1549301742-The_Theory_of_Difference_Schemes__Samarskii

190 Homogeneous Difference Schemes

and

(37) ( k u')~ = ( q u - f)o - ( k u')~ = ~ ( qu - f)o.

Substituting (37) into formula (36) and taking into account (29), we get

v = O(h^2 ).

The difference boundary condition (34) can be expressed by

a1 Y,. o

h ' - qo Yo = - fa '

*

With these, the difference scheme

0 < 7' = ih < 1'

(38)

h

h* = 4 , YN = μ2,

al Yr o

h ' - qo Yo = -fa '

*

is put in correspondence with problem (26)-(28).

The statement of the difference boundary-value problem for determi-

nation of Yi is

A; Yi-1 - C; Yi + Ai+1 Yi+1 = -F; , i=l,2,.,,N-1,

(39)

F; = 'Pi ri ,

which are supplemented with the boundary conditions

( 40)

where

and

This problem can be solved by the elimination method (see Chapter 1,
Section 1), for which the stability conditions are satisfied, because Ai > 0,
Ci >Ai+ Ai+l and 0 < x 1 ~ 1, x 2 = 0.