I read this on a book, which says we have a linear difference equation:
pk= 1/2 (pk+1 + pK-1) if 0<k<N,
s.t. boundary conditions: p0= 1, pN= 0.
The solution is also provided however I don't get it, it says:
Put bk = pk - pk-1, to obtain bk = bk-1 and hance bk = b1 for all k
[How do we get bk = bk-1 ?]
Thus, pk = b1 + pk-1 = 2b1 + pk-2 =...= kb1 + p0
Boundary conditions imply p0 = 1, b1 = -1/N
so we have pk = 1 - (k/N)
At first glance everything looks fine...but when I read it again I don't quite understand how does bk = bk-1? Could anyone explain why it is the case?
Thank you very much for your help