Example

Test problem:

$$\begin{align*} u_t &= u_{xx},\quad &x\in (0,1),\ t\in (0,T]\\ u(0,t) &= u(1,t) = 0,\quad &t\in (0,T]\\ u(x,0) & = \sin (\pi x) + 0.1\sin(100\pi x) \end{align*}$$

Exact solution:

$$\begin{equation} u(x,t) = e^{-\pi^2 t}\sin (\pi x) + 0.1e^{-\pi^2 10^4 t}\sin (100\pi x) \tag{3} \end{equation}$$