I made a very silly algorithm to "approximate" functions in order to understand partial differential equations (I'm trying to learn by myself so I'm mostly trying to draw graphs to see what is going on). We get $(x,f(x))$, compute $a_0=\{x_0,f(x_0) \}$ and then we construct the following points:
$$a_n=(x_{n-1},f(x_{n-1}))+(1,f'(x_{n-1}))\qquad n=1,2,3,...$$
So far, it seems to be working:
So I took the following partial differential equation with the following conditions:
$$\begin{array}{cc}\begin{array}{c}u_{tt}=u_{xx}\hspace{34mm}\\\end{array}&\begin{array}{c} t\in (0,+\infty) \quad x\in(0,1) \\\end{array}\\\begin{array}{c}u(0,t)=u(1,t)=0\hspace{18mm} \\\end{array}&\begin{array}{c}t\geq 0 \hspace{28mm}\\\end{array}\\\begin{array}{c}u(x,0)=f(x) \quad u_t(x,0)=g(x)\\\end{array}&\begin{array}{c}x\in[0,1] \hspace{22mm}\\\end{array}\\\end{array}$$
And used $f(x)=x(1-x) \sin (2 \pi x)$ and $g(x)=\sin(x)$. So the conditions we have in the problem seems to be saying that $u(x,t)$ must be contained in the following red curves:
I tried to do the following: Find "solution points" for these conditions and make an analogy with the algorithm I did previously. So we have:
We get $(x,t,f(x))$, compute $a_0=\{x_0,0,f(x_0 ,0) \}$ and then we construct the following points:
$$a_n=(x_{n-1},t_{n-1},\overbrace{f(x_{n-1})}^{u(x,0)})+(0,1,\overbrace{g(x_{n-1})}^{u_t (x,0)})\qquad n=1,2,3,...$$
But I am really confused: This doesn't yield anything near the actual partial differential equations I did and I tried a lot of different things. What am I missing?
