
Chapter 1 8
Duhamel’s Principles.
Some Transformations
18.1 Duhamel’s Principles in Nonstationary Problems
18.1.1 Problems for Homogeneous Linear Equations
◮ Parabolic eq uations with two independent variables.
Consider the problem for the homogeneous linear equation of parabolic type
∂w
∂t
= a(x)
∂
2
w
∂x
2
+ b(x)
∂w
∂x
+ c(x)w (18.1.1.1)
with the homogeneous initial condition
w = 0 at t = 0 (18.1.1.2)
and the boundary conditions
s
1
∂
x
w + k
1
w = g(t) at x = x
1
, (18.1.1.3)
s
2
∂
x
w + k
2
w = 0 at x = x
2
. (18.1.1.4)
By appropriately choosing the values of the coefficients s
1
, s
2
, k
1
, and k
2
in (18.1.1.3)
and (18.1.1.4), one can obtain the first, second, third, and mixed boundary ...