Inverse of discrete second derivative operator
I have a problem solving the following exercise:
We consider the boundary value problem :
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 of the problem:
I have a problem when I consider the discrete version of the problem. It can be obtained using the central difference formula for y'':
. By doing this, we find the associated discrete problem: a linear system .
is a ( 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 . The only hint I have is that I should get inspiration from the continuous case... The professor gave us the answer : . It looks like the Green's function but I have absolutely no idea on how to find this ! (Worried)
Can someone please help me ? (Nod)
Thanks for your help, and sorry for my bad English.