1549301742-The_Theory_of_Difference_Schemes__Samarskii

Mathematical apparatus in the theory of difference schemes 107

Indeed '

[Ay, v] = (-Yxx, v) + h (^2 2 )

2 - h Vo Yx,O + h 'UN Yx,N ,

J\!Iaking use of the second Green formula (10), we get

[ Ay, V] = ( Y, - V xx) + (VY x - yv x ) o - ( v Y x - yv x ) N + ( -Vo Y x, o + v N Y x, N )

= (y, -vxx) +(-Yo vx,O + YN vx,N) = [y, Av],

that is, A= A*, The first Green formula (8) gives for z = y

[Ay,y] = (y,-Yxx)+(-YoYxo+YNYxN) ' '

Problem (26) can be solved by appeal to the general theory (see Sec-

tion l), Let us determine the eigenvalues >.k and eigenfunctions μk(x) of

problem (24) accepting a solution of problem (24) in the form

y = p( x) = A cos O'.X)

Upon substituting p(x) into equation (24) we get

>. = _i_ sin^2 o:h

h^2 2

We now require the boundary conditions to be satisfied at the points
x = 0 and x = /, The condition cos o:h - 1 + ~ h^2 >. = 0 is automatically
fulfilled at the point x = 0, The case where x = l is not simple to follow
and needs investigation:

(1-~ h^2 >.) cos o:/ - cos o:(/ - h) = 0

or

cos o:h cos o:/ - cos o:/ cos o:h + sin o:/ sin o:h = sin o:/ sin o:h = 0 ,

implying that o:/ = 7rk, k = 0, 1, 2, .. , , N, Thus, we have determined the
eigenvalues

(27) >. 0 = 0,
4 ' 2 7rkh

.k = h 2 sm 21 , k = J, 2,, , , , N,