Solution for the One-dimensional Diffusion Equation
The coefficient \( A_n \) is in turn determined from the initial condition. We require
$$
\begin{equation*}
u(x,0)=g(x)=\sum_{n=1}^{\infty} A_n \sin(n\pi x/L).
\end{equation*}
$$
The coefficient \( A_n \) is the Fourier coefficients for the function \( g(x) \). Because of this, \( A_n \) is given by (from the theory on Fourier series)
$$
\begin{equation*}
A_n=\frac{2}{L}\int_0^L g(x)\sin(n\pi x/L) \mathrm{d}x.
\end{equation*}
$$
Different \( g(x) \) functions will obviously result in different results for \( A_n \).