1549301742-The_Theory_of_Difference_Schemes__Samarskii

180 Homogeneous Difference Schemes

At any point x E (0, 1) condition (6) is equivalent to the pan of

continuity conditions at the single point x = 0:

(7) u(O + 0) = u(l - 0), u'(O + 0) = u'(l - 0).

Problem (5)-(6) has a unique solution having the estirnate

II u lie ::; II f llc/qo

in complete agreement with the maximum principle.

For q 0 = 0 the statement of the problem is

u^11 = -f(x), u(O + 0) = u(l - 0), u' (0 + 0) = u' (1 - 0) ,

which is solvable under the condition fc~ f(x)dx = 0 and possesses a unique

solution u = u( x) only if

l

(8) Ju(J:)dx=O.

0

Indeed, the general solution to the equation u^11 = -f(x) aclrnits the form

:c t x

u(x) = cl x + c2 - J (! f( ex) do:) dt =cl x + c2 - J (x - t)f(t) dt

0 0 0

with arbitrary constants cl and c2.

With this, conditions (7) provide

1 l

J J(t) dt = 0, cl = - j t J(t) dt = o,

0 • 0

it being understood that the function u( x) can be recovered from condi-
tions (7) within a constant C 2. U ncle1· condition (8) we find that C 2 = 0,
extracting a unique solution of the problem.
As a first step towards the solution of the original problem, we initiate
the design of a difference scheme on an equidistant grid w h = {xi = ih, i =
0, 1, ... , N} on the segment 0::; x < 1 with step h = l/N and approximate
equation (5) and the continuity conditions (7). The first of these conditions
is satisfied if Yo = YN ·