1549301742-The_Theory_of_Difference_Schemes__Samarskii

24 Preliminaries

11. The second-order difference equations with constant coefficients. If
    the coefficients of the difference equation

(44) £ [Yi] = Ai Yi -i - Ci Yi + Bi Yi+ 1 = 0

do not depend on i, meaning Ai = a, Ci = c and Bi = b for all i = 1, 2, ... ,

a solution of the following equation

(45) byi+1-CYi+ay,_1 =0, bf:-0, af:-0,

1nay be found in explicit form. In so doing, let y?) and y~^2 ) be two solutions

of difference equation ( 44) that are linearly independent if the equalities

i=0,1,2, ... ,

hold only for C 1 = C 2 = 0. This condition can be replaced by the usual

require1nent for the determinant of the syste1n of algebraic equations

cl Yi + c2 Yi - o ,

{

(!) '(2)_

(!) '. (2) -

C1 Yi+m + C2 Yi+m - 0 , rn= 1,2, ... ,

saying that, for all 1: and in

.0.i,i+m =

( I)

Yi

(!)

Yi+m

y~2)

(2)

Yi+m

f:- 0.

By pursuing the parallel we see, in particular, that the condition

Y?)

(I )

Yi+ I

y~2)

(2)

Yi+I

f:- 0

is analogous to that of the linear independence for solutions u( x) and u( x)
of an ordinary second-order differential equation:

u(x)

(u(x))'

u(x)

(u(x))'

f:- 0.

Eliminating y~~ 1 and y~~ 1 with the aid of equation ( 44) we get

• 1 ( (2) (!) (!) (2)) - Ai ( (!) (2) (I) (2))
  t.i,i+1 - B· 1 Bi Yi+1 Yi - Bi Yi+l Yi - - B 1 Yi Yi-l - Yi-l Yi ,