1549301742-The_Theory_of_Difference_Schemes__Samarskii

158 Homogeneous Difference Schemes

Comparision of the resulting expansions with (18) permits us to assign the

values

(21) A[l] = 1, A[s] = -~

by virtue of the relations

A[l + s] + A[s] = A[l] + 2 A[s] = 0,

A[(l + s)2] - A[s^2 ] = A[(l + s)^2 - s^2 ] = A[l + 2s]

= A[l] + 2A[s] = 0.

To decide for yourself whether conditions ( 18) related to schemes (17),

(15^1 ) are met or not, we should take into account that

a(x+h)±a(x)=Cl(x)a(x+h)(-(l ± (

1

),

ax) Clx+h)

1 ( 1 1 ) _ 1 O(h^2 )

2 a(x+h)+a(x) -k(x)+ '

~ 1 __ 1_ = _1_ +O(h2),

( ) ( )

I

h a(x+h) a(;e) k(x)

a(x) a(x + h) = k^2 (x) + O(h^2 ).

We contrived to do it and give an answer to this question. When this is
the case, the functional A[k(s)] also satisfies conditions (21). As far as the
function al.'3
l/2
F[/(s)] = J f(.s) els,
-l/2
.
are concerned, we thus have

1/2

F[l] = J els= 1,

-1/2

0

A[l] = J els= 1,

-1

IJ

A [.f(.s)] = J /(.s) els

-1

1/2

F[s] = j s els= 0,

-l/2

ll

A[s]= j sels=-~,

-1