1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations^7

4. The second-order difference equations. The Cauchy problem. Bounda-
   ry-value problems. The second-order difference equation transforms into
   a more transparent form

(6) Ai Yi-l - Ci Yi+ Bi Yi+l =-Fi, i = 1, 2, ... J

Ai -::/:- 0, Bi -::/:- 0,

which in the notation b. Yi = Yi+l - Yi becomes is modified as follows

(7) Bib. Yi - Ai b. Yi-l - (Ci - Bi - Ai) Yi= -Fi.

By virtue of the relations

b. Yi - V' Yi = b. Yi - b. Yi -1 = b.^2 Yi - 1 = Yi+ 1 - 2 Yi + Yi -t ,

b. Yi- 1 = -b.^2 Yi- 1 + b. Yi ,

equation (7), in turn, can be rewritten as

Ai b.^2 Yi - t + (Bi - Ai) b. Yi - (Ci - A; - Bi) Yi = - Fi , A; -::/:- 0 ,

or, what arnounts to the same,

Bi b.^2 Yi - 1 + (Bi - Ai) b. Yi -1 - (Ci - Ai - Bi) Yi = -Fi.

The latter difference equation clarifies that (6) is an analog of a second-
order differential equation.
It is necessary to specify two conditions for the complete posing of this
or that problem. The assigned values of y and b. y suit us perfectly and
lie in the background a widespread classification which will be used in the
sequel. When equation (6) is put together with the values Yi and b. Yi given
at one point, they are referred to as the Cauchy problem. Combination
of two conditions at different nonneighboring points with equation (6) leads
to a boundary-value problem.
For the sake of definiteness, we concentrate primarily on Cauchy prob-
lem with given Yo and b. Yo. Knowing Yo and y 1 = Yo + b. Yo, Yi for
i = 2, 3, ... serves as a basis for the formulae

Yi+l =

Ci Yi - Ai Yi - 1 - Fi

Bi

Bi -::/:- 0.

Therefore, it is concluded that problem (6) has a unique solution for given
value Yo and Y 1 •