<Dd> u i + 1 − 2 u i + u i − 1 h 2 = u" (x i) + O (h 2). (\ displaystyle (\ frac (u_ (i + 1) - 2u_ (i) + u_ (i - 1)) (h ^ (2))) = u' ' (x_ (i)) + (\ mathcal (O)) (h ^ (2)).) </Dd> <P> In both of these formulae, h = x i − x i − 1 (\ displaystyle h = x_ (i) - x_ (i - 1)) is the distance between neighbouring x values on the discretized domain . One then constructs a linear system that can then be solved by standard matrix methods . For instance, suppose the equation to be solved is: </P> <Dl> <Dd> d 2 u d x 2 − u = 0, (\ displaystyle (\ frac (d ^ (2) u) (dx ^ (2))) - u = 0,) </Dd> </Dl> <Dd> d 2 u d x 2 − u = 0, (\ displaystyle (\ frac (d ^ (2) u) (dx ^ (2))) - u = 0,) </Dd>

Numerical methods for ordinary and partial differential equations