228 Homogeneous Difference Schemes- The method of the summator identities (the method of approximating
an integral identity). A solution of the problem
Lu= (ku^1 )^1 - qu = -f(x),
( 4 7)x -- (^0) , x -- (^1) ,
satisfies the integral identity
l
(48) I[u,v]= j(k1l^1 v^1 +q1lv-fv)dx+a- 1 u(O)v(O)
0
- o- 2 u(l) v(l) - μ 1 v(O) - μ 2 v(l) = 0,
where v = v( x) is an arbitrary function being continuous on the segment
0 < x < 1 and having the summable derivative in the space L 2 [0, 1]. This
identity will be used for the determination of the generalized solution of
problem (47).
The design of a difference scheme on an equidistant grid wh ={xi=
ih, i = 0, 1, ... , N, hN = 1} is based on the approximation of the integral
identity ( 48) by the summator identity for grid functions, for instance,
N N-l
(49) h[u,v] = L aiYx,ivx,;h+ L (q;Yi -f;)vih
i=l
where V; is an arbitrary grid function. Here ai is any of the coefficients
having the form a;= A[k(x; +sh)], -1 < s < 0, and providing an approx-
imation of order 2: a; = k;_ 112 + O(h^2 ). There is no difficulty to verify
that
if the trapezoids formula is in hand in computing the integrals
Xi
j (qu - f) dx
Xi-I