1549301742-The_Theory_of_Difference_Schemes__Samarskii

224 Homogeneous Difference Schemes

problems, whose ingredients have no fixed sign and turn out to be non-self-

adjoint. In this case the coefficients Yi of the approximate solution (20) are

to be determined from the orthogonality conditions for the residual Aun - f

with respect to all of the basis functions 7Ji(x):

(33) (Aun-f,7];)=0, i=l,2, ... ,n.

A non-self-adjoint boundary-value problem acquires the form:

(ku')' + r(x) u' - q(x) u = -f(x),

(34) 0 < x < 1, u(O) = u(l) = 0,

k(x) > 0, q(x) > 0.

We introduce the grid wh ={xi= ih, i = 0, 1, ... , N, hN = l}. Then the

dirnension n of the space Vn equals N - 1. The functions

7J;(x)=17(x~xi), i=l,2, ... ,N-1,
where the function 77( s) was specified by (25), provide the background for
subsequent constructions.

(35)

In this context, condition (33) becomes

N-l

L IY;j Yj - f3; = 0 , i = 1, 2, ... , N - 1 ,

j=l

where

1

J

(

di7; di7j
cxij = k(x) dx dx - r(x) -' d77· dx 7lj(x) + q(x) 7J;(x) 7lj(x) ) dx,
0
(36)
1
f3;= jf(x)17;(x)dx, i,J=l,2, ... ,N-1.
0
By definitions (25), (25') of the function 17i(x), the coefficients <-'<;,j are
nonzero only for J = i - 1, i, i + 1. Retaining notations (30) for ai and cl;
and (32) for tpi and accepting

x; u

b-:- = J:_ J r(x) (x - X;_ 1 ) dx = J r(xi +sh) (1 + s) ds,

' h2

Xi-I - l

(37)
"i+1 l
bi = : 2 J r(x) (xi+i - x) dx = ;· r(x; +sh) (l - s) ds,
Xi 0