Finite difference method on boundary value problem
I want to solve boundary value problem in n points, $\displaystyle x_{n}\in(0,1)$:
$\displaystyle -u'(x)(ku'(x))=12x^{2}$
where:
$\displaystyle u(x)\in[0,1]$
$\displaystyle u(0)=0$
$\displaystyle ku'(1)+hu'(1)=hu_{z}$
$\displaystyle x\in R$
with finite difference method instead of finite element method.
I substituted finite difference:
$\displaystyle u'(x)=\frac{u(x_{i-1})-u(x_{i+1})}{2h}$
$\displaystyle h=\frac{1}{n}$
and obtained:
$\displaystyle -kn^{2}[u(x_{i-1})-u(x_{i+1})]^{2}=3x_{i}^{2}$
$\displaystyle i=1,...,n$
And I don't know how to turn that into the matrix equation. Is there a way to do it?