1549301742-The_Theory_of_Difference_Schemes__Samarskii

516 Difference Methods for Solving Nonlinear Equations

where c 0 = const, we find the function in question

1i(x,t) = v(x)T(t) = (O"CYA)l/a ((x;o -~)2) l/a'

but minor changes are needed in con1plying it with the special boundary

regime (19):

and, therefore,

x; = 2 x 0 (0" + 2) u~ jO".

Thus, equation (15) with the special boundary regime ( 19) possesses

the solution

{

'tlo ( l - X j XI ) 2 /a ' 0 < X < XI '

1i(x,t)= vto-t

0, x>x 1 ,

(23)

where x 1 is the width of the region of the heat distri bu ti on.
As a matter of fact, the front of the tempe1·ature wave becomes im-
movable, since x 1 = const is independent of t and depends only on the
parameters x 0 , O", u 0 of the problem concerned. Moreover, at the front the
heat flow and temperature vanish for any O" > 0, while the partial deriva-
tive becomes ou/ ox = = for (} > 2 (at the front of the "travelling" wave
ou/ox ==for O" > 1).
A solution known as a "staying wave" exists during the interval of
time t < t 0. This is stipulated by the special boundary regime (19) relating
to the regi1nes with "breaking down".
For the heat conduction equation with a heat source depending 011 the
temperature in accordance with the law
oil o ( " fhl) f3
(24) Ot = OX Xo 1l OX + qoll '
there exist both types of the aforementioned solutions which fall within the
category of travelling waves for f3 < O" + 1 and the category of staying waves
for f3 = O" + l.
Numerical solutions of such problems cause some difficulties during
the course of many methods in connection with nonlinearity and tendency
to infinity of the partial derivatives at the front of the temperature way. It
is hoped that the exact solutions obtained in such a way help motivate what
is done and could serve in practical implementations as "goodness-of-fit"
tests.