1549301742-The_Theory_of_Difference_Schemes__Samarskii

28 Preliminaries

with discriminant D = 4 ( cos^2 CY - 1) = -4 sin^2 CY < 0 and

e ±i a. Then q~. 2 = e ±i k a, particular solutions are given by

y~l) = cos kCY, y~^2 ) = sin kCY

and the general solution can be written as

Yk = C\ cos kc~ + C 2 sin kCY

with arbitrary constants C 1 and C 2.

b) Let p > 1. Setting p = ch CY and Yk = qk we get

q^2 - 2 ch CY q + 1 = Q

roots q1 2 '

with discriminant D = 4 (ch^2 CY-1) > 0 and roots q1 2 = ch CY±sh CY = e ±a.

'

Then q~ 2 = e ±k a, particular solutions become

'

y~l) = ch kCY ,

and the general solution can be written as

with arbitrary constants C 1 and C 2.

c) Let p = 1. In this case, for Yk = qk, we have q^2 -2q+l = 0 and q 1 , 2 = 1.

The particular solutions y~^1 ) = 1 and y~^2 ) = k form the general solution

Yk = C 1 + C 2 k as a linear function.

Exa1nple 2 It is required to calculate the integral

7r

= J cos kijJ - cos kt.p

cos '!' of, - cos ,./, '!' di/J ' k = 0, 1, 2, ....

0

First of all we note that

7r

I 1 (t.p) = f di/J =Jr.

0

We claim that Ik is just the solution of the Cauchy problem for the second-
order difference equation:

h + 1 - 2 cos t.p h + h-1 = 0 , k=l,2,. .. , Io= 0'