Environmental Engineering Reference
In-Depth Information
Chapter 7
Potential Equations
When transients have died away and initial values have been forgotten, systems
become steady and processes are driven by boundary conditions and the source
term. Equations of wave, classical heat-conduction, hyperbolic heat-conduction and
dual-phase-lagging heat- conduction all reduce to potential equations. Their mixed
problems also reduce to the boundary-value problems of potential equations. In this
chapter we mainly discuss methods of solving boundary-value problems of potential
equations.
7.1 Fourier Method of Expansion
In this section we use examples to show the Fourier method of expansion in solving
boundary-value problems of potential equations.
Example 1.
Solve the PDS in a rectangular domain:
⎧
⎨
Δ
u
=
0
,
0
<
x
<
a
,
0
<
y
<
b
,
u
(
0
,
y
)=
ϕ
1
(
y
)
,
u
(
a
,
y
)=
ϕ
2
(
y
)
,
(7.1)
⎩
u
(
x
,
0
)=
ψ
1
(
x
)
,
u
(
x
,
b
)=
ψ
2
(
x
)
.
Solution.
Let
u
1
(
x
,
y
)
and
u
2
(
x
,
y
)
be the solutions for the case of
ψ
1
(
x
)=
ψ
2
(
x
)=
0
0, respectively. Since PDS (7.1) is linear, the
solution of PDS (7.1) is thus, by the principle of superposition,
and for the case of
ϕ
1
(
y
)=
ϕ
2
(
y
)=
u
(
x
,
y
)=
u
1
(
x
,
y
)+
u
2
(
x
,
y
)
.
Based on the given boundary conditions, consider
∞
n
=
1
X
n
(
x
)
sin
n
π
y
u
1
(
x
,
y
)=
,
(7.2)
b
where
X
n
(
x
)
is a undetermined function. Substituting it into the equation in PDS (7.1)
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