In your solutions, if iterations exist, write at least one iteration by hand. You can present your results as Matlab results inserting your Matlab implementation.
1) (25pts.) Solve the wave equation $u_{tt} = u_{xx}(x, t)$, for $0 \le x \le 1$ and $0 \le t \le 1$ with the boundary conditions
$u(0, t) = 0 = u(1, t)$, $0 \le t \le 1$
$u(x, 0) = \sin(\pi x)$, $0 \le x \le 1$
$u_t(x, 0) = 0$, $0 \le x \le 1$,
Choose $h = 0.1$ and $k = 0.1$.
2) (25pts.) Solve the heat equation $u_t(x, t) = c^2 u_{xx}(x, t)$ for $0 < x < 1$ and $0 < t < 0.1$ with the initial condition $u(x, 0) = \sin(\pi x) + \sin(2\pi x)$ for $t = 0$ and $x \in [0, 1]$, and the boundary conditions
$u(0, t) = u(1, t) = 0$. Use $h = 0.1$, $k = 0.01$, $r = 1$.
