1549301742-The_Theory_of_Difference_Schemes__Samarskii

Mathematical apparatus in the theory of difference schemes 103

1

where Jk = J J (x) uk (x) dx; in so doing

0

1 00

II f 112 = J J^2 (x) dx = I: !f.

0 k=l

We put the difference problem

Yxx + >.y = 0 x -- ih '^0 < i < N,

(14)

y(O) = y(l) = 0, y(x)tO,

h -- N' I

of searching for nontrivial solutions being the eigenfunctions and eigenvalues
of problem (14) in correspondence with the differential problem (13). For
this, the index form of (14) is

(15) Yi+1 - 2(1-~h^2 >.) Yi+ Yi-1 = 0, i=l,2, ... ,N-1.

We look for a solution of problem (14) in the form

y(x) =sin o:x

with o: to be determined. Along these lines,

Yi+ 1 + Yi-l =sin o:(x + h) +sin o:(x - h) = 2 sin o:x cos o:h.

Substituting the expression just established into (15) yields

2 sin o:x cos o:h = 2 (1 - ~ h^2 >.) sin o:x.

Since only nontrivial solutions are those to be found, that is, sin o:x t 0, it
follows from the foregoing that

1 - ~ h^2 >. = cos o: h

and
2 4. 2 o:h
A = h 2 ( 1 - cos O'. h) = h 2 S!Il 2.

The value of the parameter o: is chosen in such a way that the function
y(x) =sin o:x satisfies the boundary conditions of problem (14):

y(O) = y(l) = 0.