1549301742-The_Theory_of_Difference_Schemes__Samarskii

108 Basic Concepts of the Theory of Difference Schemes

and the orthonormal eigenfunctions {μk ( x)}, for which

where

(28)

J.lo = If'

fl 7rkX

μk = v l cos -/-)

k-- m )

k -::J m,

fl 7rNx

μ N = v f cos -/ - '

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

The normalizing multiplier Ak can be recovered from the condition

where the sum

(

7rkx 7rkx) N-l 7rkx

cos -

1

-, cos -

1

• = L h cos^2 -
  1

-s

s=l

is calculated by analogy with the preceding section. The orthonormality of
the eigenfunctions {μk} follows from the general theory, since A = A* > 0
and all the eigenvalues are simple.
Any grid function .f(x) defined on the grid wh arranges itself into a
series of {J.lk(x)}:

N

f(x) = L fk μk(x), k=O,l, ... ,N,

k=O

N

under the inner pro· d uct structure [! , fl = '\'"' i_, fk^2.

k=O

The statement of the third boundary-value problem on eigen-

values is

(29) u^11 + ..\u(x) = 0,

u'(O) = cr 1 u(O),

• u'(l) = cr 2 u(l),

O<x<l,

()] > 0)

cr 2 >0, u(x)f=.O.