temperature distribution T(x) = 1 − x. When the system has equilibrated, the box is disconnected from the heat reservoir and replaced with insulating boundary conditions, ∂T/∂x = 0 at both x = 0 and x = 1. Determine the temperature distribution inside this box, T(x, t) for t > 0 where t = 0 is the time when the boundary conditions change. Since this problem is already stated in a dimensionless form, you can take the heat equation to have the form

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