Proof: we have

But if , then

and so .

Thus the difference system is strictly diagonally dominant and the result follows.

Truncation error

To estimate the error in the numerical solution of BVP by finite difference method we first define the local truncation error in , as an approximation to L, for any smooth function , by

.

If has continuous fourth derivatives on , then

.

Here and are values in .

Thus we find that is consistent with L i.e. as for all factors having a Cont. second derivative on . Further has second order accuracy.