1549301742-The_Theory_of_Difference_Schemes__Samarskii

276 Difference Schemes for Elliptic Equations

In a similar way we find that

etc. By means of the method of separation of variables problern (8) reduces

to the two one-dimensional eigenvalue problen1s for p(x 1 ) = p(i 1 h 1 ) and

TJ(x2) = TJ(i2 h2):

Px 1 x 1 + >, (i l P--^0 >

(_,,_ r"'X1 + l 2 h I ;,(^1 l,,) ,,., I I --o. )

X1= 1

Their solutions have been already found in Chapter 2, Section 3.

Along these lines, we can in1mediately write down the eigenvalues and

the orthonormal eigenfunctions

(9)

(10)

where

;,(!) 0 = ;,(2) 0 = 0)

(11)