
3.1. Constant Coefficient Equations 281
2
◦
. On passing from t, x to the new variables t, z = x + bt, we obtain the nonhomogeneous
heat equation
∂w
∂t
= a
∂
2
w
∂z
2
+ Φ(z − bt, t),
which is treated in Section 3.1.2.
3
◦
. For all first boundary value problems, the Green’s function can be represented as
G
b
(x, ξ, t) = exp
b
2a
(ξ − x) −
b
2
4a
t
G
0
(x, ξ, t),
where G
0
(x, ξ, t) is the Green’s function for the heat equation that corresponds to b = 0.
◮ Domain: −∞ < x < ∞. Cauchy problem.
An initial condition is prescribed:
w = f (x) at t = 0.
Solution:
w(x, t) =
Z
∞
−∞
f(ξ)G(x, ξ, t) dξ +
Z
t
0
Z
∞
−∞
Φ(ξ, τ )G(x, ξ, t − τ) dξ dτ,
where
G(x, ξ, t) =
1
2
√
πat
exp
b(ξ − x)
2a
−
b
2
t
4a
−
(x − ξ)
2
4at