1549301742-The_Theory_of_Difference_Schemes__Samarskii

200 Homogeneous Difference Schemes

where ex(x) and f3(x) are solutions of the relevant Cauchy problems:

Lex= 0,

(5)

L {3 = 0,

0 < x < 1,

0 < x < 1,

ex(O) = 0,

{3(1) = 0,

k(O) ex'(O) = 1,

k( 1) {3^1 ( 1) = -1.

The functions ex( x) and {3( x) are linearly independent. This is due to

the fact that the Wronskian is nonzero 6.( x) f 0. Moreover, ex( x) > 0 for

x > 0 and {3( x) > 0 for 0 ::; x < 1.

We now turn to a difference equation of second order. We learn from

Chapter 1, Section l that any difference equation of second order A; Yi-i -

C; Yi + Bi Y;+ 1 = - F; can be treated as an equation of divergent type,

meanmg

ai+1 (Y;+1 - Y;) - ai (Y; - Yi-1) - d; Y; = -tp; ·

By replacing here tp; by h^2 tp; and di by h^2 d; we obtain

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

which is more convenient for the further comparision with the differential

equation. For the moment, we write down this equation without subscripts

Ay = (ay 5 ;)x - dy = -tp(x), x = ih'

(6)

a(x) 2 c 1 > 0, d( x) > 0' i = 1, 2, ... , N - 1.

For i = 0 ( x = 0) and i = N ( x = 1) the boundary conditions of the

first kind are imposed as usual:

(7) Yo= 0, YN = 0.

Common practice i'n numerical analysis involves the inner products

N-l N

(y, v) = L Y;V;h, (y, v] = L Y;V;h.

i=l i=l

The traditional way of covering this is to seek a solution of problem (6) in
the form
N-1
(8) Y; = L Gik tpk h = (Gik) tpk)
k=l