TABLE OF CONTENTS Let us consider the Cauchy problem for the diffusion equation in R 3
where P = ( x, y, z) ? R 3 and 
( P) is a given function.
At first observe
Suppose u 1( x, t), u 2( y, t) and u 3( z, t) are solutions of the one-dimensional dinusion equation u t ? ku ss = 0, where s ? { x, y, z}. Then u( x, y, z, t) = u 1( x, t) u 2( y, t) u 3( z, t) is a solution of u t ? k ? u = 0 in R 3.
We have
The function
is a fundamental solution of the diffusion equation ut ? ku xx = 0.
By Proposition 8.1 the function
is a solution of
| (8.1) |  |
G 3( P, t) is again called the Green's function or fundamental solution of (8.1).
Observe that
| (8.2) |  |
We consider the case when the initial data ( P) is a function with separable variables
| (8.3) |  |
Suppose that ( P) is a function with separable variables (8.3), where ?, ? and ? are bounded and continuous functions.
Then
| (8.4) |  |
with Q =...
