I have had no experience with differential equations before I was presented with this problem on a homework. The equation is:
$ -u'' + \beta u' = 0 $
$ u(0) = 0 $
$ u(1) = 1 $
I have found the general solution to be $ u = \frac{-1}{e^\beta - 1} + \frac{e^{\beta x}}{e^{\beta} - 1}$ and am now asked to derive the central, forward, and backward finite difference schemes (I am only worried about understanding central for now though).
Bear with me now, like I said I have not had any experience with differential equations. From my understanding, the central finite difference yields:
$ u''(x) \approx \frac{2u(x) - u(x+h) - u(x-h)}{h^2} $
and therefore $ -u'' + \beta u' = 0 $ becomes:
$ \frac{2u(x) - u(x+h) - u(x-h)}{h^2} + \beta u'(x) = 0 $
$ 2u(x) + u(x + h) - u(x - h) = -\beta u'(x)h^2 $
Am I on the right track so far? I am not sure what a complete "finite difference scheme" even is, so when should I stop? I have seen examples where the end result is a large matrix equation - is that required to answer this question?