1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations^29

assuming 'P to be arbitrary fixed. Indeed, the chain of the identities occur:

[cos ( k + 1) Vi - cos ( k + 1) 'Pl + [cos ( k - 1) Vi - cos ( k - 1) 'Pl

= 2 cos k'lji cos Vi - 2 cos ktp cos 'P

= 2 (cos k'lji - cos ktp) cos 'P + 2 (cos Vi - cos 'P) cos k'lji ,

from which the relations immediately follow

h+1 + h-1 = 2

7r

I

(cos 1/• - cos 'P) cos k'lji

cos 'P h + 2 d'lji

. cos Vi - cos 'P
0
7r
= 2 cos 'P h + 2 J cos k'lji d'lji = 2 cos 'P h ,
0

yielding h+1 - 2 costph + h-1 = 0.

From Example 1, case a) we know that

k > 1,

The initial conditions for k = 0 and k = 1 give cl = 0 and C2 = 7r I sin 'P
in connection with the available information that cl cos 'P + c2 sin 'P = 7r.
The outcome of this is

sin ktp

h('P) = 7r --

srn 'P

k = 0, 1, 2, ....

12. Formulae of "difference differentiation" by parts of the product and
    sum. The formula for differentiating the product of two real functions
    u(x) and v(x)
    d ) dv du
    -d x (u(x) v(x) = u(x) -d x + v(x) -d x

is well-known from differential calculus. In an attempt to establish a grid

analog of this correlation, we consider any two grid functions Yi and Vi of the

discrete variable i = 0, ±1, ±2, .... The following formulae of "difference

differentiation" are valid:

6. (Yi vi) = Yi 6. Vi + Vi+l 6. Yi = Yi+l 6. Vi +Vi 6. Yi ,
   (47)
   \7 (Yi vi) = Yi-l \7 Vi + Vi V Yi = Yi \7 Vi + Vi-l \7 Yi