Other problems^181At the nodes xi = ih, i = 1, 2, ... , N - 1, we write down the three-
point equationYxcc - qo Y = -<p(x), x = ih, i = 0, 1, ... , N - 1.
In this context, the difference derivativesux,N = u'(l - 0) - ~ h u^11 (l - 0) + O(h^2 ),
ucc,o = 1t'(O + 0) +! h u^11 (0 + 0) + O(h^2 )
will be given special investigation. Substituting here u^11 = q 0 u 0 - f recov-
ered from (5) we obtainux,N +! h(q 0 u(l) - f(l - 0)) = u'(l - 0) + O(h^2 ),
ucc,O -! h(q 0 u(O) - J(O + 0)) = u' (0 + 0) + O(h^2 ),
With these relations established, the second continuity condition u'(O+O) =
1t'(l - 0) is approximated to O(h^2 ) by the difference equation(9) ucc,O - ~ h q 0 Yo+~ h f(O + 0) = ux,N + ~ h q 0 YN - ~ h J(l - 0).
By merely setting YN+l = y 1 condition (9) becomes
<f'N = ~ (f(l - 0) + f(O + 0)).
This is a way of establishing the correspondence between problem (5 ), (7)
and the difference scheme(10) Yxcc - qo Y = -<p(x), x = ih, i=l,2, ... ,N,
with the conditions of periodicity( 11) Yo= YN,We will elaborate on this for rather complicated cases and turn to the
equation with variable coefficients
(12) (ku^1 )^1 - qu. = -f(x),
where k(x), q(x) and J(x) refer to periodic functions of period 1:(13) k(x + 1) = k(x), q(x + 1) = q(x), f(x + 1) = f(x),