1549301742-The_Theory_of_Difference_Schemes__Samarskii

208 Homogeneous Difference Schemes

where A; and B; are some numbers which will be specified in the sequel,

v;(x) and v;(x) are linearly independent solutions to the hornogeneous

equation L(p,q lu = 0 (pattern functions) and v~ ( x) is a particular solution to

the nonhomogeneous equation ( 1) subject to the homogeneous conditions:

(4) L(p,q)v! = f(x), X;_ 1 < x < X;+ 1 , v~(x;+ 1 ) = v!(x;_ 1 ) = 0.

The accepted view is that the pattern functions v; ( x) and v~ ( x)

will be declared to be solutions of the appropriate Cauchy problems

(5)

(6)

(

1

) (v;)'(x;+ 1 ) = -1.

p Xi+1

By merely setting in (3) x = x;_ 1 and x = x;+ 1 we find that

(7)

Pattern functions so defined possess a nurnber of nice properties (com-

pare with Section 6), so there is some reason to be concerned about this:

(8)

(9)

(10)

1) v;(x) is positive and rnonotonically increasing for X;_ 1 < x < X;+ 1 ;

v~ ( x) is positive and monotonically decreasing for X;_ 1 < x < X;+ 1 ;

2) the equality is certainly true:

v; (x;+ 1 ) = v;(x;_ 1 );

3) the relation occurs:

xi

v~(x;+i) = v~(x;) + v~(x;) + v;(x;) j v~ q(x) dx

Xi-.1

Ti+l

+v~(x;) j v;q(x) dx;

{(,' i

4) the recurrence relation is valid: