1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference schemes for the equation of vibrations of a string 371

This estimate remains valid for scheme ( 4a) under the constraint

(21)
4 r-? '
where E > 0 is an arbitrary number. To make sure of it, we should replace
everywhere in the above proof μk by o:k = ( 1 r^2 .Ak) ( 1 + O" r^2 .Ak)-^1.
The superposition principle unveils its potential in investigating the
stability of scheme (4) with respect to the right-hand side by considering
the problem

( 4b) Ytt - A - Y (a) + 'P' Yo=YN=O, y(x,0)=0, Yt(x,0)=0,

whose solution is sought as a sum

(22)

j. ·I

LT yJ,J

j'=O

where Y j,j' as a function of j solves for fixed j' the ho1nogeneous equation

O<j'<j,

supplied by the boundary conditions

(24)
y;J,J · •I - yJ,J · ·I -^0
o - N -

and the initial conditions

·I ·I y j'+J,jl - y j',j'

Yt J ,J --

y j'+1,j' •I

<1> J

'

·I ·I
(25) yJ,J =0,
T T

where <1> j' is so chosen as to satisfy the nonhomogeneous equation ( 4 b).
A similar. ·/ problem arises naturally for the function q,,j. By the defini-
tion of Y J,J
'

Ytt i --

1 · · j-I • •I

• yJ+l,J + ~ L_, T y_J,J tt '
  T j'=O

j-1

Ay(a) = O"T AYj+i,j +LT A(Yj,j')(a)

JI =Ci