
Chapter 1 7
Boundary Value Problems.
Green’s Function
17.1 Boundary Value Problems for Parabolic E quations
with One Space Variable. Green’s Function
17.1.1 Representation of Solutions via the Green’s Fun ction
◮ Statement of the problem (t ≥ 0, x
1
≤ x ≤ x
2
).
In general, a nonhomogeneous linear differential equation of the parabolic type with vari-
able coefficients in one dimension can be w ritten as
∂w
∂t
− L
x
[w] = Φ(x, t), (17.1.1.1)
where
L
x
[w] ≡ a(x, t)
∂
2
w
∂x
2
+ b(x, t)
∂w
∂x
+ c(x, t)w, a(x, t) > 0. (17.1.1.2)
Consider the nonstationary boundary value problem for Eq. (17.1.1.1) with an initial
condition of general form
w = f (x) at t = 0, (17.1.1.3)
and arbitrar ...