1549301742-The_Theory_of_Difference_Schemes__Samarskii

650 Methods for Solving Grid Equations

3. The inet.hod of separation of variables. The problen1 we must solve is
   problem (2) with the hon1ogeneous boundary conditions

(13) Ay = -<p,

where the function <p differs from the right-hand side f of problem (2) only

at Lhe near-boundary nodes in the following way: within the quantity

1

hi^2

for i 1 = 1, i 1 = N 1 - 1 and within the quantity h\ for i 2 = 1, 'l 2 = N 2 - l.

2

Before giving further motivations, it will be sensible to introduce the

eigenfunction μk(jh 2 ) and the eigenvalue >.k with the number k of the

problem

We learn from Chapter 2 Lhat

' - 4. 2 k 7r h2

/\ k - 2 sm I ,

h2 2 2

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

and may attempt a solution of problem (13) in the form

N2-1

(15) Yij = ~ ck(ih 1 )μk(jh 2 ),
k=I

z = 1, 2, ... , N 1 - 1, j=l,2, ... ,N 2 -l,

where the Fourier coefficient ck depends on J: 1 = 'ih 1.
Upon substituting representation ( 15) in Lo equation ( 13) we obtain

N2-1

~ [μk(jh 2 ) A1 ck(ih 1 ) + μk(jh 2 ) A2 ck(ih 1 )]

k=!

Ne-I

~ <f!k(ih1) p,:(jh2)'

k=I