1549301742-The_Theory_of_Difference_Schemes__Samarskii

222 Hmnogeneous Difference Schemes

The Ritz inethod proves to be useful in studying the problem

(24) (ku')' - q1l = -f(x), 0 < x < 1, 1l ( 0) = 0' u ( 1) = 0.

Other ideas are connected with the function

{

0,

77(.s)= l+s,

1 - s

'

(25)

s<-l, s>l,

-l<s<O,

O<s<l,

which guides the choice of the functions

(26)

where xi= ih, i = 1, 2, ... , N-l, is a node of the grid wh ={xi= ih, z =

0, 1, ... , N, hN = 1}. It follows from the fo1·egoing that

(^0) for x < xi-l and x > X;+ 1 ,
X - X· z-1
(25') 7); ( x) = for < x <xi,
h xi-1
Xi+1 - X
for Xi < X < Xi+l '
h
and, hence,
0
d77i^1
(27) -
dx h
Upon substituting
1
h
for x < xi-1 and
for Xi-l < X < X;,
for xi < x < x i+ 1.
A 11 = - -d ( k -du) + q1l
dx dx
X > Xi+1 '
into (22) we get
1
(28)
J (
d77; d77· )
CX;j = k dx d~ + q 17; 7lj dx ,
0
l
/3; = J f(x) 7J;(x) dx.
0