Example 1.

Use the finite-difference method (2.10) to solve the wave equation for a vibrating string:

$$u_{tt}=4u_{xx} \quad x\in[0,\,L] \quad t\in[0,T]$$ (2.16)

with the boundary conditions

$$u(0,t)=0\qquad u(L,t)=0.$$

Assume that the initial position and velocity are

$$u(x,0)=f(x)=\sin(\pi x),$$   and$$\quad u_t(x,0)=g(x)=0.$$

Other parameters are:

Space interval	$L$ =10
Space discretization step	$\triangle x=0.1$
Time discretization step	$\triangle t=0.05$
Amount of time steps	$T=20$

First one can find the d'Alambert solution. In the case of zero initial velocity Eq. (2.8) becomes

$$u(x,t)=\frac{f(x-2t)+f(x+2t)}{2}=\frac{\sin\pi(x-2t)+\sin\pi(x+2t)}{2}=\sin\pi x\cos 2\pi t,$$

i.e., the solution is just a sum of a travelling waves with initial form, given by $\frac{f(x)}{2}$ . Numerical solution of (2.17) is shown on Fig. (2.1.3).

Figure 2.4: Space-time evolution of the initial distribution $u(x,0)=f(x)$ , $u_t(x,0)=0$ .