1549301742-The_Theory_of_Difference_Schemes__Samarskii

206 Homogeneous Difference Schemes

Proof Putting μ = T/x and applying then formula (18), we deduce by the

preceding lemma that

(27) I Y;(x) I< (I G[(x,~) I, I TJ(~) I]~! (1, I 17(~) I].

C1

The function TJ( x) can be recovered from the condition 1)x = t_p within an

arbitrary constant. A solution to the difference equation

T/i+1 - T/i = h l.p;

can be expressed either by

N-1 N-1

T/i = T/N - L h t_p 8 = - L h t_p 8 if 1)N = 0

s =i

or by

if T/1 = 0.

s=l s=l

Substitution of these formulae for T/i into the right-hand side of inequality

(27) leads to estimates (24) and (25).

To prove inequality (26), it suffices to set cp* = μx, so that t_p = (TJ+μ)x

and to repeat. the preceding arguments.

Remark Formula (16) entails the estimate

1 1 1

llYllc < -(1,lcpl) c1 < -111.fll c1 <-Ill.file C1 _,

in the norrn II t_p II= J(cp, cp). It is plain to show the obvious inequalities
i
II 2= hcpk II < (1, I t_p I)< II t_p II <II t_p lie ..
k=l
With the aid of the estimate
2
II Ga;(x, ~)lie < -
c1
an a priori estirnate in the space C for the difference derivative of a solution
of the boundary-value problem (6)-(7) can be derived without difficulty.
Indeed, the relations
2
IY:cl= l(Ga;(x,~),cp(O)I <-(1,lcpl)
c1
lead to
2
llYxllc < -(1,lcpl).
c1