142 Basic Concepts of the Theory of Difference Schemes
so that v = Ty E H 1 if y E H;
i=l,2, ... ,N-1,
so that T*v EH ifv E H 1. Finally, (Sv)i = aivi for i = 1,2, ... ,N,
that is, S v E H 1 if v E H 1 and ( S v, v] > c 1 IIv112. Frorn such reasoning it
seems clear that(STy)i = aiY-x' i ·, i = 2, 3, ... , N - 1 ,
(T*STy)i =-(ay) x x Ji ., i= 1,2, ... ,N, for Yo= YN = 0,
which serve to motivate that the operator of problem ( 61) can be factorized
in the form ( 60) so as to have instead of (37)(62) T* STy = <p.
The fact that the operators T and T* are mutually adjoint is a corollary to
the summation by parts formula: ( y, vx) = -( v, yx], so that( y, T* v ) = -( v, T y].
Example 7 The third boundary-value problem comes next:
(63) O<x=ih<l,
In this case the operator A is of the form (see Example 2 in Section 1):
1
-0.5h (a1Y;;,1 -CT1Yo), i=O,
(64) ( Ay )i = -(ay;;)x, i=l,2, ... ,N-1,
1
0.5h(aNyr,N+cr2yN), i=N.The operator (64) can be made of the sarne type as the preceding operator
(60) if we impose another grid
xi-1/2 = ( i - ~) h,