Difference equations 33yields the first Green formula( 5 6) ( y, A u) = - (a \J u, \J y ] - (du, y) + (a y \J u) N - Yo (a \J u) 1 ,
which can be viewed as an analog of (50). After interchanging in (56) the
positions of ui and Yi we get the equation
( u, A y) = -(a \J y, \J u ] - ( d y, u) + (a y \J y) N - u 0 (a \J y) 1 ,
which will be subtracted at the next stage from (56). The final expression
leads to the second Green formulaIn the particular case when a; - 1 and d; = 0, that is, for
A - Yi : = 6. \J Yi = 6.^2 Yi - 1
Green's difference formula admits an alternative form
(y 1 6 \J U) = ( U, 6 \J Y) + (y \J U - U \J Y) N - (y 6 U - U 6 Y )o
which allows a more simpler writing of the ensuing formulae. When y 0 = 0
and yN = 0, the first Green formula becomes
(58) (y, Au)= -(a \Ju, \Jy)-(du, y),
giving for u = y
(59) (y, Ay) =-(a \J y, \J y) - (d, y^2 ).
For any y and u subject to the homogeneous boundary conditions y 0
yN = 0 and u 0 = uN = 0, the second Green formula takes a very elegant
form
(60) (Ay,u)=(y,Au).