1549301742-The_Theory_of_Difference_Schemes__Samarskii

Mathematical apparatus in the theory of difference schemes 109

As can readily be observed, the difference scheme of second-order approxi-

mation acquires the form

( 30) Yxx +A Y = 0' x = ih, i = 1, 2, ... , N - 1 ,

(Yx,O - crl Yo)+~ h,\ Yo= 0,

-(Yx,N + CT2 YN) + ~ h,\ YN = 0, y(x) f=_ 0,

yielding

ux,O - cr 1 U 0 + ~ h,\ u 0 = u~ + ~ hu~' + O(h^2 ) - cr 1 u 0 +th,\ u 0 = O(h^2 ).

It will be sensible to introduce the operator A with the values

1

• 0.5h (Y:v - cr 1 y) for x-0 - )
  (31) Ay = -Yxx for x -- ih ) O<i<N,
  1
  0.5h (Yx+cr2y) for x -- l '
  by means of which problen1 (30) admits an alternative forn1 of writing
  (32) Ap = ,\ f-l.
  The domain of operator (31) coincides with the entire space H, the domain
  of operator (25), A+ A*> 0 and, what is more,

[Ay, Y] = ( Yx, Yx] + CT1 Y; + CT2 Y~,

it being understood that A > 0 if either at least one of the coefficients cr 1

and cr 2 becomes nonzero or cr 1 + cr 2 > 0, but cr 1 > 0 and cr 2 > 0. In that

case [Ay, y] = 0 only for y(x) - 0.

Unlike the preceding two problems, we come to nothing in trying to

derive the explicit formula for O'.. The parameter O'. should be recovered

fron1 the equation

(33)

but the eigenvalues are expressed, as usual, by the formulae

(34) /\k ' = h2 4. sm^2 -2-O'.k h k=0,1,2, ... ,JV.

The eigenfunctions are determined within a constant by

h ().

(35) f-lk(x) =cos O'.k (l - x) +.^1 sm O'.k (l - x).

sm O'.k h