Ho1nogeneous difference schen1es for the heat couduction 485In this way, the difference schemeC, Yt,o -- al Yx,o (a) +^0 ·^5 h j'( o a) 1 t > 0, y( a;, 0) = u 0 ( x) ,
is responsible for problem (68). It is plain to recover from the condition at
the point x = 0 that(71)v 1 = h [0.5hr f~a) + a 1 (1-o-) TYx,O + Cy]/(C h + a 1 o-r),
yielding 0 < u 1 < 1 for o- > 0. In turn, the boundary condition of the first
kind is imposed for i = N:(72)With these, for detennination of Yi = yf +i we obtain a second-order
difference equation supplied by the boundary conditions (71 )-(72) that can
be solved by the standard elimination method.
The intuition suggests that in such a setting the governing difference
equation and the boundary condition at the point x = 0 have one and the
same order of approximation O(rm~ + h^2 ). To make sure of it, it suffices
only to evaluate the residual
of, 'l'v -- C Ut o - al Ux (a) o - Q · 5 h J( a) ·
' ' 0Substituting here the expressions
= (k tt^1 ) 0 + U.5h(k1.1^1 )~ + O(h^2 ),
yields