# Inverse of discrete second derivative operator

• Sep 11th 2010, 08:46 PM
Kapouet
Inverse of discrete second derivative operator
Hi guys,

I have a problem solving the following exercise:

We consider the boundary value problem :

$\displaystyle y''(x)=-f(x)$
$\displaystyle y(0)=y(1)=0$.

This can be solved by direct integration and with some extra calculus (inverting the order of integration of a double integral) the solution can be written with the Green's function $\displaystyle G(x,\xi)$ of the problem:

$\displaystyle y(x)=\int_0^1G(x,\xi)f(\xi)d\xi$ with $\displaystyle G(x,\xi)=min(x,\xi)[1-max(x,\xi)]$

I have a problem when I consider the discrete version of the problem. It can be obtained using the central difference formula for y'':

$\displaystyle y''(x) \approx \frac{y(x+h) - 2 y(x) + y(x-h)}{h^{2}}$. By doing this, we find the associated discrete problem: a linear system $\displaystyle Ay=f$.

$\displaystyle A$ is a $\displaystyle N \times N$ ($\displaystyle N+2$being the number of points of the discretization) tridiagonal matrix with 2 on main diagonal, and -1 on both diagonal above and below the main diagonal.

The problem here is to calculate $\displaystyle A^{-1}$. The only hint I have is that I should get inspiration from the continuous case... The professor gave us the answer :$\displaystyle (A^{-1})_{ij}=min(i,j)((N+1)-max(i,j))$. It looks like the Green's function but I have absolutely no idea on how to find this ! (Worried)