1549301742-The_Theory_of_Difference_Schemes__Samarskii

Mathe1natical apparatus in the theory of difference schemes 105

We perform simple algebraic calculations and write only the final re-

sults

N-1 N-1

(19) II y(k) 112 = L (y(k)(xs))

2

h = L h sin^2 7r~Xs

s=l s=l

~ L.., h 2 (l - cos _2Ir_kx_s) l.

s=l

Using qk = exp { i^2 7f 1 kh} and taking into account that qk = exp { i^2 7k x 8 }

and qf = 1, we find that

N-1 2 7rk N-1 N l

~ L.., h cos - R L h qks· -_ Reh qk - qk -_ Reh - qk -_ -h.

1

• J: 8 = e
  s=l s=l qk -^1 qk -^1
  Substitution of this value into (19) yields

II

(k) ll 2 = h (N - 1) h = hN = ~.

y 2 +2 2 2

Thus, the set of grid functions

(20) μCk)(x) =A y(k)(x), k=l,2, ... ,N-l,

constitutes an orthogonal and normed system in the sense of the inner

product (p(k),/-l(m)) ={;km·

5. Let a function .f(x) with the values .fo = fN = 0 be given on a grid
   wh. Then, obviously, it can be represented as a sum of the eigenfunctions
   of problem (14):
   N-1
   (21) .f(x) = L .fk μ(k)(x)
   k=l
   where the coefficients are defined by the relations .f k
   What is more, in that case
   N-1
   (22) 11 .f 112 = 2= .ff ·
   k=l
   Indeed,
   N-1
   11.r 112 = I: h .t^2 (x;) = (.t^2 , 1) =
   i=l

(.f(x), μ(k)(x)).

N-1 N-1

L .fk fm (p(k),p(m)) = L .r;,

k,m=l k=l

since (μ(k),μ(m)) ={;km as established before.