1549301742-The_Theory_of_Difference_Schemes__Samarskii

6 Preliminaries

In the general case b = bi, a = ai and c = Ci, that is, we deal with known

functions of the argument i. If bi -::/:- 0, then

Yi+l = qi Yi + 'Pi ·

It is evident that

Ci

qi = bi J 'Pi =

fi

bi , bi -::/:-^0.

Thus, we see that this problern has a unique solution if the value y is given

for some i. For the sake of sirnplicity let Yo be known in advance for ·i = 0.

With this, one can determine all the values y 1 , y 2 , ••• by the recurrence

formula just established. In the case qi = q = const and 'Pi = 0 this

provides support for the view that the whole collection of Yi constitutes a

geometric progression. If qi = q and 'Pi -::/:- 0, then

Yi+ 1 = q Yi + 'Pi = q ( q Yi -1 + 'Pi -1 ) + 'Pi = q^2 Yi -i + 'Pi + q 'Pi -i.

The outcome of this is

(3) Yi+l = q i+l Yo +'Pi + q 'Pi-l + · · · + q i-1 'Pi + q i 'Po

= qi+l Yo+ L ' qi-k 'Pk.

k=O

Adopting those ideas, it perfonns no difficulty to find a solution to the

equation Yi+l = qi Yi + 'Pi, i = 0, 1, 2, ... , in the case qi -::/:- const.

In tackling the first-order difference inequalities

(4) Yi+l < q Yi + Ii J i=0,1,2,

with known mernbers q > 0 and j~ and a given value y 0 we prefer an

alternative form of writing

(5) Vi+l = q Vi + fi , Vo =Yo·

Its solution can be most readily found with further reference to the relation

Yi <Vi. Indeed, subtracting equality (5) from inequality (4) we get

yielding Yi+l < Vi+l for any q, where vi has been expressed in q, v 0 , fi by
analogy with (3).