1549301742-The_Theory_of_Difference_Schemes__Samarskii

The quasilinear heat conduction equation^513

follows we expound some exploratory devices for obtaining the simplest

particular solutions to the equation

(15) 8u 8t = ox^8 ( Xo il^0 81l) 8x ' Xo > 0 , (J > 0 , X > 0.

Here the subsidiary information is the temperature at the point x = 0:

(16) U = U 0 tn.

With these, it is required to find a solution to equation (15) in the domain

{ J: > 0, t > 0} with the zero initial te1nperature

( 17) 1l(:r, 0) = 0.

We may atten1pt a solution of this proble1n in the fonn of a "travelling"

wave

u(x, t) = U(Dt - x), D = const,

where U(~) is the unknown function which is sought. Inserting this expres-

sion in ( 15) and taking into account that

8u clU

8x cl~ '

we derive the ordinary differential equation for the function U (0:

yielding

x 0 U^0 U^1 =DU+ const.

In the case const = 0, we obtain

or

which upon integrating once again becon1es

Since U = 0 for t = x = 0 ( ~ = 0), we find that c 0 = 0 and

(

u = U(O = -D(J ~ )l/cr = -(D2(J)l/o t l/cr( 1 - -x )l/o '

x 0 x 0 Dt