
276 SECOND-ORDER PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE
2
◦
. Fundamental solution:
E (x, t) =
1
2
√
πat
exp
−
x
2
4at
+ bt
.
◮ Reduction to the heat equation. Remarks on the Green’s functions.
The substitution w(x, t) = e
bt
u(x, t) leads to the nonhomogeneous heat equation
∂u
∂t
= a
∂
2
u
∂x
2
+ e
−bt
Φ(x, t),
which is discussed in S ection 3.1.2 in detail. The initial condition for the new variable u
remains the same, and the nonhomogeneous part in the boundary conditions is m ultiplied
by e
−bt
. Taking this into account, one can easily solve the original equation subject to the
initial and boundary conditions considered in Section 3.1.2.
In all the boundary value