The sun1n1arized approxin1ation method 603It is plain to derive for this chain oof the equations the following expressions:
00
v(l)(x, t) = j G 0 (x1' ~1' t) v(lJ(~ 1 , x 2 , x 3 , 0) d~ 1 ,
-00
00
v( 2 )(x, t) = .I G 0 (x 2 , ~ 2 , t) v( 2 )(x 1 , ~ 2 , x 3 , 0) d~ 2 ,
-N
('()
v(:3)(x, t) = j G 0 (x 3 ,~ 3 , t) v( 31 (x 1 , x 2 , ~ 3 , O) d~ 3.
-ooUpon substituting herewe obtain formula (14) for v( 3 )(x, t*) at moment t = t*, meaningv( 3 )(x, t) = u(x, t) for any t* > 0.
The property of this sort is an immediate implication of the repre-
sentation G( x, ~, t) = G( x1' x 2 , x 3 ; ~ 1 , ~ 2 , ~ 3 , t) as a product of the functions
of one variable of the special type Go ( x,.,, (.,, t). The well-established rep-
resentation of the source function is still valid for equation (13) with zero
boundary conditions of the first kind u = 0 for xo: = 0, !,., , a = 1, 2, 3, re-
lating to the boundary-value problems in the parallelepiped {O < J:°' :::; l,.,,
n = 1, 2, 3}. For this reason identity (14) is certainly true in that case.
What is 1nore, it see111s clear from the preceding examples that. u(P J
coincides with 1i( x, t) at all nodal points.
Among other things, Examples 2 and 3 show the ways of expanding
s01ne spatial process into a sequence of processes being one-dimensional
ones and continuing along the coordinate axes. Following established prac-
tice, the three-din1ensional heat conduction proble1n in a space or a paral-
lelepiped with the zero te1nperature on the lateral surface reduces on the
same grounds to the model problem concerned. If the initial distribution of
temperature is known at moment t = t 0 , then the heat conduction will be