1549301742-The_Theory_of_Difference_Schemes__Samarskii

496 Hornogeneous Difference Schemes for Time-Dependent Equations

12. A periodic problem. vVe are now interested in the proble1n of the heat
    distribution over a unifonn thin circlic ring 0 < <p < 27!" of radius l'o:

OU

ot'

a^2 [J2u

---

r5 o<p2 ,^0 < <p <^2 7l" , t' >^0 J

A unique detern1ination of a solution u( <p, t') necessitates imposing

the condition of periodicity

u( <p + 27r, t') = u( <p, t') for any <p E [O, 2 7r],

which, in turn, can be replaced by the condition of conjugation at the point

<p = 0:

u(O + 0, t') = u(2 7r - 0, t'), oul

O<p <p=IJ+U

oul

O<p <p=2rr-O.

By interchanging the variable5

the seg1nent 0 < <p < 27r is carried into the segn1ent 0 < :e < 1. In view of

this, the governing equation is 111odified into

OU

ot

O<x<l, i>O, u(x,0) = u 0 (x),

u(O + 0, t) = u(l - 0, t),

which is not surprising. On the grid

ou(O + 0, t)

ox

ou(l - 0, t)

ox

wh ={xi= ih, i = 0, 1, ... , N, h = 1/N}

we have occasion to use the simplest implicit sche1ne

Yt = ~h u , 0 < x = i h < 1 , t = j T > 0 , y( x, 0) = u 0 (a:) ,

which is supple1nentecl by the condition

Yo= YN