Is it hot?

The solution to the heat equation $u(x, t)_t = u(x, t)_{xx}$ is given as

$u(x, t) = \sum_{m=1}^{\infty} A_m e^{- \pi^2 m^2 t} \sin(\pi m x)$

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 $u(x, 0) = x$ . Then what is $\left \lfloor A_1 \right \rfloor$ ?

Notations:

• $u(x_1, x_2, ..., x_n)_{x_k},$ for $1 \leq k \leq n$ denotes for the partial derivative of $u$ with respect to $x_k.$
• $\lfloor \cdot \rfloor$ denotes the floor function .