Example 2.

Solve Eq. (2.17) with the same boundary conditions. Assume now, that initial distributions of position and velocity are

$$u(x,0)=f(x)=0$$   and$$\quad u_t(x,0)=g(x)=\begin{cases} 0,\quad x\in[0,\, x_1];\\ g_0,\quad x\in[x_1,\,x_2];\\ 0,\quad x\in[x_2,\,L]. \end{cases}$$

Other parameters are:

Initial nonzero velocity	$g_0$ =0.5
Initial space intervals	$x_1=L/4$ , $x_2=3L/4$
Space interval	$L$ =10
Space discretization step	$\triangle x=0.1$
Time discretization step	$\triangle t=0.05$
Amount of time steps	$T=400$

Numerical solution of the problem is shown on Fig. (2.1.3).

Figure 2.5: Space-time evolution of the initial distribution $u(x,0)=0$ , $u_t(x,0)=g(x)$ .