1549301742-The_Theory_of_Difference_Schemes__Samarskii

274 Difference Schemes for Elliptic Equations

where {μk, (x 1 )} is an orthonormal system of eigenfunctions: (μk, , μk;) i =

bk, k; under the inner product structure (y, v) 1 = 2=~1=~^1 y(i 1 h 1 ) v(i 1 h 1 ) h 1 •

From the governing relations (3) a similar problem arises for TJ( x 2 )

with >,(^2 ) = >, - ;,(ll:

T/o = 0, T/N2 = Q '

whose solution is given by

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

Here ( T/k 2 , ilk;) 2 = bk 2 k;, where (y, v ) 2 = L~~~^1 y( i 2 h 2 ) v( i 2 h 2 ) h 2 • Thus,

problem ( 1) is completely solved, meaning that to the eigenvalues

or

(4) >, kik2 -_ 4 ( h2 1. srn^2 7r 21 k^1 h^1 + h2 1. srn^2 7r 21 k^2 h^2 ) '

1 1 2 2

k 1 =1,2,. .. ,Ni -1, k 2 = 1, 2, ... , N2 - 1 ,

there correspond the eigenfunctions

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

The total number of the eigenfunctions is equal to (Ni - l)(N 2 - 1) = N.
These constitute an orthonormal system

(6)