Environmental Engineering Reference
In-Depth Information
Chapter 5
Cauchy Problems of Hyperbolic
Heat-Conduction Equations
In this chapter we apply the Riemann method, integral transformations and the
method of spherical means to solve Cauchy problems of hyperbolic heat-conduction
equations of one-, two- and three-dimensions. The emphasis is placed on the physics
and the methods of measuring
τ
0
following the solutions of Cauchy problems.
A comparison is also made with wave equations and classical heat-conduction
equations.
5.1 Riemann Method for Cauchy Problems
In this section we introduce the Riemann function and the Riemann method for
solving Cauchy problems of second-order equations.
5.1.1 Conjugate Operator and Green Formula
Let
L
be the linear differential operator of hyperbolic equations
2
2
=
∂
x
2
−
∂
)
∂
∂
)
∂
∂
L
y
2
+
a
(
x
,
y
x
+
b
(
x
,
y
y
+
c
(
x
,
y
)
,
(5.1)
∂
∂
where
a
,
b
and
c
are all differentiable functions of
x
and
y
. A nonhomogeneous
hyperbolic equations of second-order can thus be written as
[
]=
(
,
)
.
L
u
f
x
y
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