The solution to the heat equation is given as
Considering a rod of total length 1 such that the temperature at the extremities is kept constant and equal to 0 and at the beginning of the heat change the temperature at any point on the bar is given by . Then what is ?
Notations:
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This is just a Fourier Series problem, we have the following formula for that,
A 1 = T 2 ∫ 0 T f ( x ) s i n ( π x ) d x = ∫ 0 2 x s i n ( π x ) d x = π 2
Therefore, ⌊ A 1 ⌋ = 0