$u_{tt}(x,t) = \pi u_{xx}(x,t) \quad x \in [0,1], t > 0 \\

u(x,0) = 18sin(3\pi x) \quad \forall x \in [0,1]\\

u_{t}(x,0) = sin(\pi x) \quad \forall x \in [0,1]\\

u(0,t) = u(1,t)=0 \quad \forall t > 0$

Which of the following schemes can numerically approximate the solution?

a) $ u_{k+1,j} = r^2(u_{k,j+1} + u_{k, j-1}) + 2(1-r^2) u_{k,j} - u_{k-1,j}$

b) $ u_{k,j+1} = r^2(u_{k+1,j} + u_{k-1, j}) + 2(1-r^2) u_{k,j} - u_{k,j-1}$

c) $ u_{k+1,j} = r(u_{k,j+1} + u_{k, j-1}) + 2(1-r) u_{k,j} - u_{k-1,j}$

b) $ u_{k,j+1} = r(u_{k+1,j} + u_{k-1, j}) + 2(1-r) u_{k,j} - u_{k,j-1}$

Notes: $ \Delta x = 1/N; r = \sqrt{\pi} \Delta t / \Delta x ; k = 1,2,...,N-1 ; j = 2,... $

Can anyone not only show me the answer but also explain me a little bit? Or at least provide me some useful links with examples of similar exercises.

I'm also asked if the equation is parabolic and if it is a Laplace equation. I answered false to both of them because I consider the equation to be hiperbolic.