Difference approximation of elementary differential operators 61where~ is the mean value of x on the segment [a, x]:~=a+e(x-a), O<()<l - - ,
1J
(l-s)"ds=
1
7' + 1
0Upon substituting x + h for x and x for a into formula (11) we find that
1
( 13) v(x + h) = v(x) + h v'(x) + h^2 j (l - s) v^11 (x +sh) ds,
0(14)h2 h3
v(x + h) = v(x) + hv'(x) +
2
v^11 (x) + B v^111 (x)h4 fl
+ 6 (1 - s)^3 v(^4 l(x +sh) ds
0for r = 1 and r = 3, respectively. Replacing here h by -h and then s by
-s gives for later use the new formulae
( 15)( 16)
0
v(x - h) = v(x) - hv'(x) + h^2 j (l + s) v^11 (x +sh) ds,
-1
h2 h3
v(x - h) = v(x) - hv'(x) +
2v^11 (x) - B v^111 (x)
(I
+ ~ J^4 J (1 + s)^3 v(^4 l(x +sh) ds.
-1Adding (13) and (15), placing the term 2 v(x) on the left and then dividing
the resulting expression by h^2 , we finally get
where
1
vxx _ - v(x + h) + v(x h2 - h) -^2 v(x) -_ J g2 ( ) s v x "( + s h) d s,1 + s
l - sfor
for-1-l<s<O - ,
O<s<l. - -