1549301742-The_Theory_of_Difference_Schemes__Samarskii

364 Difference Schemes with Constant Coefficients

5.6 DIFFERENCE SCHEMES FOR THE EQUATION

OF VIBRATIONS OF A STRING

1. The statement of the difference problern and calculations of the approx-
   imation error. In this section we study the equation of vibrations of a
   string

82 u

8t^2 I

0<x 1 <l,

In this view, it seems reasonable to pass to the dimensionless variables

x = x 1 /l and t = at 1 /l, due to which the initial equation is representable

by

(1) O<t<T.

At the initial moment the supplementary conditions are specified by

(2) u(x, 0) = u 0 (x), 8u(x, at 0) __ - Uo (· .r ) '

where u 0 (x) is the initial deviation and ii 0 (x) is the initial velocity. The
string's ends n1ove in accordance with the known laws

(3) u(O, t) = μ 1 (t),

By analogy with Section 1.2 we introduce in the domain D = { 0 < x <
1, 0 < t < T} the rectangular grid whr. As equation (1) contains the
second derivative int, the number of layers cannot be any smaller than 3.
Retaining the preceding notations, we have

y = yj' f; = y}+l' fJ = yj -1' -

y-y y - fJ

Yt - T Yr= T

Ay = Yxx'

Yt - Yt iJ-2y+fJ - Yt + Yt y-y

Ytt -- T T2 1jo. t - 2 2T

As before, we replace the derivatives built into equation (1) by the formulae

f ~ <p.