Thread: Solving a linear difference equation subject to boundary conditions

Hi everyone,

I read this on a book, which says we have a linear difference equation:

p_k= 1/2 (p_k+1 + p_K-1) if 0<k<N,

s.t. boundary conditions: p₀= 1, p_N= 0.

The solution is also provided however I don't get it, it says:

Put b_k = p_k - p_k-1, to obtain b_k = b_k-1 and hance b_k = b₁ for all k

[How do we get b_k = b_k-1 ?]

Thus, p_k = b₁ + p_k-1 = 2b₁ + p_k-2 =...= kb₁ + p₀

Boundary conditions imply p₀ = 1, b₁ = -1/N

so we have p_k = 1 - (k/N)

At first glance everything looks fine...but when I read it again I don't quite understand how does b_k = b_k-1? Could anyone explain why it is the case?

Thank you very much for your help

alrite seems no one wants to answer this Q probably because it is too easy...

after read it a few more times I think I see it

so I just answer myself:

to see this: split LHS p_k in to 1/2(p_k + p_k) and the first equation becomes 1/2(p_k + p_k)= 1/2 (p_k+1 + p_K-1)

rewrite above in to: 1/2(p_k+1-p_k) = 1/2(p_k-p_K-1) which is: 1/2b_k+1 = 1/2b_k gives us b_k = b_k-1