1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations 9

In the case of boundary conditions of the second or third kinds its

order is N + 1, while for the system (6) with the supplementary conditions

(8) the order is N - 1. All the matrices of interest possess the main feature:

they have nonzero elements only on the three diagonals (the main and two

adjacent ones).

With the aid of effective Gauss method for solving linear equations

with such matrices a direct method known as the elimination method has

been designed and unveils its potential in solving difference equations.

5. The elimination inethod. The problem we must solve take now the forrn

Ai Yi-l - Ci Yi+ Bi Yi+1 =-Fi, i = 1, 2, ... , N - 1;

(9)

where Ai -::f. 0 and Bi -::f. 0 for all i = 1, 2, ... , N - l.

Other ideas are connected with reduction of the original second-order

difference equation (9) to three first-order ones, which may be, generally

speaking, nonlinear. First of all, the recurrence relation with indeterminate

coefficients O:i and f3i is supposed to be valid:

( 10)

Substituting Yi-l = O:i Yi+ (3; into (9) yields

(Ai o:i - Ci) Yi+ Ai ,Bi+ Bi Yi+1 =-Fi,

which leads, because of (10), to

If the conditions

are fulfilled simultaneously, then the equation in view holds true for any Yi.
Thus, assuming Ci - o:i Ai -::f. 0, we establish the recurrence formulae for
determination of both O:i+l and f3i+l:

( 11)

(12)
A; {3; +Fi
C, - o:; Ai '

i = 1, 2, ... , N - 1,

i. = 1, 2, ... , N - 1,