$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\tp}{\thinspace .}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\dfc}{\alpha} % diffusion coefficient
$$
Properties of the solution
The PDE
$$
u_t = \dfc u_{xx}
$$
admits solutions
$$
u(x,t) = Qe^{-\dfc k^2 t}\sin\left( kx\right)
$$
Observations from this solution:
- The initial shape \( I(x)=Q\sin kx \)
undergoes a damping \( \exp{(-\dfc k^2t)} \)
- The damping is very strong for short waves (large \( k \))
- The damping is weak for long waves (small \( k \))
- Consequence: \( u \) is smoothened with time
