512 Difference Methods for Solving Nonlinear Equations
to introduce a new variable u = J;' k(O d~ for later use in equation (14)
with further simplification of the ensuing fonnulas. The outcome of this is
u
oVJ(v) o^2 v -
ot = ox 2 +f(v), where VJ(v) = j c(O d~,
0
By merely setting u = J 0 " c( ~) d~ we are led to an alternative forn1 of
equation (14):
ov o ( 8u) -
ot = O:"C x(u) ox + f(u)'
so that v u
/' x(v) clv = /' k(v) du.
(I u
When c( v.) and k( v.) can be expressed through the power functions of
temperature v., that is,
it makes sense to introduce one more variable
and take into account that
permitting us to recast equation (14) as
ov 0 ( 0 ov ) ':'( )
Oi = OX Xo U OX + j V '
f3 - Ci
(J = Ci+ 1 '
- Some analytical solutions to the quasilinear heat conduction equation.
Nonlinearity of the coefficient of heat conductivity results in the new physi-
cal effects, the main of which is a final velocity of heat conducting. In what
