With constant positive thermal diffusivity , no internal heat source and one spatial coordinate, the heat equation isSeparation of variables makes each spatial mode decay by . The endpoint conditions determine whether the initial triangular profile is expanded in a Fourier sine series or a Fourier cosine series.
The same triangular initial temperature with cooled and insulated ends
. Original heat-equation solutions for several dimensionless times. Fixed zero-temperature ends cool the bar to zero; insulated ends preserve its initial mean and approach a uniform temperature. Details.
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