Direct methods^645The statement of the difference Dirichlet problem associated with
problem ( 1) is(2) Ay= -f(x), y/,h =p(x),
A a~ y-- y .f:o .00. '
Probletn (2) was the object of special investigations in Chapter 4,- The decontposition 111et.hod. Co111111011 practice involves the reduction
of probletn (2) to the systern of vector equations
(3)
Yo= Fa,
where Y 1 and Fj are vectors, whose con1ponents are the values of the
solution Yij = y(ih 1 , jh 2 ) and the right-hand side fij = f(ih 1 , jh 2 ) on the
jth column of the grid w h and the difference operator C will be specified
in the sequel. By re-ordering of the right-hand sides of equations (2) at the
near-boundary nodes we might agree to consider Yi j = 0 at the boundary
nodes for i = 0 and i = Ni.
An alternative forn1 of writing equation ( 2) may be useful in the fur-
ther development:
(4)
where
1 < j < N 2 - 1,
Yoj = YNi] = 0,^0 < j < N2,
Yio = Pio = 0, YiN 2 = PiN 2 , 0 < i < Ni ,
'Pij = fij for 1 < i < 1V1 - 1 ,^1 < - j -< N 2 - 1,
1
'PN1-l,j = fN1-1,j + h2. I PN1,j.