602 Econo1nical Difference Schemes for Multidimensional ProblemsExample 3 'vVe learn from Ladyzhenskaya (1968) that a solution of the
Cauchy problem for the heat conduction equation( 13) cv=l,2,3,
-oo < Xa < oo, t > 0, u(x, U) = u 0 (.t:),
is given by the formula(14) u(x, t) = u(x 1 , x 2 , x 3 , t)
00 00 00
= j j j G(x1'x2,x3;~1,~2,~3,t)ua(~1,~2,~3)d~1d~2d~3,
-00 -00 -00where G( x 1 , x 2 , x 3 ; ~ 1 , ~ 2 , ~ 3 , t) is a source function such thatHere G 0 ( xQ, (,, t) is a func:tion of t.he heat source of the Cauchy proble111
associated with the one-dimensional heat conduction ccquat.ion
The general methodology provides proper guidelines for the selection
rules in studying one-dimensional heat cond uc ti on equations
0 < t < t*'