1549301742-The_Theory_of_Difference_Schemes__Samarskii

The quasilinear heat conduction equation 515

It may happen that the temperature front is held fixed, that is, D = 0.

Such a solution always exists in a special boundary regime such as

( H1)

1

u(O, t) = U 0 (to _ t)m ,

where t 0 is an arbitrary constant, under the agreement that the initial

condition was i1nposecl fol' t = -=:

( 20) u(x,-=)=0.

Still using the framework of the n1ethod of separation of variables, a solution

to equation (15) is sought in the form

u(x, t) = v(x) T(t).

Upon substituting this product in ( 15) and separating the variables we

obtain

~ !:___ (xo v" dv) = 1 dT = ).

v dx dx T^0 +^1 dt ,

where ). is a separation paran1eter. Whence it follows that

(21) !:___ (x 0 v" dv) - >. v =^0 ,

d."C dx

(22)

Along these lines, we n1ay attempt a solution to equation (21) in the
forn1
v" -- CY (x I - x)f3 ,

where the numbers CY and (3 are free to be chosen and x 1 is an arbitrary

number. Substituting u" into (21) yields

and reveals

(3 = 2,

). ()2

CY=-----

2x0(2+0")

Having integrated the equation related to T: