Homogeneous difference schemes for the heat conduction 487- The case when the coefficient of conductivity k depends on t, k = k( x, t ).
So far we have preassun1ed for the sake of clarity that the coefficient of
conductivity k depends only on one variable :t:.
In a common setting the governing equation ( 1) of the general form
oll ot =ox o ( k(x,t) oll) ox +f(x,t),
is put together with the boundary and initial conditions ( 2)-(3). In prepa-
ration for this, the intention is to use instead of (6) another difference
scheme(75) Yt = A(t) y\a) + <p,
where A(i) v = (a(x, i) v,,) x' i = tj+o 5 and the coefficient a(x, t) for fixed t
appears by exactly the same reasoning as before (see Sections 2 or 4 of the
present chapter). The error of approximation ?jJ of the scheme concerned is
of 0( rm~ + h^2 ) if k( x, t*) E C(^3 l[o, 1] for every fixed t = t* and this is also
consistent with the results obtained.
In mastering the difficulties involved, we refer to a variable operator
0
A(t) in the space SL = H of all grid functions with the values0
Ay= -A(i)y, y ESL.In connection with its dependence on t the usual practice is to impose, in
addition, the Lipshitz condition with respect to the variable t:I (( A(t) - A(t - T)) y, y) I < T C3 ( A(t - T) y, y) )
making it possible to apply the general stability theory. As can readily be
observed, the Lipshitz condition is ensured if
I k ( x' t) - k ( x J t - T) I < T C3 k ( x' t - T) J
thereby confirming the validity of the preceding results obtained in Sections
2-4 for scheme ( 75).
Further development of this trend of research is devoted to a more
c0111plex problen1 in which the governing equation acquires the form
ou [} ( ou)
c(x, t) 8t = ox k(x, t) ox + f(x, t)