1549301742-The_Theory_of_Difference_Schemes__Samarskii

Heat conduction equation with constant coefficients 317

we find that

II

OU 112 ~ ~ II OU 112 = ( OU )

ot + 2 ot ox f ' ot ·

To majorize the right-hand side, we make use of the Cauchy-Bunyakovski1

inequality and the c:-inequality I ab I < c: a^2 + ~ c::-^1 b^2 with c: = 1, pennitting

us to arrive at the relations

(

f I ou) fit <

implying that

o II ou ll

2

ot OX < 2 1 II J(x, t) II^2.

By integrating with respect to t we thus have

ou(t) ou(O)

t l/2

< + ~ [ j 11f(t')11^2 dt']

ox ox

0

<

ou(O)

ox + fx O<tST max^11 f(t)^11 ·

With the aid of the relation

1

II

0

II u lie= o<x<l max I u(x) I< -2 oux II

we finally get

l

II u(t) lie < 2

ou(O)

o'J.:

+ -2 l fx -2 max II f(t) II.

O<tST

At the next stage a similar estimate is needed in this connection for the
difference problem (16). We proceed as usual. This amounts to introducing
the inner products and associated norms:

N-l

(y, v) = L Y;V;h, II Y II= J(Y:y),

i=l

N

(y, v] = L Y; V; h, llYll = J(Y,Y] I

i=l