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The transport of heat in a rectangular channel 0 ≤ x ≤ 1 is governed by...

The transport of heat in a rectangular channel 0 ≤ x ≤ 1 is governed by the parabolic partial differential equation ∂w/ ∂t = ∂ 2w/ ∂x2 where w(x, t) is the temperature, t is time and k is constant. If initially the left-hand side of channel is exposed to a heat source W0(x) = x(1−x) and the heat flux at the top and the bottom faces are zero (i.e. ∂w/∂x = 0 there) find a general expression for the transport in the channel for any subsequent time t > 0.

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