1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference Green's function 199

3.6 DIFFERENCE GREEN'S FUNCTION

1. Difference Green's function. Further estimation of a solution of the
   boundary-value problem for a second-order difference equation will involve
   its representation in terms of Green's function. The boundary-value prob-
   lem for the differential equation

(1)

Lu=-d ( k(x)-du) -q(x)u=-f(x),

dx dx

O<x<l,

u(O) = 0, u(l) = 0, k(x)>c 1 >0, q(x)>O,

can add interest and aid in understanding. As known, the solution of this

problem arranges itself as an integral

1

(2) u( x) = f G( x, ~) f ( 0 d~,

0

where G(x, ~) is the source function or Green's function. Function (2) is a

solution to equation (1) subject to the boundary conditions u(O) = 0 and

u( 1) = 0 if Green's function G( x, ~) as a function of x for fixed ~ satisfies

the conditions

d ( dG(x, ~))

LxG(x,~)= dx k(x) dx -q(x)G(x,~)=0,

(3) x I~,^0 < x < 1, G(O,~) = G(l,~) = 0,

[G] = G(~ + o,~) - G(~ - o,~) = 0, [ k dG] dx = -1 for X -- <, c.

The very defipition implies that Green's function is nonnegative and
symmetric:
G(x,~) > 0, G(x,~)=G(~,x).

The function C( x, ~) so defined can be written in the explicit form

(4) G(x, ~) =

D'(x) /)(~)

Cl'( 1)

Cl'(~) fJ(J:)

n( 1)

for x < ~,

for x :::: ~ ,