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