The quasilinear heat conduction equation 509
This can be solved by the standard elimination method, whose computa-
tional procedure is stable under the condition f'(y) < 0.
The convergence rate of such iterations needs investigation with regard
to the error
k+l ·u=y-y, k+l
where y is the exact solution of problem (2). Upon substituting two sub-
sequent 1terat10ns.. y k = y + v k an d k+l y = y + k+l ·u. mto equat10n. (6) we may
set up the problem for the error k+l ·u :
( 7) k+l v ,,,. + .I •/ ( Y k ) k+l v = - F k ,
(8)
k k k k
F = f(y) - f(y) + (y - Y) .f'(y) ·
Taking into account the well-established decon1position
k k k k k
f (y) = f (y) + .f' (y) (y - Y) + ~ f" (f;) (Y - y)^2 ,
k k k
where f; = y + 8(y - y ), 0 < {} < 1, we find that
Thus, it is required to evaluate a solution of the problem
(9) k+l V .z:.r + f / (y) k k+l V = 2 1 f II (y ~ )v k^2 ,
If f(y) is a concave function, that is, f"(y) > 0, then due to the
maximu1n principle we inight have
k+l k+l k+l
· v = y -y<O, y <y,
thereby clarifying that the iterations approach the exact solution of problem
(2) from below. It is plain to show that for a solution of problem (9) either
of the following esti1nates
( 10)
01'