1549301742-The_Theory_of_Difference_Schemes__Samarskii

368 Difference Schemes with Constant Coefficients

whose solutions are

srn-.? --Irk h

2 '

X (k)(x) = V2 sin Irkx.

From (9) we get the difference equation of second order for Tk(t):

( Tk ) tt + '°'k \ Tk (a) -- 0

or the equation

which can be rewritten as

~ T2 Ak

1 + ()" T^2 Ak

(10)

We may attempt a solution of the preceding equation in the form

Tk = Tk(tj) = qf. Thus the quadratic equation for q arises from (10):

q^2 - 2 (1 - a) q + 1 = 0 (the subscript k is omitted for a while). Careful

analysis of its roots q 1 2 = 1 - o; ± Ja^2 - 2 o; shows that for 0 < o; < 2,

'

the values q 1 , 2 = 1 - o; ± i Jo; (2 - o;) are complex with I q 1 , 2 I = 1. It is

sensible to pass to a new variable 'Pk for which

cos 'Pk = 1 - o;k '

making it possible to get q~k) = ei 'Pk and q~k) = e-i 'Pk, due to which the
general solution to equation (10) is representable by

where Ak and Bk stand for arbitrary constants.
After that, weJook for a solution of problem ( 4a) as a sum of particular
solutions

N-l

( 11) Yj = L ( Ak cos j<pk +Bk sin j<pk) X (k)(x).

k=l

Let u 0 k and u 0 k be the appropriate coefficients in the relevant expansions
of u 0 (x) and u 0 (x):

N-I N-l
( 12) '1l 0 (x) = L u 0 k X (k)(x), uo(x) = L Uok'y(k)(x).
k=l k=l