1549301742-The_Theory_of_Difference_Schemes__Samarskii

514 Difference Methods for Solving Nonlinear Equations

whence it follows that at the point x = 0

(

D2(J)1/o

u(U, t) = --;,;--- t^1!^0.

0

A comparison with (16) gives

1

n= -, (J

Thus, we have proved by having recourse to a "travelling" wave that prob-

lem ( 15)-(17) is solvable and its solution admits the form

{

l/o ( _ ___:_) l/o _ Uo _ l/o

(l 8 ) u(x, t) = U^0 t 1 Dt - Dl/o (Dt x) , 0 < x < Dt,

0, x > Dt,

provided the condition n = 1/ (J holds. Any solution of the fonn ( 18) ls

called a "temperature wave" with a finite velocity. vVhat is more, it

depends on three parameters x 0 , (J and u 0 in accordance with the law

At the next stage we focus our attention on the heat flow

ou u^0 +^1 u^0

W -- -"o OJ uo OX -- "o OJ (J Dl+l/o o (Dt - x)l/o = " OJo (J oD u(x, t).

In view of this, on the front of the temperature wave x = Dt the tempera-

ture and heat flow vanish for (J > 0 and the partial derivative

OH u 0 1

ox (J Dl/u (Dt - x)l-1/u

tends to= for (J > l, it is finite for (J = 1 and becomes zero for 0 < (J < 1.
Therefore, it is meaningful to speak for (J > 1 only about a generalized
solution to the heat conduction equation ( 15).
A case in point is a nonlinear dependence of the coefficient of heat
conductivity upon the te1nperature. From the formula for D it is easily
seen that we fonnally have D = = for the linear heat conductivity when
(J = O; lneaning that the velocity of heat conducting turns out to be infinite.