Hi guys,

I have a problem solving the following exercise:

We consider the boundary value problem :

y''(x)=-f(x)
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 G(x,\xi) of the problem:

y(x)=\int_0^1G(x,\xi)f(\xi)d\xi with 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'':

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 Ay=f.

A is a N \times N ( N+2being 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 A^{-1}. The only hint I have is that I should get inspiration from the continuous case... The professor gave us the answer : (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 !

Can someone please help me ?

Thanks for your help, and sorry for my bad English.