300 Difference Schernes with Constant Coefficients- The original problem. The heat diffusion process on a straight line is
described by the heat conduction equation
(1)where u = u(x, t) is the temperature, c is the specific heat, pis the density,
k is the thermal conductivity and J is the density of heat sources, that is,
the amount of heat emitted per a unit of time on a unit of length. Thermal
conductivity and specific heat may depend not only on x and t, but also
on the temperature u. In that ca5e the equation is said to be quasilinear.
If k and cp are constant, equation (1) can be written in the form
(2)OU
otk
a2 = -
cp- f
f= - ,
cp
where a^2 is the thermometric conductivity (Maxwell) or diffusivity (Lord
Kelvin).
Without loss of generality we may set a = 1 and rewrite equation (2)
as(3)Indeed, by introducing x' = x /a and denoting once again x' by x we obtain
(3). Where searching a solution to equation (2) on the segment 0 < x < l,
it is sensible to pass to the dimensionless variables
x
x' == l ,whose use permits us to rewrite equation (2) in the form (3) with 0 < x' < 1
and J = 12 J / a^2 incorporated.
vVe concentrate prirnarily on the first boundary-value problem associ-
ated with equation· (3) in the rectangle D = { 0 < :r < l, 0 < t < T}, in
which it is required to find a continuous in D solution u = u(x, t) of the
problem
OU o^2 u
ot = ox2+f(x,t),^0 < t < T,(I) u(x, 0) = u 0 (x),
u(O, t) = u 1 (t), u(l, t) = u 2 (t), 0 < - t < - T.