
16.2. Representation of the S olution of the Cauchy Problem via the Fundamental Solution 1191
where the second-order linear differential operator L
x
is defined by relation (14.2.1.2) with
x ∈ R
n
and t > 0.
The solution of the Cauchy problem for Eq. (16.2.3.1) with the general initial conditions
w = f
0
(x) at t = 0,
∂
t
w = f
1
(x) at t = 0
(16.2.3.2)
can be represented as the sum
w(x, t) =
Z
t
0
Z
R
n
Φ(y, τ)E (x, y, t, τ ) dV
y
dτ −
Z
R
n
f
0
(y)
∂
∂τ
E (x, y, t, τ )
τ=0
dV
y
+
Z
R
n
f
1
(y) + f
0
(y)ϕ(y, 0)
E (x, y, t, 0) dV
y
, dV
y
= dy
1
. . . dy
n
.
Here E = E (x, y, t, τ ) is the fundamental solution of the Cauchy problem, which satisfies,
for t > τ ≥ 0, the homogeneous linear equation