Solution for the One-dimensional Diffusion Equation
To satisfy the boundary conditions we require \( B=0 \) and \( \lambda=n\pi/L \). One solution is therefore found to be
$$
\begin{equation*}
u(x,t)=A_n\sin(n\pi x/L)e^{-n^2\pi^2 t/L^2}.
\end{equation*}
$$
But there are infinitely many possible \( n \) values (infinite number of solutions). Moreover,
the diffusion equation is linear and because of this we know that a superposition of solutions
will also be a solution of the equation. We may therefore write
$$
\begin{equation*}
u(x,t)=\sum_{n=1}^{\infty} A_n \sin(n\pi x/L) e^{-n^2\pi^2 t/L^2}.
\end{equation*}
$$