1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference Green's function 201

under the agreement that this expression solves the equation Ayi = -tp;.

From the equality Ayi = I:~:/ A(i) Gik tpk h it is easily seen that equation

(6) holds true only if A(i) Gik = -D;k/h, where Dik is Kronecker's delta:

{

1,

{jik = 0,

i = k'

i I k.

The equalities Yo = YN = 0 are certainly true for the choice Gok = GNk = 0,

thus formula (8) gives the solution of problem (6)-(7) if Gik = G(x;, xk)

as a function of i for fixed k = 1, 2, ... , N - 1 satisfies the conditions

(9)

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

We must show that Green's function specified in such a way exists and

find its explicit representation similar to expression ( 4). With this aim, the

functions CY; and f3; will be declared to be solutions of the corresponding

Cauchy problems

A CY; = 0 ' i = 1, 2, ... , N - 1,

CYo = 0 ' 1,

(10)

A f3; = 0, z= 1,2,. .. ,N-l,

It is straightforward to verify that the functions CY; and f3; possess the
following properties:

1. CY; and {3i are posi ti Ve functions, CY; being monotonically increasing
   and {3; being monotonically decreasing:

CY;> 0 for i = 1,2, ... ,N and f3; > 0 for i = 0,1, ... ,N-l. Indeed,
conditions (10) imply that

i-1

a; CYx,i = 1 + L h elk Cl!k ,

k=l