
1196 CAUCHY PROBLEM. FUNDAMENTAL SOLUTIONS
is given by the formula
w =
Z
t
0
Z
R
n
E
e
(x − y, t − τ)Φ(y, τ ) dV
y
dτ +
Z
R
n
E (x − y, t)f
m−1
(y) dV
y
. (16.2.4.13)
By taking into account relation (16.2.4.13) between the fundamental solutions, we can
successively reduce the second integral in (16.2.4.13) to the form
Z
R
n
E (x − y, t)f
m−1
(y) dV
y
=
Z
R
n
E (y, t)f
m−1
(x − y) dV
y
=
Z
R
n
L
m,y
[E
e
(y, t)]f
m−1
(x − y) dV
y
=
Z
R
n
E
e
(x − y, t)L
m,y
[f
m−1
(y)] dV
y
.
Here we have assumed that the function f
m−1
(x) sufficiently rapidly decays as |x| → ∞.
◮ On a third-order partial differential equation with mixed derivatives.
1
◦
. The constant coefficient third-order partial differential equation
L[w]