
1224 BOUNDARY VALUE PROBLEMS. GREEN’S FUNCTION
TABLE 17.9
Green’s functions of boundary value problems for equations of various types in bounded
domains. In all problems, the operators L
x
and Γ
x
are the same; x = {x
1
, . . . , x
n
}
Equation
Initial and
boundary conditions
Green’s function
Elliptic equation
−L
x
[w] = Φ(x)
Γ
x
[w] = g(x) for x ∈ S
(no initial condition required)
G(x, y) =
∞
X
k=1
u
k
(x)u
k
(y)
ku
k
k
2
λ
k
, λ
k
6= 0
Parabolic equation
∂
t
w − L
x
[w] = Φ(x, t)
w = f(x) at t = 0
Γ
x
[w] = g(x, t) f or x ∈ S
G(x, y, t) =
∞
X
k=1
u
k
(x)u
k
(y)
ku
k
k
2
exp
−λ
k
t
Hyperbolic equation
∂
tt
w − L
x
[w] = Φ(x, t)
w = f
0
(x) at t = 0
w = f
1
(x) at t = 0
Γ
x
[w] = g(x, t) f or x ∈ S
G(x, y, t) =
∞
X
k=1
u
k
(x